Spahn internal formulation of cohesion (Rev #4, changes)

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

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

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.

For plain types

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

and objects of the ambient category $H$ wrt. the reflector $\sharp\dashv \iota$:

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

$$\array{ \sharp P(x)&\rightarrow &\sharp P \\ \downarrow&&\downarrow \\ *&\xrightarrow{x}&A }$$

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

github

Lex-reflective subcategory of codiscrete objects

github

Local topos

github

Cohesive topos

github