concrete sheaf



Discrete and concrete objects

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In higher category theory




When regarding a sheaf as a space defined by how it is probed by test spaces, a concrete sheaf is a generalized space that has (at least) an underlying set of points out of which it is built.

So a concrete sheaf models a space that is given by a set of points and a choice of which morphisms of sets from concrete test spaces into it count as “structure preserving” (e.g. count as smooth, when the sheaf models a smooth space).

More in the intrinsic language of sheaves, a concrete sheaf is a sheaf on a concrete site CC that, while perhaps not representable, is “quasi-representable” in that it is a subobject of a sheaf of the form

UHom Set(|U|,S) U \mapsto Hom_{Set}(|U|, S)

where SS is a set and |U||U| is the set |U|Hom C(*,U)|U| \coloneqq Hom_C({*}, U) of points underlying the object UU in the concrete site CC.

For the category of concrete sheaves Conc(Sh(C))Conc(Sh(C)), the global sections functor

Hom Conc(Sh(C))(1,):Conc(Sh(C))SetHom_{Conc(Sh(C))}(1,-):Conc(Sh(C))\to Set

is faithful, i.e. Conc(Sh(C))Conc(Sh(C)) is a concrete category.


We discuss two definitions: the first one is more elementary and describes concrete sheaves explicitly in terms of properties of the underlying site.

The second one is more abstract and more general, and describes them entirely topos theoretically.

On a concrete site


A concrete site is a site CC with a terminal object ** such that

  1. the functor Hom C(*,):CSetHom_C(*,-) : C \to Set is a faithful functor;

  2. for every covering family {f i:U iU}\{f_i : U_i \to U\} in CC the morphism

    iHom C(*,f i): iHom C(*,U i)Hom C(*,U) \coprod_i Hom_C(*,f_i) : \coprod_i Hom_C(*, U_i) \to Hom_C(*, U)

    is surjective.

For XPSh(C)X \in PSh(C) any presheaf, write

X˜ U:X(U)Hom Set(Hom C(*,U),X(*)) \tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))

for the adjunct of the restriction map

X(U)×Hom C(*,U)X(*), X(U) \times Hom_C(*,U) \to X(*) \,,

which in turn is the adjunct of the component map of the functor

X *,U.Hom C(*,U)Hom Set(X(U),X(*)). X_{*,U} . Hom_C(*,U) \to Hom_{Set}(X(U), X(*)) \,.

A presheaf X:C opSetX : C^{op} \to Set on a concrete site is a concrete presheaf if for each UCU \in C the map X˜ U:X(U)Hom Set(Hom C(*,U),X(*))\tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*)) is injective.

A concrete sheaf is a presheaf that is both concrete and a sheaf.

So a concrete presheaf XX is a subobject of the presheaf UHom Set(Hom C(*,U),X(*))U \mapsto Hom_{Set}(Hom_C(*,U), X(*)).

Write Conc(Sh(C))Sh(C)Conc(Sh(C)) \hookrightarrow Sh(C) for the full subcategory of the category of sheaves on concrete sheaves.

In a local topos

A more abstract perspective on the previous definition is obtained by noticing the following.


Taking U=*U=* in the second condition defining a concrete site implies that any covering family of ** contains a split epimorphism, or equivalently that the only covering sieve of ** is the maximal sieve consisting of all morphisms with target **. This means that a concrete site is in particular a local site, which implies that its topos of sheaves is a local topos.

In fact, we can formulate the definition of concrete sheaf inside any local topos EE over any base topos SS:



(DiscΓCodisc):ECodiscΓDiscS(Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} S

be a local geometric morphism. Since then by definition SCodiscΓES \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} E is a subtopos the morphisms V=Γ 1(isos(S))MorEV = \Gamma^{-1}(isos(S)) \subset Mor E that are inverted by Γ\Gamma are the local isomorphisms with respect to which the objects of SS are sheaves/VV-local objects in EE.

The concrete sheaves are the objects of EE that are the VV-separated objects.


For E=Sh(C)Γ=Hom(*,)SetE = Sh(C) \stackrel{\Gamma = Hom(*,-)}{\to} Set the category of sheaves on a concrete site, this is equivalent to the previous definition.


Since CC is concrete, in the global sections geometric morphism (Disc,Γ):Sh(C)Set(Disc,\Gamma)\colon Sh(C) \to Set, the direct image Γ\Gamma is evaluation on the point: XX(*)X\mapsto X(*). The further right adjoint Codisc:SetSh(C)Codisc \colon Set\to Sh(C), sends a set AA to the functor UHom Set(Hom C(*,U),A)U\mapsto Hom_{Set}(Hom_C(*,U),A). Moreover, this right adjoint CodiscCodisc is fully faithful and thus embeds SetSet as a subtopos of Sh(C)Sh(C).

We observe that (ΓCodisc):SetSh(C) (\Gamma \dashv Codisc) : Set \to Sh(C) is the localization of Sh(C)Sh(C) at the set {DiscΓUU|UC}\{Disc \Gamma U \to U | U \in C\} of counits of the adjunction (DiscΓ)(Disc \dashv \Gamma) on representables: because if for XSh(C)X \in Sh(C) we have that

Hom Sh(C)(DiscΓUU,X) =(X(U)Hom Set(Γ(U),Γ(X))) =(X(U)Hom Set(Hom C(*,U)),X(*)) \begin{aligned} Hom_{Sh(C)}(Disc \Gamma U \to U, X) & = (X(U) \to Hom_{Set}(\Gamma(U), \Gamma(X))) \\ & = (X(U) \to Hom_{Set}(Hom_C(*,U)), X(*)) \end{aligned}

is an isomorphism, then clearly X=Codisc(X(*))X = Codisc(X(*)).

