Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory...
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Reconsidering MacLane:the foundations of categorical coherence
Peter M. Hines
York – Maths Dept. – Oct. 2013
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Previous versions
Talks is based on:
Talk at AbramskyFest (Oxford, June 2013)
Joint Maths / Computing Seminar (Oxford, March 2013)
Topic of talk:
Foundations of category theory & “MacLane’s Theorem”
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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(The ideas behind) category theory
Category Theory is simply
a calculus of mathematical1 structures.
It studies:Mathematical structures.
Structure-preserving mappings.
Transformations between structures.
1or logical, or computational, or linguistic, or . . .Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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History & prehistory
It arose from work by:
Samuel Eilenberg,
Saunders MacLane,
in Algebraic Topology.
Later applied (despite protests) in other subjects:
Theoretical Computing . . .Linguistics . . .Logic . . .Quantum Mechanics . . .Foundations of Mathematics . . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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History & prehistory
It arose from work by:
Samuel Eilenberg,
Saunders MacLane,
in Algebraic Topology.
Later applied (despite protests) in other subjects:
Theoretical Computing . . .Linguistics . . .Logic . . .Quantum Mechanics . . .Foundations of Mathematics . . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 6: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/6.jpg)
Precursors to category theory
John von Neumann (1925): Axiomatic theory of classes.
A formalism for working with proper classes:
All sets, all monoids, all lattices, &c.
Later became the von Neumann, Godel, Bernay formalism
von Neumann originated the theory. (proto-cat. theory)
Godel made it logically consistent.
Bernay rewrote it to look like ZFC set theory ....
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Category Theory textbooks
Applications of category theory in various fields
. . . a large range of texts.
The underlying theory of categories:
“Categories for the Working Mathematician”— S. MacLane (1971)
... examples & applications taken from algebraic topology.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Category Theory textbooks
Applications of category theory in various fields
. . . a large range of texts.
The underlying theory of categories:
“Categories for the Working Mathematician”— S. MacLane (1971)
... examples & applications taken from algebraic topology.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 9: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/9.jpg)
The definition ...
A category C consists of
A class of objects, Ob(C).
For all objects A,B ∈ Ob(C), a set of arrows C(A,B).
We will work diagrammatically:
An arrow f ∈ C(A,B) is drawn as
A f // B
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The axioms ...
Matching arrows can be composed
A f //
gf ��
B
g��
C
Composition is associative
h(gf ) = (hg)f
There is an identity 1A at each object A
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Examples of categories
Monoid
(Objects:) all monoids.
(Arrows:) homomorphisms.
Set
(Objects:) all sets.
(Arrows:) functions.
Poset
(Objects:) all partially ordered sets.
(Arrows:) order-preserving functions.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Diagrams in categories
Identities and equations are usually expressed graphically.
A diagram in the category Set
Z x 7→x2//
x 7→abs(x)
��
N
N
n 7→n2
::
A diagram commutes when all paths with the samesource / target describe the same arrow.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The art of diagram-chasing
Commuting diagrams can be pasted along a common edge.
Z x 7→x2//
x 7→abs(x)
��
N N
n 7→n (mod 2)
��N
n 7→n2
::
Nn 7→n (mod 2)
//
n 7→n2
99
{0,1}
Both the above diagrams commute . . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The art of diagram-chasing
Commuting diagrams can be pasted along a common edge.
Z x 7→x2//
x 7→abs(x)
��
N
n 7→n (mod 2)
��N
n 7→n (mod 2)//
n 7→n2
99
{0,1}
. . . this diagram also commutes!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The art of diagram-chasing
Edges can be deleted in commuting diagrams.
Z x 7→x2//
x 7→abs(x)
��
N
n 7→n (mod 2)
��N
n 7→n (mod 2)//
n 7→n2
99
{0,1}
. . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The art of diagram-chasing
Edges can be deleted in commuting diagrams.
Z x 7→x2//
x 7→abs(x)
��
N
n 7→n (mod 2)
��N
n 7→n (mod 2)// {0,1}
. . . this is still a commuting diagram.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Maps between categories
A mapping between categories C and D is a functor Γ : C → D.
Objects of C are mapped to objects of D.
Arrows of C are mapped to arrows of D.
A f // B Category C
��
Γ(A)Γ(f )
// Γ(B) Category D
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Functors
Functors must preserve composition and identities.
Γ(1X ) = 1Γ(X) , Γ(gf ) = Γ(g)Γ(f )
Functors preserve commutativity of diagrams.
U f //
h
��
V
g
��
k
��
commutes in C
Wj
// X
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Functors
Functors must preserve composition and identities.
