IntroductionChain closure

Tameness

Compactness and powerful images once

and for all

Michael Lieberman(Joint with Will Boney)

Masaryk University Department of Mathematics and Statisticshttpwwwmathmunicz~lieberman

Brno-Prague Algebra WorkshopMay 31 2019

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable

Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable

Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

We work in a familiar interval in the large cardinal hierarchy

[ weakly compact strongly compact ]

DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable

This gets pretty much everything

κ (κκ)-compact weakly compact

κ (δinfin)-compact δ-strongly compact

κ (lt κinfin)-compact almost strongly compact

κ (κinfin)-compact strongly compact

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

Theorem (BoneyUnger)

Let κ satisfy microω lt κ for all micro lt κ The following are equivalent

(1) κ is almost strongly compact

(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L

(3) Every AEC below κ is lt κ-tame

Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky

(2) rArr (3) Lieberman and Rosicky

(3) rArr (1) Boney and Unger

They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Large cardinal notionsand their equivalents

A careful reworking of these arguments gives

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of

H P rarr L

where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We will prove the following

TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects

ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as

RedL(T )

the category of reducts of models of T to the signature of L

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

Crucially careful reading reveals that RedL(T ) is κ-preaccessible

Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take

TM atomicnegated atomic diagram of M names cm for m isin M

It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM

involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We work in an AEC K although an arbitrary concrete accessiblecategory will do

DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with U(g0)(a0) = U(g1)(a1)

Assuming the amalgamation property which we do this is in factan equivalence relation

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0

We turn this into a question about powerful images in precisely thesame way as in LRosicky

L1 Category of diagrams witnessing equivalence of pairs

N0g0 983587983587 N

M

f0983619983619

f1983587983587 N1

g1983619983619

with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

L2 Category of pairs

N0

M

f0983619983619

f1983587983587 N1

with selected elements ai isin UNi

Let F L1 rarr L2 be the obvious forgetful functor

Facts

1 The functor F is accessible If K is below κ so is F

2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types

With these facts there is essentially nothing to do

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects

ProofSuppose M is the κ-directed colimit of θ-presentable objects

M = colimiisinIMi

with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie

N0 Mif0φi

983651983651f1φi

983587983587 N1

is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images

IntroductionChain closure

Tameness

We have nearly proven

Theorem (BoneyL)

Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent

1 κ is logically (δ lt θ)-strong compact

2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables

3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame

The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble

Lieberman Compactness and powerful images