[PPT] - One: a characterization of skeletal objects for the Aufhebung of PowerPoint Presentation

SLIDE 1

1/19

One: a characterization of skeletal objects for the Aufhebung of Level 0 in certain toposes of spaces

M. Menni

Conicet and Universidad Nacional de La Plata

SLIDE 2

2/19

Levels

Let E be a topos.

Definition

A level l (of E) is a string of adjoints E

⊣ l∗

⊣

El

l!

l∗
with fully faithful l!, l∗ : El → E.

Equivalently, a level is an essential subtopos of E. (l∗ : El → E is the full subcategory of l-sheaves.)

SLIDE 3

3/19

Levels and ‘dimensions’

SLIDE 4

3/19

Levels and ‘dimensions’

“The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular

examples. More precisely, a space may be said to have (less than
r equal to) the dimension grasped by a given level if it belongs to

the negative (left adjoint inclusion) incarnation of that level.” Lawvere’s Some thoughts on the future of category theory LNM 1488, 1991.

SLIDE 5

3/19

Levels and ‘dimensions’

“The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular

examples. More precisely, a space may be said to have (less than
r equal to) the dimension grasped by a given level if it belongs to

the negative (left adjoint inclusion) incarnation of that level.” Lawvere’s Some thoughts on the future of category theory LNM 1488, 1991. Let l! ⊣ l∗ ⊣ l∗ be a level of E. For X in E, the counit l!(l∗X) → X is the l-skeleton of X. So,

SLIDE 6

3/19

Levels and ‘dimensions’

“The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular

examples. More precisely, a space may be said to have (less than
r equal to) the dimension grasped by a given level if it belongs to

the negative (left adjoint inclusion) incarnation of that level.” Lawvere’s Some thoughts on the future of category theory LNM 1488, 1991. Let l! ⊣ l∗ ⊣ l∗ be a level of E. For X in E, the counit l!(l∗X) → X is the l-skeleton of X. So, the object X is said to be l-skeletal if its l-skeleton is an iso and

SLIDE 7

3/19

Levels and ‘dimensions’

“The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular

examples. More precisely, a space may be said to have (less than
r equal to) the dimension grasped by a given level if it belongs to

the negative (left adjoint inclusion) incarnation of that level.” Lawvere’s Some thoughts on the future of category theory LNM 1488, 1991. Let l! ⊣ l∗ ⊣ l∗ be a level of E. For X in E, the counit l!(l∗X) → X is the l-skeleton of X. So, the object X is said to be l-skeletal if its l-skeleton is an iso and l! : El → E is the full subcat of l-skeletal objects Intuition:

SLIDE 8

3/19

Levels and ‘dimensions’

“The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular

examples. More precisely, a space may be said to have (less than
r equal to) the dimension grasped by a given level if it belongs to

the negative (left adjoint inclusion) incarnation of that level.” Lawvere’s Some thoughts on the future of category theory LNM 1488, 1991. Let l! ⊣ l∗ ⊣ l∗ be a level of E. For X in E, the counit l!(l∗X) → X is the l-skeleton of X. So, the object X is said to be l-skeletal if its l-skeleton is an iso and l! : El → E is the full subcat of l-skeletal objects Intuition: X is l-skeletal iff dim X ≤ l. Also:

SLIDE 9

3/19

Levels and ‘dimensions’

“The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular

examples. More precisely, a space may be said to have (less than
r equal to) the dimension grasped by a given level if it belongs to

the negative (left adjoint inclusion) incarnation of that level.” Lawvere’s Some thoughts on the future of category theory LNM 1488, 1991. Let l! ⊣ l∗ ⊣ l∗ be a level of E. For X in E, the counit l!(l∗X) → X is the l-skeleton of X. So, the object X is said to be l-skeletal if its l-skeleton is an iso and l! : El → E is the full subcat of l-skeletal objects Intuition: X is l-skeletal iff dim X ≤ l. Also: l has monic skeleta if the l-skeleton of every object is monic.

SLIDE 10

4/19

Some simple examples

SLIDE 11

4/19

Some simple examples

Proposition

If C is a small category then restriction along a full inclusion B → C is the inverse image of a level B → C.

SLIDE 12

4/19

Some simple examples

Proposition

If C is a small category then restriction along a full inclusion B → C is the inverse image of a level B → C.

1. (Simplicial sets) Every truncation ∆n → ∆ induces a level
∆n →

∆. (Every level of ∆ is of this form.)