Γ(1X ) = 1Γ(X) , Γ(gf ) = Γ(g)Γ(f )
Functors preserve commutativity of diagrams.
U f //
h
��
V
g
��
k
��
commutes in C
Wj
// X
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 20: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/20.jpg)
Functors
Functors must preserve composition and identities.
Γ(1X ) = 1Γ(X) , Γ(gf ) = Γ(g)Γ(f )
Functors preserve commutativity of diagrams.
Γ(U)Γ(f ) //
Γ(h)
��
Γ(V )
Γ(g)
��
Γ(k)
~~
commutes in D
Γ(W )Γ(j)
// Γ(X )
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Examples of functors (I)
A functor from Set to Monoid.
Take a set X .
Form the free monoid X ∗ (All finite words over X).
Every function f : X → Y
induces a homomorphism
map(f ) : X ∗ → Y ∗
This is a functor Free : Set→ Monoid.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Examples of functors (II)
A functor from Top∗ to Group.
Take a pointed topological space T
Form its fundamental group π1(T )
Every continuous mapc : S → T
induces a homomorphism
π(f ) : π1(S)→ π1(T )
This is a functor π : Top∗ → Group.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A sweeping generalisation
In general:
finding invariants (e.g. fundamental group, K0 group, &c.)
using constructors (e.g. monoid semi-ring construction)
type re-assignments (e.g. Int→ Real)
forming algebraic models(e.g. Brouwer-Heyting-Kolmogorov interpretation)
. . .
are all examples of functors.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Monoidal Categories
and
MacLane’s Theorem
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Categories with additional structure:
Monoidal Categories ≡ Categories with Tensors.
A tensor ⊗ on a category is:
a way of combining two objects / arrowsto make a new object / arrow of the same category.
Objects: Given X ,Y , we can form X ⊗ Y .
Arrows: Given f ,g, we can form f ⊗ g.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
1/ Types match:
A f // B
X h // Y
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
1/ Types match:
A⊗ X f⊗h // B ⊗ Y
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
2/ Composition is preserved:
A f // Bg // C
X h // Y k // Z
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
2/ Composition is preserved:
Agf // C
X kh // Z
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
2/ Composition is preserved:
A⊗ Xgf⊗kh // C ⊗ Z
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
2/ Composition is preserved:
A f // Bg // C
X h // Y k // Z
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
2/ Composition is preserved:
A⊗ X f⊗h // B ⊗ Yg⊗k // C ⊗ Z
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Properties of tensors:
A tensor is a functor:
⊗ : C × C → C
Functoriality implies:
2/ Composition is preserved:
A⊗ X(g⊗k)(f⊗h)=gf⊗kh // C ⊗ Z
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Familiar examples
Tensor product of Hilbert spaces / bounded linear maps
Cartesian product (pairing) of Sets / functions
Direct sum of Vector spaces / matrices
Disjoint union of Sets / functions
Combining Binary trees
. . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The final conditions
We also require:
Associativity
f ⊗ (g ⊗ h) = (f ⊗ g)⊗ h
A unit object I ∈ Ob(C)
X ⊗ I = X = I ⊗ X for all objects X ∈ Ob(C)
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Trivial objects
Monoidal categories usually2 have a unit object I ∈ Ob(C)
A⊗ I = A = I ⊗ A for all objects A ∈ Ob(C)
These are trivial objects within a category:
The single-element set.
The trivial monoid.
The empty space.
The underlying scalar field.
The trivially true proposition.
2Part of the original definition. Later shown not to be essential (Saavedra72 / Kock08 / PH13).
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A problem, and MacLane’s solution
The problem ...In real-world examples, the condition
f ⊗ (g ⊗ h) = (f ⊗ g)⊗ h
is almost never satisfied.
... and its solution.MacLane’s theorem lets us pretend that
f ⊗ (g ⊗ h) = (f ⊗ g)⊗ h
with no harmful side-effects.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A problem, and MacLane’s solution
The problem ...In real-world examples, the condition
f ⊗ (g ⊗ h) = (f ⊗ g)⊗ h
is almost never satisfied.
... and its solution.MacLane’s theorem lets us pretend that
f ⊗ (g ⊗ h) = (f ⊗ g)⊗ h
with no harmful side-effects.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Failure of associativity - an example
Associativity often fails, in a trivial way!
The disjoint union of setsGiven sets A,B,
A ] B = {(a,0)} ∪ {(b,1)}
This is not associative . . . for ridiculous reasons.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Non-associativity of disjoint union
A ] (B ] C) =
{(a,0)} ∪ {(b,01)} ∪ {(c,11)}
(A ] B) ] C =
{(a,00)} ∪ {(b,10)} ∪ {(c,1)}
These are not the same set – for annoying syntactical reasons.
