Spahn internal formulation of cohesion (Rev #9, changes)

Showing changes from revision #8 to #9: Added | Removed | Changed

For generalities on the translation Ho CT-Ho TT see dictionary HoCT-HoTT.

For generalities on the formulation of Ho TT in Coq see HoTT-Coq - references.

For generalities on cohesive toposes see Urs Schreiber, differential cohomology in a cohesive topos

For a blog comment on the internal formulation of cohesion, see Mike Shulman

• internalizing the external - the joy of codiscreteness, ncafé

• discreteness, concreteness, fibrations, and scones, ncafé

Notation

We assume that

is an adjoint quadruple, where we assume that $\Pi$ preserves finite products and $Disc$ and $co Disc$ are full and faithful.

This adjoint quadruple gives rise to an adjoint triple

The Ho CT-axioms of cohesion

The last statement follows from the $(\infty,1)$-category analog of the discussion here.

Coq

This section is on the Coq formalization of axiomatic homotopy cohesion. It draws from Mike Shulman, Internalizing the external, or The Joys of Codiscreteness (blog post). The corresponding Coq-code library is at Mike Shulman, HoTT/Coq/Subcatgeories

Subtopos

Reflective subfibration

This section comments on the code ReflectiveSubfibration.v.

It codes the axiom that

exhibits the codiscrete objects as reflective subcategory of $H$.

For plain types

The following axioms characterize in type theoretical language the relations between objects of a reflective subcategory by means of the endofunctor

1. $A:Type\vdash in Rsc(A):Prop.$ This says that for any type $A$, we have a proposition $in Rsc(A)$ (expressing the assertion that $A$ is in our putative reflective subcategory).

2. $A:Type\dashv \sharp A:Type.$ This says that for any $A$, we have another type $\sharp A$ (its reflection into the subcategory).

3. $A:Type\dashv in Rsc(\sharp A).$ This says that for any $A$, the type $\sharp A$ is in the subcategory.

4. $A:Type\dashv \eta_A:A\to \sharp A$. This says that for any $A$, we have a map $\eta_A:A\to \sharp A$ (which we can think of as the unit of the adjunction).

5. $A,B:Type,inRsc(B)\dashv is Equiv(\lambda g.g\circ\eta_A)$. This says that for any types A and B, where B is in the subcategory, precomposing with $\eta_A$ yields an equivalence between $B^{\sharp A}$ and $B^A$ (which we can think of as the adjunction isomorphism since precomposition $g\mapsto g\circ \eta_A$ with the adjunction-unit assigns the adjunct).

For dependent types

The previous axioms are given for plain types but one can impose them also on dependent types $x:A\dashv P(x):Type$ in that we require

In category theory this means that

is a pullback and the right arrow is a fibration. The operation $\sharp$ commutes with pullbacks and gives an operation on every overcategory $H/A$ . There are axioms expressing the requirement that $\sharp$ is a reflection.

Stable factorization system

github

Lex-reflective subcategory

Theorem

Let $C$ be a ccc. Let $I\hookrightarrow C$ be a reflective subcategory.

Then $I$ is an exponential ideal iff the reflector preserves finite products.

github

Lex-reflective subcategory of codiscrete objects

github

Local topos

$H$ being local means that we have a geometric morphism with extra right adjoint

github

Cohesive topos

(Discrete objects are reflectively- and coreflectively embedded.)

github

References

• Mike Shulman, Discreteness, Concreteness, Fibrations, and Scones, blog

• Mike Shulman, Internalizing the External, or The Joys of Codiscreteness, blog

• Mike Shulman, Reflective Subfibrations, Factorization Systems, and Stable Units, blog