SLIDE 13

4/19

Some simple examples

Proposition

If C is a small category then restriction along a full inclusion B → C is the inverse image of a level B → C.

1. (Simplicial sets) Every truncation ∆n → ∆ induces a level
∆n →

∆. (Every level of ∆ is of this form.)

2. (The classifier of (non-trivial) Boolean algebras.) Let F be the

category of nonempty finite sets and let Fn → F be the

bvious truncation.

SLIDE 14

4/19

Some simple examples

Proposition

If C is a small category then restriction along a full inclusion B → C is the inverse image of a level B → C.

1. (Simplicial sets) Every truncation ∆n → ∆ induces a level
∆n →

∆. (Every level of ∆ is of this form.)

2. (The classifier of (non-trivial) Boolean algebras.) Let F be the

category of nonempty finite sets and let Fn → F be the

bvious truncation.
3. (Cubical? sets/The classifier of strictly bipointed objects)

[A, Set] where A is the category of finite strictly bipointed

sets. Truncations.

SLIDE 15

4/19

Some simple examples

Proposition

If C is a small category then restriction along a full inclusion B → C is the inverse image of a level B → C.

1. (Simplicial sets) Every truncation ∆n → ∆ induces a level
∆n →

∆. (Every level of ∆ is of this form.)

2. (The classifier of (non-trivial) Boolean algebras.) Let F be the

category of nonempty finite sets and let Fn → F be the

bvious truncation.
3. (Cubical? sets/The classifier of strictly bipointed objects)

[A, Set] where A is the category of finite strictly bipointed

sets. Truncations.
4. Other.

Remark:

SLIDE 16

4/19

Some simple examples

Proposition

If C is a small category then restriction along a full inclusion B → C is the inverse image of a level B → C.

1. (Simplicial sets) Every truncation ∆n → ∆ induces a level
∆n →

∆. (Every level of ∆ is of this form.)

2. (The classifier of (non-trivial) Boolean algebras.) Let F be the

category of nonempty finite sets and let Fn → F be the

bvious truncation.
3. (Cubical? sets/The classifier of strictly bipointed objects)

[A, Set] where A is the category of finite strictly bipointed

sets. Truncations.
4. Other.

Remark: All the levels indicated above have monic skeleta.

SLIDE 17

5/19

Aufhebung

The levels of E may be partially ordered as subtoposes. That is, m is above l if and only if l∗ factors through m∗ (I.e. l-sheaves are m-sheaves)

SLIDE 18

5/19

Aufhebung

The levels of E may be partially ordered as subtoposes. That is, m is above l if and only if l∗ factors through m∗ (I.e. l-sheaves are m-sheaves) A level m is way above level l if both subcategories l!, l∗ : El → E factor through m∗ : Em → E. Em

m∗ E

El

l!
l∗
(I.e. both l-sheaves and l-skeletal objects are m-sheaves.)

SLIDE 19

5/19

Aufhebung

The levels of E may be partially ordered as subtoposes. That is, m is above l if and only if l∗ factors through m∗ (I.e. l-sheaves are m-sheaves) A level m is way above level l if both subcategories l!, l∗ : El → E factor through m∗ : Em → E. Em

m∗ E

El

l!
l∗
(I.e. both l-sheaves and l-skeletal objects are m-sheaves.)

Definition

The Aufhebung of level l is the least level of E that is way above l.

SLIDE 20

6/19

Pre-cohesive geometric morphisms and level 0

SLIDE 21

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful,

SLIDE 22

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

SLIDE 23

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition:

SLIDE 24

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples:

SLIDE 25

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set,

SLIDE 26

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set,

SLIDE 27

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations,

SLIDE 28

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations, Cubical Sets,

SLIDE 29

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations, Cubical Sets, SDG,

SLIDE 30

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations, Cubical Sets, SDG, AG, etc.

SLIDE 31

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations, Cubical Sets, SDG, AG, etc. The level p∗ ⊣ p∗ ⊣ p! : S → E will be called Level 0.

SLIDE 32

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations, Cubical Sets, SDG, AG, etc. The level p∗ ⊣ p∗ ⊣ p! : S → E will be called Level 0. Notice that level 0 has monic skeleta (monic β).

SLIDE 33

6/19

Pre-cohesive geometric morphisms and level 0

Let p : E → S be a pre-cohesive geometric morphism, so that E

p!