There is an obvious isomorphism between them ...
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Non-associativity of disjoint union
A ] (B ] C) =
{(a,0)} ∪ {(b,01)} ∪ {(c,11)}
(A ] B) ] C =
{(a,00)} ∪ {(b,10)} ∪ {(c,1)}
These are not the same set – for annoying syntactical reasons.
There is an obvious isomorphism between them ...
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Replacing equality by isomorphism:
Strict associativity:
A⊗ (B ⊗ C)=
(A⊗ B)⊗ C
Associativity up to isomorphism
A⊗ (B ⊗ C)
τABC //(A⊗ B)⊗ C
τ−1ABC
oo
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How to ignore isomorphisms
Provided the associativity isomorphisms satisfy:
1 naturality
2 A coherence condition
we can ignore them completely.
Natural examples generally satisfy these conditions!
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Naturality
We can ‘push arrows through associativity isomorphisms’
A f
⊗
X
⊗
X
Y
⊗B g Y
C h Z Z
τ(f ⊗ (g ⊗ h)) = ((f ⊗ g)⊗ h)τ
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Naturality
We can ‘push arrows through associativity isomorphisms’
A
⊗
A f X
B g
⊗
Y
⊗B
C C h Z
τ(f ⊗ (g ⊗ h)) = ((f ⊗ g)⊗ h)τ
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Coherence
MacLane’s coherence condition
The two ways of re-arranging
A⊗ (B ⊗ (C ⊗ D))
into((A⊗ B)⊗ C)⊗ D
must be identical.
Also called MacLane’s Pentagon condition
τ τ = (τ ⊗ 1) τ (1⊗ τ)
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Coherence
MacLane’s coherence condition
The two ways of re-arranging
A⊗ (B ⊗ (C ⊗ D))
into((A⊗ B)⊗ C)⊗ D
must be identical.
Also called MacLane’s Pentagon condition
τ τ = (τ ⊗ 1) τ (1⊗ τ)
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Rebracketing four symbols
A⊗ (B ⊗ (C ⊗ D))
A⊗ ((B ⊗ C)⊗ D)
(A⊗ B)⊗ (C ⊗ D)
(A⊗ (B ⊗ C))⊗ D
((A⊗ B)⊗ C)⊗ D
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Yes, there are two paths you can go by, but ...
MacLane’s pentagon
A⊗ (B ⊗ (C ⊗ D))1 ⊗τ , ,
))τ , ,
zz
A⊗ ((B ⊗ C)⊗ D)
τ , ,
��
(A⊗ B)⊗ (C ⊗ D)
τ , ,
$$
commutes!
(A⊗ (B ⊗ C))⊗ D
τ , , ⊗1uu((A⊗ B)⊗ C)⊗ D
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MacLane’s coherence theorem:
When we have1 Naturality2 Coherence
every canonical diagram – built up using
τ , , , ⊗ and 1
is guaranteed to commute.
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A consequence:
Given a tensor that is associative up to isomorphism,
A⊗ (B ⊗ C)
τABC //A⊗ (B ⊗ C)
τ−1ABC
oo
We can ‘pretend it is strictly associative’
A⊗ (B ⊗ C)= A⊗ (B ⊗ C)
with no “harmful side-effects”.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The conclusion
The theory of coherence has written
itself out of existence!
By appealing to MacLane’s theorem ...
We can completely ignore questions of coherence,
naturality, pentagons, canonical diagrams, &c.
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Two common descriptions of MacLane’s theorem:
1 Every canonical diagram commutes.
2 We can treat
A⊗ (B ⊗ C)
τA,B,C
++(A⊗ B)⊗ C
τ−1A,B,C
kk
as a strict identity
A⊗ B ⊗ C = A⊗ B ⊗ C
with no ‘harmful side-effects’.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Two inaccurate descriptions of MacLane’s theorem:
1 Every canonical diagram commutes.
2 We can treat
A⊗ (B ⊗ C)
τA,B,C
++(A⊗ B)⊗ C
τ−1A,B,C
kk
as a strict identity
A⊗ B ⊗ C = A⊗ B ⊗ C
with no ‘harmful side-effects’.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Two contrary claims:
Not every canonical diagram commutes.
(Claim 1)
Treating associativity isomorphisms asstrict identities can have major consequences.3
(Claim 2)
3everything collapses to a triviality ...Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A simple example:
The Cantor monoid U (single-object category).Single object N.Arrows: all bijections on N.
