Contents

category theory

# Contents

## Definition

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

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

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

Here morphisms on top are left adjoint to morphisms below, hence

$conc \dashv \iota_{conc}$

exhibits the concrete objects as a reflective subcategory, the reflector $conc$ being “concretification”.

###### 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 $\eta^\sharp$:

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

hence to the object which exhibits the epi/mono-factorization of $\eta^\sharp_X$

(1)$\eta^\sharp_X \;\colon\; X \underoverset{epi}{ \eta^{conc}_X }{\longrightarrow} conc X \underoverset{mono}{}{\longrightarrow} \sharp X \,.$

First we need to show that $conc X$, thus defined, is indeed concrete, hence that $\eta^\sharp_{im(\eta^\sharp_X)}$ is a monomorphism. For this, consider the following naturality square of the $\Gamma \dashv coDisc$-adjunction hom-isomorphism

(2)$\array{ {\phantom{A}} \\ Hom_{Set}( \Gamma im(\eta^\sharp_X), \Gamma im(\eta^\sharp_X) ) &\simeq& Hom_{\mathbf{H}}( im(\eta^\sharp_X), \sharp im(\eta^\sharp_X) ) \\ {}^{ \mathllap{ (-) \circ \Gamma(\eta^{conc}_X) } }\big\downarrow && \big\downarrow^{ \mathrlap{ (-) \circ \eta^{conc}_X } } \\ Hom_{Set}( \Gamma X, \Gamma im(\eta^\sharp_X) ) &\simeq& Hom_{\mathbf{H}}( X, \sharp im(\eta^\sharp_X) ) } \phantom{AAAA} \array{ {\phantom{A}} \\ \left\{ id_{\Gamma im(\eta^\sharp_X)} \right\} &\longrightarrow& \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }} \left\{ \eta^{\sharp}_{im(\eta^\sharp_X)} \right\} \\ \big\downarrow && \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }} \big\downarrow \\ \left\{ \Gamma(\eta^{conc}_X) \right\} &\longrightarrow& \left\{ \underset{ iso }{ \underbrace{ \sharp(\eta^{conc}_X) }} \circ \eta^\sharp_{ X } \;=\; \eta^{\sharp}_{ im(\eta^\sharp_X) } \circ \eta^{conc}_X \right\} }$

By chasing the identity morphism on $\Gamma im(\eta^\sharp_X)$ through this diagram, as shown by the diagram on the right, we obtain the equality displayed in the bottom right entry, where we used the general formula for adjuncts and the definition $\sharp \coloneqq coDisc \circ \Gamma$.

But observe that $\Gamma (\eta^{conc}_X)$, and hence also $\sharp(\eta^{conc}_X)$, is an isomorphism, as indicated above: Since $\Gamma$ is both a left adjoint as well as a right adjoint, it preserves both epimorphisms as well as monomorphisms, hence it preserves image factorizations. This implies that $\Gamma \eta^{conc}_X$ is the epimorphism onto the image of $\Gamma( \eta^\sharp_X )$. But by idempotency of $\sharp$, the latter is an isomorphism, and hence so is the epimorphism in its image factorization.

Therefore the equality in (2) says that

\begin{aligned} \eta^\sharp_{ X } & = \left( iso \circ \eta^{\sharp}_{ im(\eta^\sharp_X)} \right) \circ \eta^{conc}_X \\ & = mono \circ \eta^{conc}_X \,, \end{aligned}

where in the second line we remembered that $\eta^{conc}_X$ is, by definition, the epimorphism in the epi/mono-factorization of $\eta^\sharp_X$.

Now the defining property of epimorphisms allows to cancel this commmon factor on both sides, which yields

$\eta^{\sharp}_{ im(\eta^\sharp_X) } \;=\; iso \circ mono \;=\; mono.$

This shows that $conc X \coloneqq im(\eta^\sharp_X)$ is indeed concrete.

It remains to show that this construction is left adjoint to the inclusion. We claim that the adjunction unit of $(conc \dashv \iota_{conc})$ is provided by $\eta^{conc}$ (1).

To see this, first notice that, since the epi/mono-factorization is orthogonal and hence functorial, we have commuting diagrams of the form

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

Now to demonstrate the adjunction, 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 (3), 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.

### Further properties

###### Proposition

(cohesive maps to concrete objects glue)
Let $\mathbf{H}$ be a cohesive $\infty$-topos, and consider $Y \,\in\, \mathbf{H}_{\sharp_1} \xhookrightarrow{\;} \mathbf{H}$ a concrete object, in that its sharp modality unit is (-1)-truncated/monomorphic: $Y \xhookrightarrow{ \;\;\eta^\sharp_Y\;\; } \sharp Y$. Then cohesive maps to $X$ glue (satisfy the respective sheaf property):

For any $Y \,\in\, \mathbf{H}$ and any open cover, namely any (-1)-connected/effective epi-morphism $U \twoheadrightarrow X$, cohesive maps $U \xrightarrow{\;} X$ whose maps of underlying $\infty$-groupoids descend/extend to $Y$, then they also descend/extend to $Y$ as cohesive maps, in an essentially unique way, in that all solid homotopy-commutative square as follows have essentially unique dashed lifts:

The point here is the given interpretation of this lifting problem, but the proof of the latter is immediate:
###### Proof

Under the given assumptions, the essentially unique existence of the lift is an instance of the (n-connected, n-truncated) factorization system for $n = (-1)$.

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

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