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Internal Neighbourhood Spaces

Partha Pratim Ghosh

Department of Mathematical Sciences, UNISAEmail: ghoshpp@unisa.ac.za

CT 2017, July 16-22 2017University of British Columbia

Vancouver

Internal Neighbourhood Spaces Partha Pratim Ghosh Frame 1 of 20. . .

Motivation

In [Bentley et al., 1991] the authors state in their Introduction:

. . . to obtain useful results, topologists have often forcedproperties of spaces or maps in the sense of addingsupplementary and extraneous conditions or even chang-ing the definition of a property altogether, and left cat-egorically defined constructions well alone, i.e.,essentially continued working in Top.Our aim in this paper is to provide evidence that doing pre-cisely the opposite, i.e., leaving concepts as they are butstepping outside Top and thereby changing constructionsin an appropriate way will illuminate the situation and pro-vide a natural setting or solution for problems for which nodecent solution in Top or any reasonable subcategory ofTop seems to exist. . . .

[Bentley et al., 1991]

Bentley, L., Herrlich, H., and Lowen, R. (1991).

Improving constructions in topology.In Herrlich, H. and Porst, H., editors, Category Theory at Work, volume 18 of Research Expositions in Mathematics, pages 3–20.Heldermann, Berlin.

Motivation

In [Bentley et al., 1991] the authors state in their Introduction:

. . . to obtain useful results, topologists have often forcedproperties of spaces or maps in the sense of addingsupplementary and extraneous conditions or even chang-ing the definition of a property altogether, and left cat-egorically defined constructions well alone, i.e.,essentially continued working in Top.Our aim in this paper is to provide evidence that doing pre-cisely the opposite, i.e., leaving concepts as they are butstepping outside Top and thereby changing constructionsin an appropriate way will illuminate the situation and pro-vide a natural setting or solution for problems for which nodecent solution in Top or any reasonable subcategory ofTop seems to exist. . . .

[Bentley et al., 1991]

Motivation

Extensions of Top

psTop prTop Top

reflective reflective

topological topological topological

prTop is the category of pretopological spacespsTop is the category of pseudotopological spaces

Motivation

Extensions of Top

psTop prTop Top

prTop is the category of pretopological spaces

psTop is the category of pseudotopological spaces

Motivation

Extensions of Top

psTop prTop Top

prTop is the category of pretopological spacespsTop is the category of pseudotopological spaces

Motivation

Purpose of this TalkExtend this construction to a more general context of a well behaved finitelycomplete category A.

pse[A] pre[A] wNbd[A] Nbd[A]

Top[A]

A Appj

reflective

topological

topological topological

reflective

topological

reflective

topological

non-full

subcategory

non-full

subcategory

reflective

topological

in a specialcontext

includingwhen

A = Set

Motivation

Purpose of this Talk

Extend this construction to a more general context of a well behaved finitelycomplete category A.

Top[A]

A Appj

reflective

topological

reflective

topological

reflective

topological

non-full

subcategory

non-full

subcategory

reflective

topological

in a specialcontext

includingwhen

A = Set

Motivation

Purpose of this Talk

Extend this construction to a more general context of a well behaved finitelycomplete category A.

Top[A]

A Appj

reflective

topological

reflective

topological

reflective

topological

non-full

subcategory

non-full

subcategory

reflective

topological

in a specialcontext

includingwhen

A = Set

Brief Review of Known Facts

Topological Spaces I

Theorem 2.1Given any set X and a function X

ξ−→ Fil(X ) from the set X to the set Fil(X )of all filters on X with the properties:

U ∈ ξ(x)⇒ x ∈ U (1)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

there exists a unique topology Ξ on X such that for each x ∈ X , ξ(x) is the setof all neighbourhoods of the point x in the topological space (X , Ξ).

Moral:There is a one-to-one correspondence between topologies on X and functions

Xξ−→ Fil(X ) satisfying (1), (2).

Topological Spaces I

Theorem 2.1Given any set X and a function X

ξ−→ Fil(X ) from the set X to the set Fil(X )of all filters on X with the properties:

U ∈ ξ(x)⇒ x ∈ U (1)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

there exists a unique topology Ξ on X such that for each x ∈ X , ξ(x) is the setof all neighbourhoods of the point x in the topological space (X , Ξ).

Moral:There is a one-to-one correspondence between topologies on X and functions

Xξ−→ Fil(X ) satisfying (1), (2).

Understanding Condition (2)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

Condition (2) is equivalent to:

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

in topological spaces, define F → x if F ⊇ ξ(x){F : F → x

}is a filter of filters

(3) is now completely in terms of lattice operations on Fil(X ) . . .