⊣

⊣ ⊣ p∗

⊣

S

p∗

p!
p∗, p! : S → E are fully faithful, p! preserves finite products and

the counit βX : p∗(p∗X) → X is monic. Intuition: pieces

⊣

discrete

⊣

points

⊣

codiscrete Examples: ∆ → Set, its truncations ∆n → Set, the classifier of BAs and its truncations, Cubical Sets, SDG, AG, etc. The level p∗ ⊣ p∗ ⊣ p! : S → E will be called Level 0. Notice that level 0 has monic skeleta (monic β). 0-skeletal objects will be called discrete.

SLIDE 34

7/19

Level 1

SLIDE 35

7/19

Level 1

Let p : E → S be pre-cohesive with its associated level 0.

SLIDE 36

7/19

Level 1

Let p : E → S be pre-cohesive with its associated level 0.

Definition

Level 1 (of p) is the Aufhebung of level 0. That is,

SLIDE 37

7/19

Level 1

Let p : E → S be pre-cohesive with its associated level 0.

Definition

Level 1 (of p) is the Aufhebung of level 0. That is, the least level of E that is way above level 0. That is,

SLIDE 38

7/19

Level 1

Let p : E → S be pre-cohesive with its associated level 0.

Definition

Level 1 (of p) is the Aufhebung of level 0. That is, the least level of E that is way above level 0. That is, the least level l of E such that discrete and codiscrete spaces are l-sheaves.

SLIDE 39

8/19

Lawvere’s characterization of Level 1

“

SLIDE 40

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition:

SLIDE 41

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91]

SLIDE 42

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

SLIDE 43

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

For any level l of E above level 0,

SLIDE 44

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

For any level l of E above level 0, l is way above 0 if and only if,

SLIDE 45

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

For any level l of E above level 0, l is way above 0 if and only if, for every X in E,

SLIDE 46

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

For any level l of E above level 0, l is way above 0 if and only if, for every X in E, p!(l!(l∗X)) → p!X is an iso (

SLIDE 47

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

For any level l of E above level 0, l is way above 0 if and only if, for every X in E, p!(l!(l∗X)) → p!X is an iso (where l!(l∗X) → X is the l-skeleton of X). “more pictorially:

SLIDE 48

8/19

Lawvere’s characterization of Level 1

“Because of the special feature of dimension zero of having a components functor to it [...], the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of components of the skeleton at that dimension of the space” [L’91] Let p : E → S be pre-cohesive with its associated level 0.

Theorem (Lawvere)

For any level l of E above level 0, l is way above 0 if and only if, for every X in E, p!(l!(l∗X)) → p!X is an iso (where l!(l∗X) → X is the l-skeleton of X). “more pictorially: if two points of any space can be connected by anything, then they can be connected by a curve.”

SLIDE 49

9/19

Application: 1-dense subobjects with discrete domain

SLIDE 50

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

SLIDE 51

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

Lemma

If v is 1-dense then

SLIDE 52

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

Lemma

If v is 1-dense then v is split.

Proof.

SLIDE 53

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

Lemma

If v is 1-dense then v is split.

Proof.

By L’s Thm. p!v : p!(p∗A) → p!Y is an iso.

SLIDE 54

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

Lemma

If v is 1-dense then v is split.

Proof.

By L’s Thm. p!v : p!(p∗A) → p!Y is an iso. Take the composite Y

σ

p∗(p!Y )

p∗(p!v)−1

p∗(p!(p∗A))

p∗τ p∗A

where σ and τ are the unit and counit of p! ⊣ p∗.

SLIDE 55

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

Lemma

If v is 1-dense then v is split.

Proof.

By L’s Thm. p!v : p!(p∗A) → p!Y is an iso. Take the composite Y

σ

p∗(p!Y )

p∗(p!v)−1

p∗(p!(p∗A))

p∗τ p∗A

where σ and τ are the unit and counit of p! ⊣ p∗. An object X in E is called (0-)separated if it is separated for the subtopos p∗ ⊣ p! : S → E. (I.e. a subobject of a codiscrete object.)

SLIDE 56

9/19

Application: 1-dense subobjects with discrete domain

Let A in S and v : p∗A → Y monic in E.

Lemma

If v is 1-dense then v is split.

Proof.

By L’s Thm. p!v : p!(p∗A) → p!Y is an iso. Take the composite Y

σ

p∗(p!Y )

p∗(p!v)−1

p∗(p!(p∗A))

p∗τ p∗A

where σ and τ are the unit and counit of p! ⊣ p∗. An object X in E is called (0-)separated if it is separated for the subtopos p∗ ⊣ p! : S → E. (I.e. a subobject of a codiscrete object.)

