The canonical intensive quality of a pre-cohesive topos Francisco - - PowerPoint PPT Presentation

The canonical intensive quality of a pre-cohesive topos

Francisco Marmolejo Instituto de Matem´ aticas Universidad Nacional Aut´

• noma de M´

exico Joint work with Mat´ ıas Menni Monday, July 17, 2017

The canonical intensive quality of a pre-cohesive topos

Francisco Marmolejo Instituto de Matem´ aticas Universidad Nacional Aut´

• noma de M´

exico Joint work with Mat´ ıas Menni Monday, July 17, 2017 and help from F.W. Lawvere

Axiomatic Cohesion

• I. Categories of space as cohesive backgrounds
• II. Cohesion versus non-cohesion; quality types
• III. Extensive quality; intensive quality in its rarefied and condensed

aspects; the canonical qualities form and substance

• IV. Non-cohesion within cohesion via constancy on infinitesimals
• V. The example of reflexive graphs and their atomic numbers
• VI. Sufficient cohesion and the Grothendieck condition
• VII. Weak generation of a subtopos by a quotient topos

Axiomatic Cohesion

• I. Categories of space as cohesive backgrounds
• II. Cohesion versus non-cohesion; quality types
• III. Extensive quality; intensive quality in its rarefied and condensed

aspects; the canonical qualities form and substance

• IV. Non-cohesion within cohesion via constancy on infinitesimals
• V. The example of reflexive graphs and their atomic numbers
• VI. Sufficient cohesion and the Grothendieck condition
• VII. Weak generation of a subtopos by a quotient topos

“I look forward to further work on each of these aspects”

Axiomatic Cohesion

E and S are toposes.

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism.

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E S

p! ⊣p∗ ⊣p∗ p! ⊣

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E S

p! ⊣p∗ ⊣p∗ p! ⊣

i) p∗ full and faithful

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E S

p! ⊣p∗ ⊣p∗ p! ⊣

i) p∗ full and faithful ii) p! preserves finite products

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E S

p! ⊣p∗ ⊣p∗ p! ⊣

i) p∗ full and faithful ii) p! preserves finite products iii) θ : p∗ → p! is epi

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E S

p! ⊣p∗ ⊣p∗ p! ⊣

i) p∗ full and faithful ii) p! preserves finite products iii) θ : p∗ → p! is epi (the Nullstellensatz)

Axiomatic Cohesion

E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E S

p! ⊣p∗ ⊣p∗ p! ⊣

i) p∗ full and faithful ii) p! preserves finite products iii) θ : p∗ → p! is epi (the Nullstellensatz) Continuity Axiom: iv) p!(E p∗S) → (p!E)S iso.

Quality type

E S

p∗ ⊣p∗

p : E → S is a quality type if

Quality type

E S

p∗ ⊣p∗

p : E → S is a quality type if p∗ is full and faithful,

Quality type

E S

p! ⊣p∗ ⊣p∗

p : E → S is a quality type if p∗ is full and faithful, p! ⊣ p∗ exists

Quality type

E S

p! ⊣p∗ ⊣p∗

p : E → S is a quality type if p∗ is full and faithful, p! ⊣ p∗ exists and θ : p∗ → p! is iso.

Quality type

E S

p! ⊣p∗ ⊣p∗

p : E → S is a quality type if p∗ is full and faithful, p! ⊣ p∗ exists and θ : p∗ → p! is iso. “A quality type is a category of cohesion in one extreme sense”

Canonical Quality Type

L the full subcategory of E of those objects X for which θX : p∗ → p! is iso

Canonical Quality Type

L the full subcategory of E of those objects X for which θX : p∗ → p! is iso L E S

s∗ p! ⊣p∗ ⊣p∗ p! ⊣

Canonical Quality Type

Reflexive Graphs Sets

Canonical Quality Type

p! connected components Reflexive Graphs Sets

p!

Canonical Quality Type

p! connected components p∗ discrete Reflexive Graphs Sets

p! ⊣p∗

Canonical Quality Type

p! connected components p∗ discrete p∗ points Reflexive Graphs Sets

p! ⊣p∗ ⊣p∗

Canonical Quality Type

p! connected components p∗ discrete p∗ points p! codiscrete Reflexive Graphs Sets

p! ⊣p∗ ⊣p∗ p! ⊣

Canonical Quality Type

p! connected components p∗ discrete p∗ points p! codiscrete L Reflexive Graphs Sets

s∗ p! ⊣p∗ ⊣p∗ p! ⊣

Canonical Quality Type

L E S

s∗ p! ⊣p∗ ⊣p∗ p! ⊣

Canonical Quality Type

L E S

s∗ p! ⊣p∗ ⊣p∗ p! ⊣ q∗

Canonical Quality Type

E L E S

s∗ s∗ p! ⊣p∗ ⊣p∗ p! ⊣ q∗

Canonical Quality Type

E L E S

p! s∗ s∗ p! ⊣p∗ ⊣p∗ p! ⊣ q∗

Canonical Quality Type

E E L E S

p! s∗ s∗ s∗ p! ⊣p∗ ⊣p∗ p! ⊣ q∗

Canonical Quality Type

E E L E S

p∗ p! s∗ s∗ s∗ p! ⊣p∗ ⊣p∗ p! ⊣ q∗

Theorem from Axiomatic Cohesion

Theorem Any category of cohesion satisfying reasonable completeness conditions has a canonical intensive quality s whose codomain is the subcategory s∗ : L → E consisting of those X for which the map θX : p∗X → p!X is an isomorphism. Moreover, s∗ has a left adjoint s! and a coproduct-preserving right adjoint s∗.