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

)x11 x12 x13 . . . x1n . . . → p1

x21 x22 x23 . . . x2n . . . → p2

x31 x32 x33 . . . x3n . . . → p3

xm1 xm2 xm3 xmn. . . . . . → pm

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

)x11 x12 x13 . . . x1n . . . → p1

x21 x22 x23 . . . x2n . . . → p2

x31 x32 x33 . . . x3n . . . → p3

xm1 xm2 xm3 xmn. . . . . . → pm

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

)x11 x12 x13 . . . x1n . . . → p1

x21 x22 x23 . . . x2n . . . → p2

x31 x32 x33 . . . x3n . . . → p3

xm1 xm2 xm3 xmn. . . . . . → pm

Condition (2) is the fil-ter version of this famil-iar fact for sequences ofnumbershenceforth calledthe sequence condition

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

in topological spaces, define F → x if F ⊇ ξ(x)

{F : F → x

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

U ∈ ξ(x)⇒ (∃V ∈ ξ(x))(y ∈ V ⇒ U ∈ ξ(y)

F ⊇ ξ(x),Gp ⊇ ξ(p)⇒⋃

⋂p∈FGp ⊇ ξ(x) (3)

Topological Spaces IITheorem 2.2 ([Kowalsky, 1965, §5])

Given any set X and a function XΓ−→ Fil(Fil(X )) such that:

x ={A ⊆ X : x ∈ A

}∈ Γ (x), (4)

a ⊆ Γ (x)⇒⋂

a ∈ Γ (x), (5)

F ∈ Γ (x),Gp ∈ Γ (p)⇒⋃

⋂p∈FGp ∈ Γ (x) (6)

there exists a unique topology Ξ on X such that for each x ∈ X the setγ(x) =

⋂Γ (x) ∈ Γ (x) is the set of all neighbourhoods of x in the topological

space (X , Ξ).

this formulation leads to successive extensions of Top . . .

[Kowalsky, 1965]

Kowalsky, H. (1965).

General Topology.Academic Press, New York.

Topological Spaces IITheorem 2.2 ([Kowalsky, 1965, §5])

Given any set X and a function XΓ−→ Fil(Fil(X )) such that:

x ={A ⊆ X : x ∈ A

}∈ Γ (x), (4)

a ⊆ Γ (x)⇒⋂

a ∈ Γ (x), (5)

F ∈ Γ (x),Gp ∈ Γ (p)⇒⋃

⋂p∈FGp ∈ Γ (x) (6)

there exists a unique topology Ξ on X such that for each x ∈ X the setγ(x) =

⋂Γ (x) ∈ Γ (x) is the set of all neighbourhoods of x in the topological

space (X , Ξ).

this formulation leads to successive extensions of Top . . .[Kowalsky, 1965]

Kowalsky, H. (1965).

General Topology.Academic Press, New York.

Pretopological Spaces

Definition 2.3 (Pretopological Space (see [Bentley et al., 1991],[Herrlich et al., 1991]))

A pretopological space is a pair (X , Γ ),

where X is a set, XΓ−→ Fil(Fil(X )) is

a function such that for each x ∈ X , x ∈ Γ (x) and Γ (x) is closed underintersections.

[Bentley et al., 1991] [Herrlich et al., 1991]

Herrlich, H., Lowen-Colebunders, E., and Schwarz, F. (1991).

Improving Top: PrTop and PsTop.In Herrlich, H. and Porst, H., editors, Category Theory at Work, volume 18 of Research Expositions in Mathematics, pages 21–34.Heldermann, Berlin.

A pretopological space is a pair (X , Γ ), where X is a set, XΓ−→ Fil(Fil(X )) is

a function such that

for each x ∈ X , x ∈ Γ (x) and Γ (x) is closed underintersections.

a function such that for each x ∈ X , x ∈ Γ (x)

and Γ (x) is closed underintersections.

a function such that for each x ∈ X , x ∈ Γ (x) and Γ (x) is closed underintersections.

For any function Xf−→ Y and any F ∈ Fil(X ):

→f F =

{B ⊆ Y : f −1B ∈ F

For any function Xf−→ Y and any F ∈ Fil(X ):

→f F =

{B ⊆ Y : f −1B ∈ F

Clearly,→f F is the filter on Y generated by the images f (A), A ∈ F .

Definition 2.3 (Pretopological Morphism (see [Bentley et al., 1991],[Herrlich et al., 1991]))

Given the pretopological spaces (X , Γ ) and (Y , Φ),

a function Xf−→ Y is said to

be a pretopological morphism, if

Given the pretopological spaces (X , Γ ) and (Y , Φ),a function Xf−→ Y is said to

F ∈ Γ (x)⇒→f F ∈ Φ(f (x)).

prTop is the category of pretopological spaces and pretopological morphisms.

