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´ onoma 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´ onoma 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 p ! p ! ⊣ p ∗ ⊣ p ∗ ⊣ S

Axiomatic Cohesion E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E i) p ∗ full and faithful p ! p ! ⊣ p ∗ ⊣ p ∗ ⊣ S

Axiomatic Cohesion E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E i) p ∗ full and faithful ii) p ! preserves finite products p ! p ! ⊣ p ∗ ⊣ p ∗ ⊣ S

Axiomatic Cohesion E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E i) p ∗ full and faithful ii) p ! preserves finite products p ! iii) θ : p ∗ → p ! is epi p ! ⊣ p ∗ ⊣ p ∗ ⊣ S

Axiomatic Cohesion E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E i) p ∗ full and faithful ii) p ! preserves finite products p ! iii) θ : p ∗ → p ! is epi p ! ⊣ p ∗ ⊣ p ∗ ⊣ (the Nullstellensatz) S

Axiomatic Cohesion E and S are toposes. p : E → S geometric morphism. E is pre-cohesive over S if E i) p ∗ full and faithful ii) p ! preserves finite products p ! iii) θ : p ∗ → p ! is epi p ! ⊣ p ∗ ⊣ p ∗ ⊣ (the Nullstellensatz) S Continuity Axiom: iv) p ! ( E p ∗ S ) → ( p ! E ) S iso.

Quality type E p : E → S is a quality type if p ∗ ⊣ p ∗ S

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

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

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

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

Canonical Quality Type L the full subcategory of E of those objects X for which θ X : p ∗ → p ! is iso s ∗ L E p ! p ! ⊣ p ∗ ⊣ p ∗ ⊣ S

Canonical Quality Type Reflexive Graphs Sets

Canonical Quality Type p ! connected components Reflexive Graphs p ! Sets

Canonical Quality Type p ∗ p ! connected discrete components Reflexive Graphs p ! ⊣ p ∗ Sets

Canonical Quality Type p ∗ p ! p ∗ connected discrete points components Reflexive Graphs p ! ⊣ p ∗ ⊣ p ∗ Sets

Canonical Quality Type p ! p ∗ p ! p ∗ connected discrete points codiscrete components Reflexive Graphs p ! p ! ⊣ p ∗ ⊣ p ∗ ⊣ Sets

Canonical Quality Type p ! p ∗ p ! p ∗ connected discrete points codiscrete components s ∗ Reflexive Graphs L p ! p ! ⊣ p ∗ ⊣ p ∗ ⊣ Sets

Canonical Quality Type s ∗ L E p ! ⊣ p ∗ ⊣ p ∗ p ! ⊣ S

Canonical Quality Type s ∗ L E p ! ⊣ p ∗ ⊣ p ∗ q ∗ p ! ⊣ S

Canonical Quality Type s ∗ L E p ! s ∗ ⊣ p ∗ ⊣ p ∗ q ∗ p ! ⊣ E S

Canonical Quality Type s ∗ L E p ! s ∗ ⊣ p ∗ ⊣ p ∗ q ∗ p ! ⊣ E p ! S

Canonical Quality Type s ∗ L E s ∗ p ! s ∗ E ⊣ p ∗ ⊣ p ∗ q ∗ p ! ⊣ E p ! S

Canonical Quality Type s ∗ L E s ∗ p ! s ∗ E ⊣ p ∗ ⊣ p ∗ p ∗ q ∗ p ! ⊣ E p ! 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 θ X : 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 θ X : 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 s ∗ Reflexive Graphs L

Reflexive Graphs Again s ! s ∗ ⊥ Reflexive Graphs L

Reflexive Graphs Again s ! s ∗ ⊥ Reflexive Graphs L ⊥ s ∗

Reflexive Graphs Again super-cooling s ! s ∗ ⊥ Reflexive Graphs L ⊥ s ∗

Reflexive Graphs Again super-cooling s ! s ∗ ⊥ Reflexive Graphs L ⊥ s ∗ super-heating

The Actual Construction of the Adjoints s ! : E → L . p ∗ θ X p ∗ p ∗ X p ∗ p ! X β X s ∗ s ! X X a pushout.

The Actual Construction of the Adjoints For the right adjoint s ∗ : E → L we need φ : p ∗ → p ! . s ∗ s ∗ X X η X p ∗ p ∗ X p ! p ∗ X φ p ∗ 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 ∗ (Ω E ) = Ω L .

� � � � � � � L in E . Proof. L ∈ L , m : X θ X p ∗ X p ! X p ∗ m p ! m ≃ p ∗ L p ! L θ L

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 β X : 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 β X : 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 : Con C op → Con is precohesive if and only if C has a terminal object and every object has a point.