The tensor
We have a tensor ( ? ) : U × U → U .
(f ? g)(n) =
2.f(n
2
)n even,
2.g(n−1
2
)+ 1 n odd.
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Properties of the Cantor monoid (I)
The Cantor monoid has only one object —
N ? (N ? N) = N = (N ? N) ? N
( ? ) : U × U → U is associative up to a natural isomorphism
τ(n) =
2n n (mod 2) = 0,n + 1 n (mod 4) = 1,n−1
2 n (mod 4) = 3.
that satisfies MacLane’s pentagon condition.
This is not the identity map!
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Properties of the Cantor monoid (II)
Not all canonical diagrams commute:
N
τ
��
N
id?τ88
τ?id&&N
This diagram does not commute.
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Properties of the Cantor monoid (II)
Using an actual number:
1
n 7→n+1
��
1
n 7→2n−188
2
On the upper path, 1 7→ 2.
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Properties of the Cantor monoid (II)
Taking the right hand path:
1
n 7→n&& 1
1 6= 2, so this diagram does not commute.
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What does MacLane’s thm. actually say?
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A recent (May 2013) report
“Hines uses MacLane’s theorem – the fact
that all canonical diagrams commute – to
construct a large class of examples where . . . ”
— Anonymous Referee
(Category Theory / Theoretical Computing journal).
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If in doubt ...
. . . ask the experts:
http://en.wikipedia.org/wiki/Monoidal category
“It follows that any diagram whosemorphisms are built using [canonicalisomorphisms], identities and tensorproduct commutes.”
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Tinker, Tailor, Soldier, Sarcasm
Untangling The Web – N.S.A. guide to internet use
Do not as a rule rely on Wikipediaas your sole source of information.The best thing about Wikipedia arethe external links from entries.
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MacLane, on MacLane’s theorem
Categories for the working mathematician (1st ed.)
Moreover all diagrams involving [canonical iso.s] mustcommute. (p. 158)
These three [coherence] diagrams imply that “all” suchdiagrams commute. (p. 159)
We can only prove that every “formal” diagram commutes.(p. 161)
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MacLane, on MacLane’s theorem
Categories for the working mathematician (1st ed.)
Moreover all diagrams involving [canonical iso.s] mustcommute. (p. 158)
These three [coherence] diagrams imply that “all” suchdiagrams commute. (p. 159)
We can only prove that every “formal” diagram commutes.(p. 161)
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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MacLane, on MacLane’s theorem
Categories for the working mathematician (1st ed.)
Moreover all diagrams involving [canonical iso.s] mustcommute. (p. 158)
These three [coherence] diagrams imply that “all” suchdiagrams commute. (p. 159)
We can only prove that every “formal” diagram commutes.(p. 161)
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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MacLane, on MacLane’s theorem
Categories for the working mathematician (1st ed.)
Moreover all diagrams involving [canonical iso.s] mustcommute. (p. 158)
These three [coherence] diagrams imply that “all” suchdiagrams commute. (p. 159)
We can only prove that every “formal” diagram commutes.(p. 161)
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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What does his theorem say?
MacLane’s coherence theorem for associativity
All diagrams within the image of a certainfunctor are guaranteed to commute.
This usually means all canonical diagrams.
In some circumstances, this is not the case.
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Dissecting MacLane’s theorem
— a closer look
A technicality:In common with MacLane, we study monogenic categories.
Objects are generated by:
Some object S,
The tensor ( ⊗ ).
This is not a restriction — S is thought of as a ‘variable symbol’.
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Dissecting MacLane’s theorem
— a closer look
A technicality:In common with MacLane, we study monogenic categories.
Objects are generated by:
Some object S,
The tensor ( ⊗ ).
This is not a restriction — S is thought of as a ‘variable symbol’.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The source of the functor (Buxus Sempervirens)
This is based on (non-empty) binary trees.
�
x �
� x
x x
Leaves labelled by x ,
Branchings labelled by �.
The rank of a tree is the number of leaves.Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A posetal category of trees
MacLane’s categoryW.
(Objects) All non-empty binary trees.
(Arrows) A unique arrow between any two treesof the same rank.
— write this as (v ← u) ∈ W(u, v).
Key points:
1 ( � ) is a tensor onW.
2 W is posetal — all diagrams overW commute.
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MacLane’s Substitution Functor
MacLane’s theorem relies on a monoidal(i.e. tensor-preserving) functor
WSub : (W,�)→ (C,⊗)
This is based on a notion of substitution.
i.e. mapping formal symbols to concrete objects & arrows.