Theorem from Axiomatic Cohesion

Theorem Any category of cohesion satisfying reasonable completeness conditions has a canonical intensive quality s whose codomain is the subcategory s∗ : L → E consisting of those X for which the map θX : p∗X → p!X is an isomorphism. Moreover, s∗ has a left adjoint s! and a coproduct-preserving right adjoint s∗. Thus L is a topos. (Algebras for a left exact comonad.)

Reflexive Graphs Again

Reflexive Graphs

Reflexive Graphs Again

L Reflexive Graphs s∗

Reflexive Graphs Again

L Reflexive Graphs

⊥

s∗ s!

Reflexive Graphs Again

L Reflexive Graphs

⊥ ⊥

s∗ s! s∗

Reflexive Graphs Again

L Reflexive Graphs

⊥ ⊥

super-cooling s∗ s! s∗

Reflexive Graphs Again

L Reflexive Graphs

⊥ ⊥

super-cooling super-heating s∗ s! s∗

The Actual Construction of the Adjoints

s! : E → L. p∗p∗X p∗p!X X s∗s!X p∗θX βX a pushout.

The Actual Construction of the Adjoints

For the right adjoint s∗ : E → L we need φ : p∗ → p!. s∗s∗X X p∗p∗X p!p∗X ηX φp∗X a pullback.

Theorem Let p : E → S be an essential and local geometric morphism between toposes such that the Nullstellensatz holds. Then then the inclusion s∗ : L → E of Leibniz objects has a right adjoint. It follows that L is a topos and p induces an hyperconnected essential geometric morphism s : E → L.

Basically consequence of

Basically consequence of Lemma If p : E → S satisfies the Nullstellensatz, then the image of s∗ : L → E is closed under subobjects.

Basically consequence of Lemma If p : E → S satisfies the Nullstellensatz, then the image of s∗ : L → E is closed under subobjects. As a consequence s∗(ΩE) = ΩL.

• Proof. L ∈ L, m : X
• L in E.

p∗X p!X p∗L p!L

θX

• p∗m
• p!m
• θL

≃

p : E → S essential and local. The Nullstellensatz holds.

p : E → S essential and local. The Nullstellensatz holds.

Lemma If X ∈ E is separated for the topology induced by p∗ ⊣ p!, then s∗s∗X is discrete.

p : E → S essential and local. The Nullstellensatz holds.

Lemma If X ∈ E is separated for the topology induced by p∗ ⊣ p!, then s∗s∗X is discrete. Lemma X ∈ E is Leibniz if and only if βX : p∗p∗X → X has a retraction.

p : E → S essential and local. The Nullstellensatz holds.

Lemma If X ∈ E is separated for the topology induced by p∗ ⊣ p!, then s∗s∗X is discrete. Lemma X ∈ E is Leibniz if and only if βX : p∗p∗X → X has a retraction. Lemma Let Ω be the subobject classifier of E. Then s∗s∗Ω is discrete if and only if p : E → S is an equivalence.

Proposition Boolean objects in E are discrete. Thus, E Boolean implies that p : E → S is an equivalence.

Proposition Boolean objects in E are discrete. Thus, E Boolean implies that p : E → S is an equivalence. Proposition L Boolean implies that p : E → S is an equivalence.

Pre-cohesive presheaf topos

C a small category whose idempotents split.

Pre-cohesive presheaf topos

C a small category whose idempotents split. Proposition p : ConCop → Con is precohesive if and only if C has a terminal

• bject and every object has a point.

Pre-cohesive presheaf topos

C a small category whose idempotents split. Proposition p : ConCop → Con is precohesive if and only if C has a terminal

• bject and every object has a point.

Lemma For any X in ConCop, the counit s∗(s∗X) → X is (s∗(s∗X))C = {x ∈ QC | for all a, b : 1 → C, x · a = x · b} for every C ∈ C.

C with terminal and every object has a point

C with terminal and every object has a point

The lemma gives no information as to the nature of L for p : ConCop → Con.

C with terminal and every object has a point

The lemma gives no information as to the nature of L for p : ConCop → Con. Proposition L is a presheaf topos.

C with terminal and every object has a point

The lemma gives no information as to the nature of L for p : ConCop → Con. Proposition L is a presheaf topos. s : ConCop → L is essentially the geometric morphism r : ConCop → Con(C/≡)op

C with terminal and every object has a point

The lemma gives no information as to the nature of L for p : ConCop → Con. Proposition L is a presheaf topos. s : ConCop → L is essentially the geometric morphism r : ConCop → Con(C/≡)op induced by r : C → C/≡.