Lemma

If v is 1-dense and Y is separated then v is an iso.

SLIDE 57

10/19

Characterizations of skeletal objects

SLIDE 58

10/19

Characterizations of skeletal objects

Let m be the Aufhebung of level l.

SLIDE 59

10/19

Characterizations of skeletal objects

Let m be the Aufhebung of level l. Can we characterize m-skeletal objects in terms of l?

SLIDE 60

10/19

Characterizations of skeletal objects

Let m be the Aufhebung of level l. Can we characterize m-skeletal objects in terms of l? Case l = −∞ (so m = 0)

SLIDE 61

10/19

Characterizations of skeletal objects

Let m be the Aufhebung of level l. Can we characterize m-skeletal objects in terms of l? Case l = −∞ (so m = 0) (ct2018) If p : E → S is pre-cohesive, l.c. and S is Boolean then

SLIDE 62

10/19

Characterizations of skeletal objects

Let m be the Aufhebung of level l. Can we characterize m-skeletal objects in terms of l? Case l = −∞ (so m = 0) (ct2018) If p : E → S is pre-cohesive, l.c. and S is Boolean then X is 0-skeletal iff X is decidable.

SLIDE 63

10/19

Characterizations of skeletal objects

Let m be the Aufhebung of level l. Can we characterize m-skeletal objects in terms of l? Case l = −∞ (so m = 0) (ct2018) If p : E → S is pre-cohesive, l.c. and S is Boolean then X is 0-skeletal iff X is decidable. Case l = 0 (so m = 1) ???

SLIDE 64

11/19

Bounded depth formulas

SLIDE 65

11/19

Bounded depth formulas

Consider the bounded depth formula (BD1) x1 ∨ (x1 ⇒ (x0 ∨ ¬x0)) (Bezhanishvili, Marra, McNeill, Pedrini, Tarski’s theorem on intuitionistic logic, for polyhedra, APAL 2018)

SLIDE 66

11/19

Bounded depth formulas

Consider the bounded depth formula (BD1) x1 ∨ (x1 ⇒ (x0 ∨ ¬x0)) (Bezhanishvili, Marra, McNeill, Pedrini, Tarski’s theorem on intuitionistic logic, for polyhedra, APAL 2018) Notice that, assuming coHeyting operations one has ⊤ ≤ x1 ∨ (x1 ⇒ (x0 ∨ ¬x0)) iff ⊤/x1 ≤ x1 ⇒ (x0 ∨ ¬x0) iff (⊤/x1) ∧ x1 ≤ (x0 ∨ ¬x0) iff ∂x1 ≤ (x0 ∨ ¬x0)

SLIDE 67

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall:

SLIDE 68

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall: every X has monic 0-skeleton βX : p∗(p∗X) → X. So,

SLIDE 69

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall: every X has monic 0-skeleton βX : p∗(p∗X) → X. So, for any subobject u : U → X we may build the implication u ⇒ βX : (U ⇒ βX) → X.

SLIDE 70

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall: every X has monic 0-skeleton βX : p∗(p∗X) → X. So, for any subobject u : U → X we may build the implication u ⇒ βX : (U ⇒ βX) → X.

Definition

A subobject u : U → X has discrete boundary if ⊤X ≤ u ∨ (u ⇒ βX) as subobjects of X. (Intuition:

SLIDE 71

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall: every X has monic 0-skeleton βX : p∗(p∗X) → X. So, for any subobject u : U → X we may build the implication u ⇒ βX : (U ⇒ βX) → X.

Definition

A subobject u : U → X has discrete boundary if ⊤X ≤ u ∨ (u ⇒ βX) as subobjects of X. (Intuition: ∂u ≤ βX.)

SLIDE 72

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall: every X has monic 0-skeleton βX : p∗(p∗X) → X. So, for any subobject u : U → X we may build the implication u ⇒ βX : (U ⇒ βX) → X.

Definition

A subobject u : U → X has discrete boundary if ⊤X ≤ u ∨ (u ⇒ βX) as subobjects of X. (Intuition: ∂u ≤ βX.) p∗A = U ∩ (U ⇒ β)

U

u

(U ⇒ β)

u⇒β

X

SLIDE 73

12/19

Discrete boundaries

Let p : E → S be pre-cohesive with associated level 0. Recall: every X has monic 0-skeleton βX : p∗(p∗X) → X. So, for any subobject u : U → X we may build the implication u ⇒ βX : (U ⇒ βX) → X.