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The functor itself
On objects:
WSub(x) = S,
WSub(u�v) =WSub(u)⊗WSub(v).
An object ofW:
�
x �
� x
x x
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An inductively defined functor (I)
On objects:
WSub(x) = S,
WSub(u�v) =WSub(u)⊗WSub(v).
An object of C:⊗
S ⊗
⊗ S
S S
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An inductively defined functor (II)
On arrows:
WSub(u ← u) = 1 .
WSub(a�v ← a�u) = 1 ⊗WSub(v ← u).
WSub(v�b ← u�b) =WSub(v ← u)⊗ 1 .
WSub((a�b)�c ← a�(b�c)) = τ , , .
The role of the Pentagon
The Pentagon condition =⇒ WSub is a monoidal functor.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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An inductively defined functor (II)
On arrows:
WSub(u ← u) = 1 .
WSub(a�v ← a�u) = 1 ⊗WSub(v ← u).
WSub(v�b ← u�b) =WSub(v ← u)⊗ 1 .
WSub((a�b)�c ← a�(b�c)) = τ , , .
The role of the Pentagon
The Pentagon condition =⇒ WSub is a monoidal functor.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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An inductively defined functor (II)
On arrows:
WSub(u ← u) = 1 .
WSub(a�v ← a�u) = 1 ⊗WSub(v ← u).
WSub(v�b ← u�b) =WSub(v ← u)⊗ 1 .
WSub((a�b)�c ← a�(b�c)) = τ , , .
The role of the Pentagon
The Pentagon condition =⇒ WSub is a monoidal functor.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 80: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/80.jpg)
An inductively defined functor (II)
On arrows:
WSub(u ← u) = 1 .
WSub(a�v ← a�u) = 1 ⊗WSub(v ← u).
WSub(v�b ← u�b) =WSub(v ← u)⊗ 1 .
WSub((a�b)�c ← a�(b�c)) = τ , , .
The role of the Pentagon
The Pentagon condition =⇒ WSub is a monoidal functor.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 81: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/81.jpg)
The story so far ...
We have a functorWSub : (W,�)→ (C,⊗).
Every object of C is the image of an object ofW
Every canonical arrow of C is the image of an arrow ofW
Every diagram overW commutes.
As a corollary:
The image of every diagram in (W ,�) commutes in (C,⊗).
Question: Are all canonical diagrams in the image ofWSub?
– This is only the case whenWSub is an embedding!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The story so far ...
We have a functorWSub : (W,�)→ (C,⊗).
Every object of C is the image of an object ofW
Every canonical arrow of C is the image of an arrow ofW
Every diagram overW commutes.
As a corollary:
The image of every diagram in (W ,�) commutes in (C,⊗).
Question: Are all canonical diagrams in the image ofWSub?
– This is only the case whenWSub is an embedding!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 83: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/83.jpg)
The story so far ...
We have a functorWSub : (W,�)→ (C,⊗).
Every object of C is the image of an object ofW
Every canonical arrow of C is the image of an arrow ofW
Every diagram overW commutes.
As a corollary:
The image of every diagram in (W ,�) commutes in (C,⊗).
Question: Are all canonical diagrams in the image ofWSub?
– This is only the case whenWSub is an embedding!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 84: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/84.jpg)
The story so far ...
We have a functorWSub : (W,�)→ (C,⊗).
Every object of C is the image of an object ofW
Every canonical arrow of C is the image of an arrow ofW
Every diagram overW commutes.
As a corollary:
The image of every diagram in (W ,�) commutes in (C,⊗).
Question: Are all canonical diagrams in the image ofWSub?
– This is only the case whenWSub is an embedding!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 85: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/85.jpg)
The story so far ...
We have a functorWSub : (W,�)→ (C,⊗).
Every object of C is the image of an object ofW
Every canonical arrow of C is the image of an arrow ofW
Every diagram overW commutes.
As a corollary:
The image of every diagram in (W ,�) commutes in (C,⊗).
Question: Are all canonical diagrams in the image ofWSub?
– This is only the case whenWSub is an embedding!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 86: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/86.jpg)
The story so far ...
We have a functorWSub : (W,�)→ (C,⊗).
Every object of C is the image of an object ofW
Every canonical arrow of C is the image of an arrow ofW
Every diagram overW commutes.
As a corollary:
The image of every diagram in (W ,�) commutes in (C,⊗).
Question: Are all canonical diagrams in the image ofWSub?
– This is only the case whenWSub is an embedding!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 87: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/87.jpg)
“A beautiful (useful) theory slain byan ugly counterexample”?