Pseudotopological Spaces

Definition 2.4 (Pseudotopological Space (see [Bentley et al., 1991],[Herrlich et al., 1991]))

A pseudotopological space is a pair (X , Γ ),

where X is a set,

XΓ−→ Fil(Fil(X )) is a function such that for each x ∈ X , x ∈ Γ (x) and for

each filter F on X , maximal filter U on X :(F ⊆ U ⇒ U ∈ Γ (x)

)⇒ F ∈ Γ (x).

A pseudotopological space is a pair (X , Γ ), where X is a set,

XΓ−→ Fil(Fil(X )) is a function such that

for each x ∈ X , x ∈ Γ (x) and foreach filter F on X , maximal filter U on X :(

F ⊆ U ⇒ U ∈ Γ (x))⇒ F ∈ Γ (x).

XΓ−→ Fil(Fil(X )) is a function such that for each x ∈ X , x ∈ Γ (x) and

foreach filter F on X , maximal filter U on X :(

F ⊆ U ⇒ U ∈ Γ (x))⇒ F ∈ Γ (x).

XΓ−→ Fil(Fil(X )) is a function such that for each x ∈ X , x ∈ Γ (x) and for

each filter F on X , maximal filter U on X :(F ⊆ U ⇒ U ∈ Γ (x)

)⇒ F ∈ Γ (x).

Definition 2.4 (Pseudotopological Morphism (see [Bentley et al., 1991],[Herrlich et al., 1991]))

Given the pseudotopological spaces (X , Γ ) and (Y , Φ),

a function Xf−→ Y is

said to be a pseudotopological morphism, if

F ∈ Γ (x)⇒→f F ∈ Φ(f (x)).

Given the pseudotopological spaces (X , Γ ) and (Y , Φ), a function Xf−→ Y is

F ∈ Γ (x)⇒→f F ∈ Φ(f (x)).

psTop is the category of pseudotopological spaces and pseudotopologicalmorphisms.

Extending to the General Case

Key Issues for Generalisation

A. main focus is on the lattice of subobjects

B. one needs to avoid the atomicity present for sets

C. filters can be defined on any semilattice

D. known from [Iberkleid and McGovern, 2009]: a lattice L is distributive, ifand only if, Fil(L) is a coherent frame

A frame is coherent if it is compact, algebraic and binary meets of compactelements is compact.

E. one needs a nice factorisation of morphisms

[Iberkleid and McGovern, 2009]

Iberkleid, W. and McGovern, W. W. (2009).

A Natural Equivalence for the Category of Coherent Frames.Alg. Univ., 62:247–258.available at: http://home.fau.edu/wmcgove1/web/Papers/WolfNatEq.pdf.

D. known from [Iberkleid and McGovern, 2009]: a lattice L is distributive, ifand only if, Fil(L) is a coherent frameA frame is coherent if it is compact, algebraic and binary meets of compactelements is compact.

Our Setup

1. A is a finitely complete category

2. A has a proper factorisation system (E,M) (see [Carboni et al., 1997, §2]for details on factorisation systems)

3. Each SubM (X ) is a distributive complete lattice

[Carboni et al., 1997]

Carboni, A., Janelidze, G., Kelly, G. M., and Pare, R. (1997).

On localization and stabilization for factorization systems.Appl. Categ. Str., 5(1):1 – 58.

Carboni, A., Lack, S., and Walters, R. (1993).

Introduction to Extensive and Distributive Categories.Jr. Pure Appl. Alg., 84:145 – 158.

[Clementino et al., 2004]

Clementino, M. M., Giuli, E., and Tholen, W. (2004).

A Functional Approach to General Topology.In Categorical Foundations, number 97 in Encyclopedia Math. Appl., pages 103–163. Cambridge University Press.

Our Setup

1. A is a finitely complete category2. A has a proper factorisation system (E,M) (see [Carboni et al., 1997, §2]

for details on factorisation systems)3. Each SubM (X ) is a distributive complete lattice

Definition 3.1 (Proper Factorisations)

A proper factorisation system for A is (E,M), where:

(a) E is a set of epimorphisms of A,

(b) M is a set of monomorphisms of A,

(c) for every morphism Xf−→ Y of A,

(d) if v ◦ e = m ◦ u for some e ∈ E and m ∈M

Our Setup

(c) for every morphism Xf−→ Y of A, there exists a X

e−→ I in E

Our Setup

e−→ I in E and aI

m−→ Y in M such that

Our Setup

e−→ I in E and a

Im−→ Y in M such that X

99Im // Y , and

Our Setup

99Im // Y , and

Our Setup

99Im // Y , and

(d) if v ◦ e = m ◦ u for some e ∈ E and m ∈M then there exists a uniquemorphism w such that

Our Setup

99Im // Y , and

(d) if v ◦ e = m ◦ u for some e ∈ E and m ∈M then there exists a unique

morphism w such that · e //

��

��!w��· m

commutes.

