nLab
concrete object

Contents

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

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

concι conc conc \dashv \iota_{conc}

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

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)

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

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

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

(2)Hom Set(Γim(η X ),Γim(η X )) Hom H(im(η X ),im(η X )) ()Γ(η X conc) ()η X conc Hom Set(ΓX,Γim(η X )) Hom H(X,im(η X ))AAAA{id Γim(η X )} (η X conc)η X {η im(η X ) } (η X conc)η X {Γ(η X conc)} {(η X conc)isoη X =η im(η X ) η X conc} \array{ 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{ \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 Γim(η X )\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 coDiscΓ\sharp \coloneqq coDisc \circ \Gamma.

But observe that Γ(η X conc)\Gamma (\eta^{conc}_X), and hence also (η X conc)\sharp(\eta^{conc}_X), is an isomorphism, as indicated above: Since Γ\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 Γη X conc\Gamma \eta^{conc}_X is the epimorphism onto the image of Γ(η X )\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

η X =(isoη im(η X ) )η X conc =monoη X conc, \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 η X conc\eta^{conc}_X is, by definition, the epimorphism in the epi/mono-factorization of η X \eta^\sharp_X.

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

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

This shows that concXim(η X )conc X \coloneqq im(\eta^\sharp_X) is indeed concret.

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

Now to demonstrate the adjunction, 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 (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 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

See also

Last revised on June 23, 2018 at 02:14:27. See the history of this page for a list of all contributions to it.