A full theory of coherence for associativity is:
more mathematically elegant,
much more practically useful!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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single-object categories
WSub : (W,�)→ (C,⊗) can never be an
embedding when C has a finite set of objects.
The Cantor monoid has precisely one object
Where did this come from?
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Hilbert’s Hotel
A children’s story about infinity.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Hilbert’s “Grand Hotel”
An infinite corridor, with rooms numbered 0,1,2,3, . . .
N ↪→ N the successor function.
N ∼= N ] N the Cantor pairing.
N ∼= N× N an exercise!
[N→ {0,1}] is not isomorphic to N
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Hilbert’s “Grand Hotel”
An infinite corridor, with rooms numbered 0,1,2,3, . . .
N ↪→ N the successor function.
N ∼= N ] N the Cantor pairing.
N ∼= N× N an exercise!
[N→ {0,1}] is not isomorphic to N
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 92: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/92.jpg)
Hilbert’s “Grand Hotel”
An infinite corridor, with rooms numbered 0,1,2,3, . . .
N ↪→ N the successor function.
N ∼= N ] N the Cantor pairing.
N ∼= N× N an exercise!
[N→ {0,1}] is not isomorphic to N
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 93: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/93.jpg)
Hilbert’s “Grand Hotel”
An infinite corridor, with rooms numbered 0,1,2,3, . . .
N ↪→ N the successor function.
N ∼= N ] N the Cantor pairing.
N ∼= N× N an exercise!
[N→ {0,1}] is not isomorphic to N
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 94: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/94.jpg)
Hilbert’s “Grand Hotel”
An infinite corridor, with rooms numbered 0,1,2,3, . . .
N ↪→ N the successor function.
N ∼= N ] N the Cantor pairing.
N ∼= N× N an exercise!
[N→ {0,1}] is not isomorphic to N
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
![Page 95: Reconsidering MacLane: the foundations of categorical ...varg1/MATHS1.pdf · The underlying theory of categories: “Categories for the Working Mathematician” — S. MacLane (1971)...](https://reader033.fdocuments.us/reader033/viewer/2022060503/5f1d3f195d580f0e85688293/html5/thumbnails/95.jpg)
Self-similarity
The categorical identity S ∼= S ⊗ S
Exhibited by two canonical isomorphisms:
(Code) C : S ⊗ S → S
(Decode) B : S → S ⊗ S
These are unique (up to unique isomorphism).
Examples
The natural numbers N, Separable Hilbert spaces,Infinite matrices, Cantor set & other fractals, &c.
C-monoids, and other untyped (single-object) categories withtensors
Any unit object I of a monoidal category . . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Self-similarity
The categorical identity S ∼= S ⊗ S
Exhibited by two canonical isomorphisms:
(Code) C : S ⊗ S → S
(Decode) B : S → S ⊗ S
These are unique (up to unique isomorphism).
Examples
The natural numbers N, Separable Hilbert spaces,Infinite matrices, Cantor set & other fractals, &c.
C-monoids, and other untyped (single-object) categories withtensors
Any unit object I of a monoidal category . . .
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A tensor on a single object
At a self-similar object S, we may define a tensor by
S ⊗ S t⊗u // S ⊗ S
C��
S
B
OO
t?u// S
( ? ) makes C(S,S) a single-object monoidal category!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Associativity at a single object
The tensor ( ? ) is associative up to isomorphism.
S B //
τ
��
S ⊗ S1S⊗B // S ⊗ (S ⊗ S)
τS,S,S
��S S ⊗ S
Coo (S ⊗ S)⊗ S
C⊗1S
oo
Claim: This is the identity arrow
precisely when
the object S is trivial.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Associativity at a single object
The tensor ( ? ) is associative up to isomorphism.
S B //
τ
��
S ⊗ S1S⊗B // S ⊗ (S ⊗ S)
τS,S,S
��S S ⊗ S
Coo (S ⊗ S)⊗ S
C⊗1S
oo
Claim: This is the identity arrow
precisely when
the object S is trivial.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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constructing
categories where all
canonical diagrams commute
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How to Rectify the Anomaly
Given a badly-behaved category (C,⊗), we can
build a well-behaved (non-strict) version.
Think of this as the Platonic Ideal of (C,⊗).
We (still) assume C is monogenic, with objects generated by {S, ⊗ }
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Building the ‘Platonic Ideal’
We will construct PlatC
A version of C for whichWSub is an embedding.
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Constructing PlatC
Objects are free binary trees�
S �
� S
S S
Leaves labelled by S ∈ Ob(C),
Branchings labelled by �.