Definition

A subobject u : U → X has discrete boundary if ⊤X ≤ u ∨ (u ⇒ βX) as subobjects of X. (Intuition: ∂u ≤ βX.) p∗A = U ∩ (U ⇒ β)

U

u

(U ⇒ β)

u⇒β

X

An object X in E has discrete boundaries if every subobject of X has discrete boundary.

SLIDE 74

13/19

The case of reflexive graphs

SLIDE 75

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

SLIDE 76

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement)

SLIDE 77

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement) A graph is 1-skeletal iff it has discrete boundaries.

SLIDE 78

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement) A graph is 1-skeletal iff it has discrete boundaries.

Level 1 is the whole of ∆1.

SLIDE 79

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement) A graph is 1-skeletal iff it has discrete boundaries.

Level 1 is the whole of ∆1. Every graph has discrete boundaries.

SLIDE 80

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement) A graph is 1-skeletal iff it has discrete boundaries.

Level 1 is the whole of ∆1. Every graph has discrete boundaries. Naive question: is it true in general that 1-skeletal iff discrete boundaries ?

SLIDE 81

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement) A graph is 1-skeletal iff it has discrete boundaries.

Level 1 is the whole of ∆1. Every graph has discrete boundaries. Naive question: is it true in general that 1-skeletal iff discrete boundaries ?

No. See drawing.

SLIDE 82

13/19

The case of reflexive graphs

Consider p : ∆1 → Set.

Proposition (somewhat misleading but suggestive statement) A graph is 1-skeletal iff it has discrete boundaries.

Level 1 is the whole of ∆1. Every graph has discrete boundaries. Naive question: is it true in general that 1-skeletal iff discrete boundaries ?

No. See drawing.

Does anything survive the passage to the elementary setting?

SLIDE 83

14/19

Curves

Let p : E → S be pre-cohesive with associated level 0.

SLIDE 84

14/19

Curves

Let p : E → S be pre-cohesive with associated level 0.

Definition

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries. (Hence, curves have discrete boundaries.)

SLIDE 85

14/19

Curves

Let p : E → S be pre-cohesive with associated level 0.

Definition

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries. (Hence, curves have discrete boundaries.) Consider Level 1 of E. (The least level s.t. codiscrete and discrete objects are sheaves.)

SLIDE 86

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono.

SLIDE 87

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono. If Y is separated and has discrete boundaries then u is an isomorphism.

Proof.

SLIDE 88

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono. If Y is separated and has discrete boundaries then u is an isomorphism.

Proof.

p∗A

u′

U

u

U ⇒ β

u⇒β

Y

p.b. for some A in S so

SLIDE 89

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono. If Y is separated and has discrete boundaries then u is an isomorphism.

Proof.

p∗A

u′

U

u

U ⇒ β

u⇒β

Y

p.b. for some A in S so u′ is an 1-dense subobject of (U ⇒ β).

SLIDE 90

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono. If Y is separated and has discrete boundaries then u is an isomorphism.

Proof.

p∗A

u′

U

u

U ⇒ β

u⇒β

Y

p.b. for some A in S so u′ is an 1-dense subobject of (U ⇒ β). (U ⇒ β) is separated (it is a subobject of Y ). So

SLIDE 91

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono. If Y is separated and has discrete boundaries then u is an isomorphism.

Proof.

p∗A

u′

U

u

U ⇒ β

u⇒β

Y

p.b. for some A in S so u′ is an 1-dense subobject of (U ⇒ β). (U ⇒ β) is separated (it is a subobject of Y ). So u′ is an iso by the previous Lemma.

SLIDE 92

15/19

1-dense subobjects of sep. spaces with discrete boundaries

Lemma

Let u : U → Y be a 1-dense mono. If Y is separated and has discrete boundaries then u is an isomorphism.

Proof.

p∗A

u′

U

u

U ⇒ β

u⇒β

Y

p.b. for some A in S so u′ is an 1-dense subobject of (U ⇒ β). (U ⇒ β) is separated (it is a subobject of Y ). So u′ is an iso by the previous Lemma. As Y has discrete boundaries (square is a p.o.) u is an iso.