Our Setup

Definition 3.1 (Admissible Subobjects)

Any Mm−→ X (m ∈M) is an admissible subobject of X .

Our Setup

Let SubM (X ) be the set of admissible subobjects.

(a) For m, n ∈ SubM (X ), m ≤ n if there exists a morphism k making thediagram:

! k // N

n��X

to commute, i.e., m = n ◦ k .(b) In presence of finite completeness of A, SubM (X ) is a meet semilattice.[Carboni et al., 1997]

Our Setup

! k // N

n��X

to commute, i.e., m = n ◦ k .

(b) In presence of finite completeness of A, SubM (X ) is a meet semilattice.[Carboni et al., 1997]

Our Setup

! k // N

n��X

to commute, i.e., m = n ◦ k .(b) In presence of finite completeness of A, SubM (X ) is a meet semilattice.

Our Setup

Example 3.1 (Examples of Setup)

(a) Every Grothendieck topos

(b) Every small complete, small cocomplete and extensive category (see[Carboni et al., 1993] for extensive and distributive categories)

(c) If A be such and A be an object, then so also is (A ↓ A) (see[Clementino et al., 2004, §2.10])

(d) Hence many of the examples of [Clementino et al., 2004] is an example ofthis context

Our Setup

Neighbourhoods of all Sorts

Definition 3.1 (Pre- & Weak Neighbourhoods on an Object)

A pre-neighbourhood on an object X is an order preserving map

SubM (X )opµ−→ Fil(X ) such that:

n ∈ µ(m)⇒ m ≤ n.

A pre-neighbourhood µ on X such that:

µ(m) =⋃

p∈µ(m)

is a weak neighbourhood on X .

A pre-neighbourhood on an object X is an order preserving mapSubM (X )op

µ−→ Fil(X ) such that:

n ∈ µ(m)⇒ m ≤ n.

µ(m) =⋃

p∈µ(m)

n ∈ µ(m)⇒ m ≤ n.

µ(m) =⋃

p∈µ(m)

n ∈ µ(m)⇒ m ≤ n.

µ(m) =⋃

p∈µ(m)

n ∈ µ(m)⇒ m ≤ n.

µ(m) =⋃

p∈µ(m)

p weak neighbourhoodsare precisely theinterpolative pre-neighbourhoods

Definition 3.1 (Neighbourhoods on an Object)

A weak neighbourhood µ on X such that:

µ(∨

G ) =⋂g∈G

µ(g), for all G ⊆ SubM (X )

µ(∨

G ) =⋂g∈G

is a neighbourhood on X .

µ(∨

G ) =⋂g∈G

is a neighbourhood on X .

Neighbourhoods are meet preserving weak neighbourhoods.

Interior and Open

Theorem 3.2 (µ-Interior and µ-Open)

Given a pre-neighbourhood µ on X let:

Oµ = {p ∈ SubM (X ) : p ∈ µ(p)}, (7)

intµ(m) =∨{

p ∈ Oµ : p ≤ m}. (8)

(a) Oµ is closed under finite meets.

(b) Oµ is closed under arbitrary joins, if and only if, µ preserve meets.

(c) If µ is a neighbourhood on X then m 7→ intµ(m) is a Kuratowski interioroperator and:

p ∈ µ(m)⇔ m ≤ intµ(p).

Interior and Open

Oµ = {p ∈ SubM (X ) : p ∈ µ(p)}, (7)

intµ(m) =∨{

p ∈ Oµ : p ≤ m}. (8)

Interior and Open

Oµ = {p ∈ SubM (X ) : p ∈ µ(p)}, (7)

intµ(m) =∨{

p ∈ Oµ : p ≤ m}. (8)

Interior and Open

Oµ = {p ∈ SubM (X ) : p ∈ µ(p)}, (7)

intµ(m) =∨{

p ∈ Oµ : p ≤ m}. (8)

Interior and Open

Oµ = {p ∈ SubM (X ) : p ∈ µ(p)}, (7)

intµ(m) =∨{

p ∈ Oµ : p ≤ m}. (8)

Interior and Open

weak neighbourhoods are not neighbourhoods. . .

