nLab
concrete object

Contents

Definition

For Γ:\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:coDisc : \mathcal{B} \hookrightarrow \mathcal{E}.

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

In this situation, we say that a concrete object XX \in \mathcal{E} is one for which the (ΓcoDisc)(\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

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

Concretification factorization

Proposition

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

H concAAAAH \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 (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction as

ΓcoDisc:HAAconcAA AAι concAAH concAAA AAASet \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 SSetS \in Set, the codiscrete object coDisc(S)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 concconc (concretification), is given by sending each object to the image of its (ΓcoDisc)(\Gamma \dashv coDisc) adjunction unit η \eta^\sharp:

conc:Xim(η X ). conc \;\colon\; X \mapsto im(\eta^\sharp_X) \,.

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

(1)X 1 epiη X 1 conc im(η X 1 ) mono X 1 X 2 epiη X 2 conc im(η X 2 ) mono X 2 \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 η conc\eta^{conc} is a universal morphism in the sense discussed at adjoint functors. Hence consider any morphism f:X 1X 2f \;\colon\; X_1 \to X_2 with X 2H concHX_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:

X 1 AAfAA X 2 epi η X 1 conc ! mono im(η X 1 ) X 2 \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 2X_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

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }

References

Last revised on June 16, 2018 at 14:42:47. See the history of this page for a list of all contributions to it.