On the other hand, comparison with the previous definition shows that this is a monomorphism precisely if XX is a concrete sheaf. But this is also the definition of a separated object.

So the concrete sheaves on CC are precisely the separated objects for this Lawvere-Tierney topology on Sh(C)Sh(C) that corresponds to the subtopos Codisc:SetSh(C)Codisc : Set \hookrightarrow Sh(C).



Let Γ:ES\Gamma : E \to S be a local topos. From the definition of concrete sheaves as separated presheaves it follows immediately that


The category of concrete sheaves Conc Γ(E)Conc_\Gamma(E) forms a reflective subcategory of EE

SCodiscΓConc Γ(E)ConcE S \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} Conc_\Gamma(E) \stackrel{\overset{Conc}{\leftarrow}}{\hookrightarrow} E

which is a quasitopos.

The left adjoint ConcConc is concretization which sends a sheaf XX to the image sheaf

ConcX:UIm(X(U)Hom Set(Hom C(*,U),X(*))). Conc X : U \mapsto Im(X(U) \to Hom_{Set}(Hom_C(*,U), X(*))) \,.

Slice topos over a concrete object

Let (DiscΓcoDisc):𝒮(Disc \dashv \Gamma \dashv coDisc) : \mathcal{E} \to \mathcal{S} be a Grothendieck topos that is a local topos over 𝒮\mathcal{S} and let XX \in \mathcal{E} be a concrete object, equivalently an object such that the (ΓcoDisc)(\Gamma \dashv coDisc)-counit XcoDiscΓUX \to coDisc \Gamma U is a monomorphism.

We discuss properties of the over-topos /X\mathcal{E}/X.

Notice that

e 0(ΓcoDisc):𝒮coDiscΓ e_0 \coloneqq (\Gamma \dashv coDisc) : \mathcal{S} \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}

is the canonical topos point of \mathcal{E}.


For every global element (xΓ(X)):*X(x \in \Gamma(X)) : * \to X (for every XX \in \mathcal{E}) there is a topos point of the form

(e 0,x):𝒮x *x *𝒮/Γ(X)coDisc/XΓ/X/X. (e_0,x) : \mathcal{S} \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} \mathcal{S}/\Gamma(X) \stackrel{\overset{\Gamma/X}{\leftarrow}}{\underset{coDisc/X}{\to}} \mathcal{E}/X \,.

This is discussed in detail at over-topos – points.


(relative concretization)

Let XX \in \mathcal{E} be concrete. Then the image under the coDisc/XΓ/XcoDisc/X \circ \Gamma/X-monad of any object (AX)/X(A \to X) \in \mathcal{E}/X is an object (A˜X)(\tilde A \to X) with A˜\tilde A being concrete.

This A˜\tilde A is the finest concrete sheaf structure on ΓA\Gamma A that extends ΓAΓX\Gamma A \to \Gamma X to a morphism of concrete sheaves.


By definition of the slice geometric morphism we have that coDisc/XΓ/X(AfX)coDisc/X \circ \Gamma/X (A \stackrel{f}{\to} X) is the pullback A˜X\tilde A \to X in

A˜ coDiscΓA coDiscΓA X coDiscΓX, \array{ \tilde A &\to & coDisc \Gamma A \\ \downarrow && \downarrow^{\mathrlap{coDisc \Gamma A}} \\ X &\to& coDisc \Gamma X } \,,

where the bottom morphism is the (ΓcoDisc)(\Gamma \dashv coDisc)-unit. Since this is a monomorphism by assumption on XX it follows that A˜coDiscΓA\tilde A\to coDisc \Gamma A is a monomorphism. Since coDisccoDisc is a full and faithful functor by assumption on \mathcal{E} and Γ\Gamma is a right adjoint it follows that the adjunct ΓA˜ΓcoDiscΓAΓA\Gamma \tilde A \to \Gamma coDisc\Gamma A \stackrel{\simeq}{\to} \Gamma A is a monomorphism, as is its image coDiscΓA˜coDiscΓAcoDisc \Gamma \tilde A\to coDisc \Gamma A under the right adjoint coDisccoDisc.

Then by the universal property of the unit we have a commuting diagram

coDiscΓA˜ A˜ coDiscΓA, \array{ && coDisc \Gamma \tilde A \\ & \nearrow & \downarrow \\ \tilde A &\to& coDisc \Gamma A } \,,

where the bottom and the right morphisms are monomorphisms. Therefore also the diagonal morphism, the (ΓcoDisc)(\Gamma \dashv coDisc)-unit on A˜\tilde A, is a monomorphism, and hence A˜\tilde A is concrete.



The notion of quasitoposes of concrete sheaves goes back to

and is further developed in

A review of categories of concrete sheaves, with special attention to sheaves on CartSp, i.e. to diffeological spaces is in

The characterization of concrete sheaves in terms of the extra right adjoint of a local topos originated in discussion with David Carchedi.

Concrete sheaves over trivial sites are called extensional presheaves in the following references.

Last revised on March 12, 2021 at 09:03:56. See the history of this page for a list of all contributions to it.