Interior and Open

Example 3.2 (Weak neighbourhood not a neighbourhood)

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

For A ⊆ X , if supA < 1 let nA be the smallest n ≥ 1 such that:

x ∈ A⇒ x ≤ 1− 1

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

x ∈ A⇒ x ≤ 1− 1

µ(A) =

{{X}, if A ∩ [1, 2] 6= ∅ or supA = 1{T ⊆ X : [0, 1− 1

nA] ⊆ T

}, otherwise

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

x ∈ A⇒ x ≤ 1− 1

µ(A) =

{{X}, if A ∩ [1, 2] 6= ∅ or supA = 1{T ⊆ X : [0, 1− 1

nA] ⊆ T

}, otherwise

Clearly µ is a pre-neighbourhood on X .

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

x ∈ A⇒ x ≤ 1− 1

µ(A) =

{{X}, if A ∩ [1, 2] 6= ∅ or supA = 1{T ⊆ X : [0, 1− 1

nA] ⊆ T

}, otherwise

Further:

{[0, 1− 1

]: n ≥ 1

}∪ {X}.

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

x ∈ A⇒ x ≤ 1− 1

µ(A) =

{{X}, if A ∩ [1, 2] 6= ∅ or supA = 1{T ⊆ X : [0, 1− 1

nA] ⊆ T

}, otherwise

Further:

{[0, 1− 1

]: n ≥ 1

}∪ {X}.

Hence: µ is a weak neighbourhood.

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

x ∈ A⇒ x ≤ 1− 1

µ(A) =

{{X}, if A ∩ [1, 2] 6= ∅ or supA = 1{T ⊆ X : [0, 1− 1

nA] ⊆ T

}, otherwise

Further:

{[0, 1− 1

]: n ≥ 1

}∪ {X}.

{X} = µ([0, 1)) = µ(⋃n≥1

[0, 1− 1

n])⊂⋂n≥1

µ([0, 1− 1

Interior and Open

Let X = [0, 2] and a0 = 0, an = 1− 1n , n ≥ 1.

x ∈ A⇒ x ≤ 1− 1

µ(A) =

{{X}, if A ∩ [1, 2] 6= ∅ or supA = 1{T ⊆ X : [0, 1− 1

nA] ⊆ T

}, otherwise

Further:

{[0, 1− 1

]: n ≥ 1

}∪ {X}.

Hence µ is not a neighbourhood.

Interior and Open

a pre-neighbourhood µ such that:

Interior and Open

p ∈ µ(m)⇔ m ≤ intµ(p)

Interior and Open

p ∈ µ(m)⇔ m ≤ intµ(p)

is a weak neighbourhood, but may not be a neighbourhood!

Interior and Open

Example 3.2 (A weak neighbourhood with interior property not aneighbourhood)

Interior and Open

Let (X , T ) be a topological space with C its set of closed sets.

Interior and Open

Let (X , T ) be a topological space with C its set of closed sets.Define:

µ(A) ={V ⊆ X : (∃C ∈ C)(A ⊆ C ⊆ V )

Interior and Open

µ(A) ={V ⊆ X : (∃C ∈ C)(A ⊆ C ⊆ V )

µ is a pre-neighbourhood.

Interior and Open

µ(A) ={V ⊆ X : (∃C ∈ C)(A ⊆ C ⊆ V )

Oµ = C.

Interior and Open

µ(A) ={V ⊆ X : (∃C ∈ C)(A ⊆ C ⊆ V )

Oµ = C.Hence µ satisfies the interior property.

Interior and Open

µ(A) ={V ⊆ X : (∃C ∈ C)(A ⊆ C ⊆ V )

If C be not closed under arbitrary joins then µ is not a neighbourhood.

Morphisms

Given any morphism Xf−→ Y one obtains:

[Holgate and Slapal, 2011]

Holgate, D. and Slapal, J. (2011).

Categorical neighborhood operators.Top. Appl., 158:2356–2365.

Morphisms

(a) from the (E,M) factorisation the image-preimage adjunction:

Morphisms

(a) from the (E,M) factorisation the image-preimage adjunction:

SubM (X )

∃f ,,

⊥ SubM (Y )

Morphisms

(a) given m ∈ SubM (X ), y ∈ SubM (Y ):

Morphisms

f−1N

f−1n

��

��X

pullback along f defines n 7→ f−1n

MorphismsGiven any morphism X

f−→ Y one obtains:

Mf∣∣M //

��

��X

the (E, M) factorisation of f ◦m produces m 7→ ∃fm

Morphisms

(b) since ∃f

and f−1

are both order preserving maps, one has the furtheradjunction:

Morphisms

(b) since ∃f

and f−1

are both order preserving maps, one has the furtheradjunction:

Fil(X )ss

44⊥ Fil(Y ) ,

Morphisms

(b) where:

→f A =

{y ∈ SubM (Y ) : f

−1y ∈ A

}, for A ∈ Fil(X ) (7)

←f B =

{x ∈ SubM (X ) : (∃b ∈ B)(f

−1b ≤ x)

}, for B ∈ Fil(Y ), (8)

Clearly:→f A is the filter generated by the images ∃

fa (a ∈ A), and

←f B is

the filter generated by the preimages f−1b (b ∈ B).