SLIDE 93

16/19

The elementary result: ‘curves are 1-dimensional’

SLIDE 94

16/19

The elementary result: ‘curves are 1-dimensional’

Definition (Recall)

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries.

SLIDE 95

16/19

The elementary result: ‘curves are 1-dimensional’

Definition (Recall)

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries.

Proposition

If level 1 of E has monic skeleta then curves 1-skeletal.

Proof.

SLIDE 96

16/19

The elementary result: ‘curves are 1-dimensional’

Definition (Recall)

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries.

Proposition

If level 1 of E has monic skeleta then curves 1-skeletal.

Proof.

The previous Lemma implies that:

SLIDE 97

16/19

The elementary result: ‘curves are 1-dimensional’

Definition (Recall)

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries.

Proposition

If level 1 of E has monic skeleta then curves 1-skeletal.

Proof.

The previous Lemma implies that: if Y is separated and has discrete boundaries then the 1-skeleton of Y is epic.

SLIDE 98

16/19

The elementary result: ‘curves are 1-dimensional’

Definition (Recall)

An object X in E is a curve if there is an epic Y → X such that Y is separated and has discrete boundaries.

Proposition

If level 1 of E has monic skeleta then curves 1-skeletal.

Proof.

The previous Lemma implies that: if Y is separated and has discrete boundaries then the 1-skeleton of Y is epic.

‘Proposition’

In the examples, an object is 1-skeletal if and only if it is a curve.

SLIDE 99

17/19

The case of presheaf toposes

Let C be a small category with terminal object and such that every

bject has a point so that p :

C → Set is pre-cohesive.

Lemma

For any object C in C, the following are equivalent:

1. The representable C( , C) in

C has discrete boundaries.

SLIDE 100

17/19

The case of presheaf toposes

Let C be a small category with terminal object and such that every

bject has a point so that p :

C → Set is pre-cohesive.

Lemma

For any object C in C, the following are equivalent:

1. The representable C( , C) in

C has discrete boundaries.

2. For every f : B → C in C, f is constant or f has a section.

Such objects of C will be called edge types.

SLIDE 101

17/19

The case of presheaf toposes

Let C be a small category with terminal object and such that every

bject has a point so that p :

C → Set is pre-cohesive.

Lemma

For any object C in C, the following are equivalent:

1. The representable C( , C) in

C has discrete boundaries.

2. For every f : B → C in C, f is constant or f has a section.

Such objects of C will be called edge types. Let Ce → C be the full subcategory of edge-types.

Proposition

SLIDE 102

17/19

The case of presheaf toposes

Let C be a small category with terminal object and such that every

bject has a point so that p :

C → Set is pre-cohesive.

Lemma

For any object C in C, the following are equivalent:

1. The representable C( , C) in

C has discrete boundaries.

2. For every f : B → C in C, f is constant or f has a section.

Such objects of C will be called edge types. Let Ce → C be the full subcategory of edge-types.

Proposition

If points in C separate maps with edge type codomain and every

bject is edge-wise connected then

Ce → C is level 1.

SLIDE 103

17/19

The case of presheaf toposes

Let C be a small category with terminal object and such that every

bject has a point so that p :

C → Set is pre-cohesive.

Lemma

For any object C in C, the following are equivalent:

1. The representable C( , C) in

C has discrete boundaries.

2. For every f : B → C in C, f is constant or f has a section.

Such objects of C will be called edge types. Let Ce → C be the full subcategory of edge-types.

Proposition

If points in C separate maps with edge type codomain and every

bject is edge-wise connected then

Ce → C is level 1. If, moreover, this level has monic skeleta then 1-skeletal objects coincide with curves.

SLIDE 104

18/19

Bibliography I

F. W. Lawvere.

Some thoughts on the future of category theory. SLNM 1488, 1991.

N. Bezhanishvili, V. Marra, D. McNeill, and A. Pedrini.

Tarskis theorem on intuitionistic logic, for polyhedra.

Ann. Pure Appl. Logic, 2018.
F. W. Lawvere.

Linearization of graphic toposes via Coxeter groups. JPAA, 2002.

G. M. Kelly and F. W. Lawvere.

On the complete lattice of essential localizations.

Bull. Soc. Math. Belg. Ser. A, 1989.

SLIDE 105

19/19

Bibliography II

C. Kennett, E. Riehl, M. Roy, and M. Zaks.

Levels in the toposes of simplicial sets and cubical sets.

J. Pure Appl. Algebra, 2011.
M. Menni