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 $coDisc : \mathcal{B} \hookrightarrow \mathcal{E}$.

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

In this situation, we say that a concrete object $X \in \mathcal{E}$ is one for which the $(\Gamma \dashv coDisc)$-unit of an adjunction is a monomorphism.

If $\mathcal{E}$ is a sheaf topos, this is called a concrete sheaf.

If $\mathcal{E}$ is a cohesive (∞,1)-topos then this is called a concrete (∞,1)-sheaf or the like.

The dual notion is that of a co-concrete object.

Properties

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

Concretification factorization

Proposition

For $\mathbf{H}$ a local topos, write

$\mathbf{H}_{conc} \overset{ \phantom{AAAA} }{\hookrightarrow} \mathbf{H}$

for its full subcategory of concrete objects.

Then there is a sequence of reflective subcategory-inclusions that factor the $(\Gamma \dashv coDisc)$-adjunction as

$\Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set$
Proof

For the adjunction on the right, we just need to observe that for every set $S \in Set$, the codiscrete object $coDisc(S)$ is concrete, which is immediate by idempotency of $\sharp$ and the fact that every isomorphism is also a monomorphism.

For the adjunction on the left we claim that the left adjoint $conc$ (concretification), is given by sending each object to the image of its $(\Gamma \dashv coDisc)$ adjunction unit $\eta^\sharp$:

$conc \;\colon\; X \mapsto im(\eta^\sharp_X) \,.$

The adjunction unit of $(conc \dashv \iota_{conc})$ is provided by the epimorphism in the epi/mono-factorization, which, being orthogonal, is functorial:

(1)$\array{ X_1 &\underoverset{epi}{\eta^{conc}_{X_1}}{\longrightarrow}& im(\eta^\sharp_{X_1}) &\underset{mono}{\longrightarrow}& \sharp X_1 \\ \big\downarrow && \big\downarrow && \big\downarrow \\ X_2 &\underoverset{epi}{\eta^{conc}_{X_2}}{\longrightarrow}& im(\eta^\sharp_{X_2}) &\underset{mono}{\longrightarrow}& \sharp X_2 }$

To see this, it is sufficient, to show that $\eta^{conc}$ is a universal morphism in the sense discussed at adjoint functors. Hence consider any morphism $f \;\colon\; X_1 \to X_2$ with $X_2 \in \mathbf{H}_{conc} \hookrightarrow \mathbf{H}$. Then we need to show that there is a unique diagonal morphism as below, that makes the following top left triangle commute:

$\array{ X_1 &\overset{\phantom{AA} f \phantom{AA}}{\longrightarrow}& X_2 \\ {}^{\mathllap{epi}}\big\downarrow^{\mathrlap{\eta^{conc}_{X_1}}} &{}^{\mathllap{\exists !}}\nearrow& \big\downarrow^{\mathrlap{mono}} \\ im(\eta^\sharp_{X_1}) &\underset{}{\longrightarrow}& \sharp X_2 }$

Now, from (1), we have a commuting square as shown. Here the left morphism is an epimorphism by construction, while the right morphism is a monomorphism by assumption on $X_2$. With this, the epi/mono-factorization says that there is a diagonal lift which makes both triangles commute.

It remains to see that the lift is unique with just the property of making the top left triangle commute. But this is equivalently the statement that the left morphism is an epimorphism.

cohesion

tangent cohesion

differential cohesion

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