Morphisms

Definition 3.3 (Pre-neighbourhood Morphisms)

Given the pre-neighbourhoods µ on X and φ on Y , a morphism Xf−→ Y is a

pre-neighbourhood morphism if:

Morphisms

Definition 3.3 (Pre-neighbourhood Morphisms)

Given the pre-neighbourhoods µ on X and φ on Y , a morphism Xf−→ Y is a

pre-neighbourhood morphism if:

p ∈ φ(n)⇒ f−1p ∈ µ(f

−1n). (7)

Morphisms

Theorem 3.3 ([Holgate and Slapal, 2011, §3])

Given pre-neighbourhood structures µ on X and ν on Y , the following areequivalent :

(a) for each n ∈ SubM (Y ), p ∈ φ(n)⇒ f−1p ∈ µ(f

−1n)

(b) for each n ∈ SubM (Y ),←f φ(n) ⊆ µ(f

−1n)

(c) for each n ∈ SubM (Y ), φ(n) ⊆→f µ(f

−1n)

(d) for each m ∈ SubM (X ),←f φ(∃

fm) ≤ µ(m)

Morphisms

Theorem 3.3 ([Holgate and Slapal, 2011, §3])

Given pre-neighbourhood structures µ on X and ν on Y , the following areequivalent :

(a) for each n ∈ SubM (Y ), p ∈ φ(n)⇒ f−1p ∈ µ(f

−1n)

(b) for each n ∈ SubM (Y ),←f φ(n) ⊆ µ(f

−1n)

(c) for each n ∈ SubM (Y ), φ(n) ⊆→f µ(f

−1n)

(d) for each m ∈ SubM (X ),←f φ(∃

fm) ≤ µ(m)

MorphismsTheorem 3.3 ([Holgate and Slapal, 2011, §3])

given the order preserving maps:

Fil(X ) Fil(Y )

SubM (X )op SubM (Y )op

Fil(X ) Fil(Y )

Categories of Internal Neighbourhood Structures

Internal Neighbourhood Structures

Definition 4.1 (Internal Weak Neighbourhood Spaces)

wNbd[A] is the category whose objects are (X , µ), where µ is a weak

neighbourhood on X and morphisms are (X , µ)f−→ (X , φ) where f is a

pre-neighbourhood morphism.

Definition 4.1 (Internal Pretopological Spaces)

pre[A] is the category whose objects are (X , µ), where µ is a pre-neighbourhood

on X and morphisms are (X , µ)f−→ (X , φ) where f is a pre-neighbourhood

morphism.

Definition 4.1 (Internal Pseudotopological Spaces)

pse[A] is the category whose:

(b) morphisms are (X , Γ )f−→ (Y , Φ) where:

A ∈ Γ (m)⇒→f A ∈ Φ(∃

(a) objects are (X , Γ ),

A ∈ Γ (m)⇒→f A ∈ Φ(∃

(a) objects are (X , Γ ), where SubM (X )Γ−→ Fil(Fil(X )) is an order

preserving map such that for each m ∈ SubM (X ):

A ∈ Γ (m)⇒→f A ∈ Φ(∃

↑ m ∈ Γ (m) (7)

A ∈ Γ (m)⇒→f A ∈ Φ(∃

Internal Neighbourhood StructuresDefinition 4.1 (Internal Pseudotopological Spaces)

↑ m ∈ Γ (m) (7)

(∀A ∈ Fil(X ))

[(∀U ∈Max[X ])(

A ⊆ U ⇒ U ∈ Γ (m))

⇒ A ∈ Γ (m)

A ∈ Γ (m)⇒→f A ∈ Φ(∃

Definition 4.1 (Internal Neighbourhood Spaces)

Nbd[A] is the category whose objects are (X , µ), where µ is a neighbourhood

on X and morphisms are (X , µ)f−→ (Y , φ), where f is a pre-neighbourhood

morphism such that f−1

preserve arbitrary joins.

Theorem 4.1 (Equivalents of Pre-image Preserving Joins)

the following are equivalent for any morphism Xf−→ Y :

(a) SubM (Y )

⊥ SubM (X )

(b) f−1

preserve joins.

(c)←f preserve meets.

(d) Fil(X )

ff←f

⊥ Fil(Y )

(a) SubM (Y )

⊥ SubM (X )

(b) f−1

preserve joins.

(d) Fil(X )

ff←f

⊥ Fil(Y )

(a) SubM (Y )

⊥ SubM (X )

(b) f−1

preserve joins.

