category theory

# Contents

## Definition

For $\Gamma : \mathcal{E} \to \mathcal{B}$ a functor we say that it has codiscrete objects if it has a full and faithful right adjoint $coDisc : \mathcal{B} \hookrightarrow \mathcal{E}$.

An object in the essential image of $coDisc$ is called a codiscrete object.

This is for instance the case for the global section geometric morphism of a local topos $(Disc \dashv \Gamma \dashv coDisc) \mathcal{E} \to \mathcal{B}$.

If one thinks of $\mathcal{E}$ as a category of spaces, then the codiscrete objects are called codiscrete spaces.

The dual notion is that of discrete objects.

## Properties

$\Gamma$ is a faithful functor on morphisms whose codomain is concrete.

## Properties

###### Proposition

If $\mathcal{E}$ has a terminal object that is preserved by $\Gamma$, then $\mathcal{E}$ has concrete objects.

This is (Shulman, theorem 1).

###### Proposition

If $\mathcal{E}$ has codiscrete objects and has pullbacks that are preserved by $\Gamma$ and , then $\Gamma$ is a Grothendieck fibration.

This is (Shulman, theorem 2).

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$