There is an instantiation map Inst : Ob(PlatC)→ Ob(C)
S�((S�S)�S) 7→ S ⊗ ((S ⊗ S)⊗ S)
This is not just a matter of syntax!
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Constructing PlatC
What about arrows?
Homsets are copies of homsets of C
Given trees T1,T2,
PlatC(T1,T2) = C(Inst(T1), Inst(T2))
Composition is inherited from C in the obvious way.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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The tensor ( � ) : PlatC × PlatC → PlatC
A f // X
A�Xf�g // B�Y
B g // Y
The tensor of PlatC is
(Objects) A free formal pairing, A�B,
(Arrows) Inherited from (C,⊗), so f�g def .= f ⊗ g.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Some properties of the platonic ideal ...
1 The functor
WSub : (W,�)→ (PlatC ,�)
is always monic.
2 As a corollary:
All canonical diagrams of (PlatC ,�) commute.
3 Instantiation defines an epic monoidal functor
Inst : (PlatC ,�)→ (C,⊗)
through which McL’.s substitution functor always factors.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Some properties of the platonic ideal ...
1 The functor
WSub : (W,�)→ (PlatC ,�)
is always monic.
2 As a corollary:
All canonical diagrams of (PlatC ,�) commute.
3 Instantiation defines an epic monoidal functor
Inst : (PlatC ,�)→ (C,⊗)
through which McL’.s substitution functor always factors.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Some properties of the platonic ideal ...
1 The functor
WSub : (W,�)→ (PlatC ,�)
is always monic.
2 As a corollary:
All canonical diagrams of (PlatC ,�) commute.
3 Instantiation defines an epic monoidal functor
Inst : (PlatC ,�)→ (C,⊗)
through which McL’.s substitution functor always factors.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A monic / epic decomposition
MacLane’s substitution functor always factorsthrough the platonic ideal:
(W,�)
(monic)WSub //
WSub
$$
(PlatC ,�)
Inst (epic)
��(C,⊗)
This gives a monic / epic decomposition of his functor.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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A highly relevant question ...
What does the Platonic Ideal of a single-object category
actually look like?
The simplest possible case:
The trivial monoidal category (I,⊗).
Objects: Ob(I) = {x}.
Arrows: I(x , x) = {1x}.
Tensor:x ⊗ x = x , 1x ⊗ 1x = 1x
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What is the platonic ideal of I?
(Objects) All non-empty binary trees:
�
x �
� x
x x
(Arrows) For all trees T1,T2,
PlatI(T1,T2) is a single-element set.
There is a unique arrow between any two trees!
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A la recherche du tensors perdu
(PhD Thesis) The prototypical self-similar category (X ,�)
Objects: All non-empty binary trees.
Arrows: A unique arrow between any two objects.
This monoidal category:
1 was introduced to study self-similarity S ∼= S ⊗ S,
2 contains MacLane’s (W,�) as a subcategory.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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Coherence for Self-Similarity
(a special case of a much more general theory)
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A straightforward coherence theorem
We base this on the category (X ,�)
Objects All non-empty binary trees.
Arrows A unique arrow between any two trees.
This category is posetal — all diagrams over X commute.
We will define a monoidal substitution functor:
XSub : (X ,�)→ (C,⊗)
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The self-similarity substitution functor
An inductive definition of XSub : (X ,�)→ (C,⊗)
On objects:
x 7→ Su�v 7→ XSub(u)⊗XSub(v)
On arrows:
(x ← x) 7→ 1S ∈ C(S,S)
(x ← x�x) 7→ C ∈ C(S ⊗ S,S)(x�x ← x) 7→ B ∈ C(S,S ⊗ S)
(b�v ← a�u) 7→ XSub(b ← a)⊗XSub(v ← u)
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Interesting properties:
1 XSub : (X ,�)→ (C,⊗) is always functorial.
2 Every arrow built up from
{C , B , 1S , ⊗ }
is the image of an arrow in X .
3 The image of every diagram in X commutes.
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Interesting properties:
1 XSub : (X ,�)→ (C,⊗) is always functorial.
2 Every arrow built up from
{C , B , 1S , ⊗ }
is the image of an arrow in X .
3 The image of every diagram in X commutes.
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Interesting properties:
1 XSub : (X ,�)→ (C,⊗) is always functorial.
2 Every arrow built up from
{C , B , 1S , ⊗ }
is the image of an arrow in X .
3 The image of every diagram in X commutes.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net
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XSub factors through the Platonic ideal
There is a monic-epic decomposition of XSub.
(X ,�)XSub //
XSub
$$
(PlatC ,�)
Inst
��(C,⊗)
Every canonical (for self-similarity) diagramin (PlatC ,�) commutes.