(d) Fil(X )

ff←f

⊥ Fil(Y )

(a) SubM (Y )

⊥ SubM (X )

(b) f−1

preserve joins.

(d) Fil(X )

ff←f

⊥ Fil(Y )

(a) SubM (Y )

⊥ SubM (X )

(b) f−1

preserve joins.

(d) Fil(X )

ff←f

⊥ Fil(Y )

SubM (X )SubM (Y ) f−1

Fil(X )Fil(Y )←f

∀f⊥ t

Fil(X )Fil(Y )←f

∀f⊥ t

∀f x =∨{

y ∈ SubM (Y ) : f−1y ≤ x

tf A =

{y ∈ SubM (Y ) : (∃a ∈ A)(∀f a ≤ y)

Definition 4.1 (Internal Topological Spaces)

An internal topological space is an internal neighbourhood space (X , µ) in whichOµ is a frame in the partial order of SubM (X ).

Definition 4.1 (Internal Topological Spaces)

Top[A] is the full subcategory of Nbd[A] consisting of internal topologicalspaces.

Definition 4.1 (The Non-full Subcategory of Preimage Preserve Joins)

Appj is the (non-full) subcategory of A whose objects are same as of A and

morphisms are those morphisms f from A for which f−1

preserve joins.

Reflective Extensions and Topologicity

Reflectivity & Topologicity Chart

Top[A]

A Appj

non-full

subcategory

non-fullsubcategory

Top[A]

A Appj

non-full

subcategory

non-fullsubcategory

reflective

topological

Top[A]

A Appj

non-full

subcategory

non-fullsubcategory

reflective

topological

reflective

x ∧ ¬x = 0

Theorem 5.1 (Reflective Inclusion of pre[A] in pse[A])

If: each SubM (X ) is further assumed to be pseudocomplemented,

Theorem 5.1 (Reflective Inclusion of pre[A] in pse[A])

If: each SubM (X ) is further assumed to be pseudocomplemented,then: pre[A] is a reflective full subcategory of pse[A].

Top[A]

A Appj

non-full

subcategory

non-fullsubcategory

reflective

topological

reflective

x ∧ ¬x = 0

topological

x ∧ f−1y = 0 ⇒

∃fx ∧ y = 0

Theorem 5.1 (Topologicity of pse[A])

If for every morphism Xf−→ Y of A, each x ∈ SubM (X ) and y ∈ SubM (Y ):

x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

then, pse[A] is topological over A.

x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

Condition x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

fx ∧ y = 0 yields:

(a) For every proper filter A ∈ Fil(X ), B ∈ Fil(Y ):

→f A ⊆ B ⇒ (∃C ∈ Fil(X ))

(A ⊆ C and B ⊆

→f C).

(b) For every maximal filter U ∈Max[X ],→f U is a maximal filter on Y .

(c) (X , Γ )f−→ (Y , Φ) is a pseudotopological morphism, if and only if:

U ∈Max[X ] ∩ Γ (m)⇒→f U ∈Max[Y ] ∩ Φ(∃

x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

→f A ⊆ B ⇒ (∃C ∈ Fil(X ))

(A ⊆ C and B ⊆

→f C).

U ∈Max[X ] ∩ Γ (m)⇒→f U ∈Max[Y ] ∩ Φ(∃

x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

→f A ⊆ B ⇒ (∃C ∈ Fil(X ))

(A ⊆ C and B ⊆

→f C).

U ∈Max[X ] ∩ Γ (m)⇒→f U ∈Max[Y ] ∩ Φ(∃

x ∧ f−1y = 0⇒ ∃

fx ∧ y = 0

→f A ⊆ B ⇒ (∃C ∈ Fil(X ))

(A ⊆ C and B ⊆

→f C).

U ∈Max[X ] ∩ Γ (m)⇒→f U ∈Max[Y ] ∩ Φ(∃

Top[A]

A Appj

non-full

subcategory

non-fullsubcategory

reflective

topological

reflective

x ∧ ¬x = 0

topological

x ∧ f−1y = 0 ⇒

∃fx ∧ y = 0

reflective

topological

existenceof largesttopology

Theorem 5.1 (Reflective Inclusion of Top[A] and topologicity)

the following are equivalent :

(a) For every object X , there exists a largest internal topology on X .

(b) Top[A] is a full reflective subcategory of Nbd[A].

(c) Top[A] is topological over Appj .

Top[A]

A Appj

reflective

topological

reflective

x ∧ ¬x = 0

topological

x ∧ f−1y = 0 ⇒

∃fx ∧ y = 0

reflective

topological

existenceof largesttopology

reflective

each f−1

preserve join

Theorem 5.1 (Reflectivity of Nbd[A] in wNbd[A])

If for every morphism Xf−→ Y of A f

−1preserve joins then Nbd[A] is a full

reflective subcategory of wNbd[A].