C with terminal and every object has a point

The lemma gives no information as to the nature of L for p : ConCop → Con. Proposition L is a presheaf topos. s : ConCop → L is essentially the geometric morphism r : ConCop → Con(C/≡)op induced by r : C → C/≡. C D

f g

C with terminal and every object has a point

The lemma gives no information as to the nature of L for p : ConCop → Con. Proposition L is a presheaf topos. s : ConCop → L is essentially the geometric morphism r : ConCop → Con(C/≡)op induced by r : C → C/≡. C D

f g

f ≡ g if f = g or both f and g are constant C D 1

f g p q

s : ConCop → L is in general not local. s! : L → ConCop does not in general preserve finite products.

Sites

Theorem A bounded essential connected geometric morphism p : E → Con satisfies the Nullstellensatz iff E has a connected and locally connected site of definition (C, J) such that every object of C has a point.

Sites

Theorem A bounded essential connected geometric morphism p : E → Con satisfies the Nullstellensatz iff E has a connected and locally connected site of definition (C, J) such that every object of C has a point. The site (C, J) is locally connected if each J-covering sieve on C is connected as a full subcategory of C/C. If furthermore C has a terminal object, then we say that (C, J) is connected and locally connected.

(C, J) connected and locally connected

Theorem Let C/≡ be the category that results from identifying all the points, and let r : C → C/≡ be the quotient functor. If r+J is the largest topology on C/≡ such that r reflects covers, then L(Sh(C, J)) ≃ Sh(C/≡, r+J).

(C, J) connected and locally connected

Theorem Let C/≡ be the category that results from identifying all the points, and let r : C → C/≡ be the quotient functor. If r+J is the largest topology on C/≡ such that r reflects covers, then L(Sh(C, J)) ≃ Sh(C/≡, r+J). A sieve S on C in the category C/≡ is in (r+J)C if and only if the sieve {g : domg → C in C|r(g) ∈ S} is in JC.

Even if (C, J) is subcanonical, (C/≡, r+C) is not subcanonical in general.

Even if (C, J) is subcanonical, (C/≡, r+C) is not subcanonical in general. One can use Giraud’s theorem to produce a subcanonical site from Sh(C/≡, r+J).

Even if (C, J) is subcanonical, (C/≡, r+C) is not subcanonical in general. One can use Giraud’s theorem to produce a subcanonical site from Sh(C/≡, r+J). If, furthermore, one assumes that every representable is separable, then one application of ( )+ construction suffices.

Closed intervals and piecewise linear functions

The category C.

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R.

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R. Morphisms: Continuous functions f : [a, b] → [c, d],

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R. Morphisms: Continuous functions f : [a, b] → [c, d], such that f is piecewise linear.

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R. Morphisms: Continuous functions f : [a, b] → [c, d], such that f is piecewise linear. The topology is given by a basis K:

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R. Morphisms: Continuous functions f : [a, b] → [c, d], such that f is piecewise linear. The topology is given by a basis K: for a = b, only the total sieve covers.

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R. Morphisms: Continuous functions f : [a, b] → [c, d], such that f is piecewise linear. The topology is given by a basis K: for a = b, only the total sieve covers. for a < b, the covering families are of the form {[ri, ri+1]

[a, b]|a = r0 < · · · < rk = b is a partition of [a, b]}

Closed intervals and piecewise linear functions

The category C. Objects: [a, b] with a ≤ b, a, b ∈ R. Morphisms: Continuous functions f : [a, b] → [c, d], such that f is piecewise linear. The topology is given by a basis K: for a = b, only the total sieve covers. for a < b, the covering families are of the form {[ri, ri+1]

[a, b]|a = r0 < · · · < rk = b is a partition of [a, b]}

p : Sh(C, K) → Con is cohesive.

The category D.

The category D. Objects: open intervals (a, b), with a ≤ b ∈ R.

The category D. Objects: open intervals (a, b), with a ≤ b ∈ R. Morphisms: f : (a, b) → (c, d) clases of equivalence

The category D. Objects: open intervals (a, b), with a ≤ b ∈ R. Morphisms: f : (a, b) → (c, d) clases of equivalence − − | | c d a b ( )

J1

( )

J2

( )

J3

( )

J4

( )

J5

( )

J6

There is an obvious functor F : C → D Sh(D, F+K) = L(Sh(C, K)) and it is subcanonical.

Since the sites are subcanonical, F! : Sh(C, K) → Sh(D, F+K) preserves representables.

Since the sites are subcanonical, F! : Sh(C, K) → Sh(D, F+K) preserves representables. F! preserves colimits.

Since the sites are subcanonical, F! : Sh(C, K) → Sh(D, F+K) preserves representables. F! preserves colimits. So ConCop ConDop Sh(C, J) Sh(D, F+J)

F!

• F!
• commutes.