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Relating associativity and self-similarity
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A tale of two functors
Comparing the associativity and self-similarity categories.
MacLane’s (W,�)
Objects: Binary trees.
Arrows: Unique arrow between
two trees of the same rank.
The category (X ,�)
Objects: Binary trees.
Arrows: Unique arrow between
any two trees.
There is an obvious inclusion (W,�) ↪→ (X ,�)
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Is associativity a restriction of self-similarity?
Does the following diagram commute?
(W,�) �� //
WSub
��
(X ,�)
XSub
��(C,⊗)
Does the associativity functor
factor through
the self-similarity functor?
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Proof by contradiction:
Let’s assume this is the case.
Special arrows of (X ,�)
For arbitrary trees u,e, v ,
tuev = ((u�e)�v ← u�(e�v))
lv = (v ← e�v)
ru = (u ← u�e)
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Since all diagrams over X commute:
The following diagram over (X ,�) commutes:
u�(e�v)tuev //
1u�lv
��
(u�e)�v
ru�1v
��u�v
Let’s apply XSub to this diagram.
By Assumption: tuev 7→ τU,E ,V (assoc. iso.)
Notation: u 7→ U , v 7→ V , e 7→ E , lv 7→ λV , ru 7→ ρU
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Since all diagrams over X commute:
The following diagram over (X ,�) commutes:
u�(e�v)tuev //
1u�lv
��
(u�e)�v
ru�1v
��u�v
Let’s apply XSub to this diagram.
By Assumption: tuev 7→ τU,E ,V (assoc. iso.)
Notation: u 7→ U , v 7→ V , e 7→ E , lv 7→ λV , ru 7→ ρU
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Since all diagrams over X commute:
The following diagram over (C,⊗) commutes:
U ⊗ (E ⊗ V )τUEV //
1U⊗λU
(U ⊗ E)⊗ V
ρU⊗1V
~~U ⊗ V
This is MacLane’s units triangle— the defining equation for a unit (trivial) object.
The choice of e was arbitrary — every object is trivial!
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Since all diagrams over X commute:
The following diagram over (C,⊗) commutes:
U ⊗ (E ⊗ V )τUEV //
1U⊗λU
(U ⊗ E)⊗ V
ρU⊗1V
~~U ⊗ V
This is MacLane’s units triangle— the defining equation for a unit (trivial) object.
The choice of e was arbitrary — every object is trivial!
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A general result
The following diagram commutes
(W,�) �� //
WSub
��
(X ,�)
WSub
��(C,⊗)
exactly when (C,⊗) is degenerate —
i.e. all objects are trivial.
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An important special case:
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What is strict self-similarity?
Can the code / decode maps
C : S ⊗ S → S , B : S → S ⊗ S
be strict identities?
In single-object monoidal categories:
We only have one object, so S ⊗ S = S.
S
Id**S ⊗ S
Id
gg
Take the identity as both the code and decode arrows.
Untyped ≡ Strictly Self-Similar.
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What is strict self-similarity?
Can the code / decode maps
C : S ⊗ S → S , B : S → S ⊗ S
be strict identities?
In single-object monoidal categories:
We only have one object, so S ⊗ S = S.
S
Id**S ⊗ S
Id
gg
Take the identity as both the code and decode arrows.
Untyped ≡ Strictly Self-Similar.
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Generalising Isbell’s argument
1 Strict associativity: A⊗ (B ⊗ C)=(A⊗ B)⊗ CAll arrows of (W,�) are mapped toidentities of (C,⊗)
2 Strict self-similarity: S ⊗ S=S.All arrows of (X ,�) are mapped tothe identity of (C,⊗).
WSub trivially factors through XSub.
The conclusion
Strictly associative untyped monoidal categories are degenerate.
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This is seen in various fields ...
We see special cases of this in many areas:
(Monoid Theory)
Congruence-freeness (e.g. the polycyclic monoids).
(Group Theory)
No normal subgroups (e.g. Thompson’s group F).
(λ calculus / Logic)
Hilbert-Post completeness / Girard’s dynamical algebra.
(Linguistics)
Recently (re)discovered ... not yet named!
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Another perspective ...
Another way of looking at things:
The ‘No Simultaneous Strictness’ Theorem
One cannot have both
(I) Associativity A⊗ (B ⊗ C) ∼= (A⊗ B)⊗ C
(II) Self-Similarity S ∼= S ⊗ S
as strict equalities.
Coherence in Hilbert’s hotel arXiv[math.CT]:1304.5954 www.peter.hines.net