Top[A]

A Appj

reflective

topological

reflective

x ∧ ¬x = 0

topological

x ∧ f−1y = 0 ⇒

∃fx ∧ y = 0

reflective

each f−1

preserve join

topological

subobjectlattices

are frames

Regular Epimorphisms

Regular Epimorphisms of Internal PretopologicalSpaces

Theorem 6.1If the forgetful functor pre[A]

V−→ A create kernel pairs and preserve coequalisers

then a morphism (X , γ)f−→ (Y , φ) of pre[A] is a regular epimorphism, if and

only if, Xf−→ Y is a regular epimorphism of A and:

φ(y) ={u ∈ SubM (Y ) : y ≤ u and f

−1u ∈ γ(f

−1y)}.

Regular Epimorphisms

Regular Epimorphisms of InternalPseudotopological Spaces

Theorem 6.2If the forgetful functor pre[A]

W−→ A is topological then a morphism

(X , Γ )f−→ (Y , Φ) of pre[A] is a regular epimorphism, if and only if, X

f−→ Y is aregular epimorphism of A and:

(∀y ∈ SubM (Y ))(∀V ∈Max[Y ] ∩ Φ(y))

(∃x ∈ f−1y)(∃U ∈Max[X ] ∩ Γ (x)) (

V =→f U).

Concluding Remarks

Way Forward. . .

(a) regular epimorphisms of internal neighbourhood spaces. . .

(b) how good are the extensions — topological hull? or quasi-topos hull? . . .

(c) effective descent morphisms of the internal neighbourhood spaces

(d) possibilities of a general structure (B,B F−→ A . . . ) where F is a fibrationand . . .

Concluding Remarks

Way Forward. . .

Concluding Remarks

Way Forward. . .

Concluding Remarks

Way Forward. . .

Concluding Remarks

Way Forward. . .

monoid poset

Concluding Remarks

Way Forward. . .

monoid poset

category

Concluding Remarks

Way Forward. . .

monoid poset

category

monoidal category

Concluding Remarks

Way Forward. . .

monoid poset

category

monoidal category

cartesian cocartesian

Concluding Remarks

Way Forward. . .

monoid poset

category

monoidal category

linear

Concluding Remarks

Way Forward. . .

monoid poset

category

monoidal category

linear

subobject quotient object

Concluding Remarks

Way Forward. . .

(a) regular epimorphisms of internal neighbourhood spaces. . .(b) how good are the extensions — topological hull? or quasi-topos hull? . . .(c) effective descent morphisms of the internal neighbourhood spaces

monoid poset

category

monoidal category

linear

vector spacesInternal Neighbourhood Spaces Partha Pratim Ghosh Frame 19 of 20. . .

Concluding Remarks

Way Forward. . .

(a) regular epimorphisms of internal neighbourhood spaces. . .(b) how good are the extensions — topological hull? or quasi-topos hull? . . .(c) effective descent morphisms of the internal neighbourhood spaces

monoid poset

category

monoidal category

linear

vector spaces

form (A,B F−→ A). . .

Concluding Remarks

THANK YOU

Concluding Remarks

Bentley, L., Herrlich, H., and Lowen, R. (1991).Improving constructions in topology.In Herrlich, H. and Porst, H., editors, Category Theory at Work, volume 18of Research Expositions in Mathematics, pages 3–20. Heldermann, Berlin.

Carboni, A., Janelidze, G., Kelly, G. M., and Pare, R. (1997).On localization and stabilization for factorization systems.Appl. Categ. Str., 5(1):1 – 58.

Carboni, A., Lack, S., and Walters, R. (1993).Introduction to Extensive and Distributive Categories.Jr. Pure Appl. Alg., 84:145 – 158.

Clementino, M. M., Giuli, E., and Tholen, W. (2004).A Functional Approach to General Topology.In Categorical Foundations, number 97 in Encyclopedia Math. Appl., pages103–163. Cambridge University Press.

Herrlich, H., Lowen-Colebunders, E., and Schwarz, F. (1991).Improving Top: PrTop and PsTop.In Herrlich, H. and Porst, H., editors, Category Theory at Work, volume 18of Research Expositions in Mathematics, pages 21–34. Heldermann, Berlin.

Concluding RemarksCategorical neighborhood operators.Top. Appl., 158:2356–2365.

Iberkleid, W. and McGovern, W. W. (2009).A Natural Equivalence for the Category of Coherent Frames.Alg. Univ., 62:247–258.available at: http://home.fau.edu/wmcgove1/web/Papers/WolfNatEq.pdf.

Kowalsky, H. (1965).General Topology.Academic Press, New York.

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