typical contexts
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 $C$ that, while perhaps not representable, is “quasi-representable” in that it is a subobject of a sheaf of the form
where $S$ is a set and $|U|$ is the set $|U| := Hom_C({*}, U)$ of points underlying the object $U$ in the concrete site $C$.
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.
A concrete site is a site $C$ with a terminal object $*$ such that
the functor $Hom_C(*,-) : C \to Set$ is a faithful functor;
for every covering family $\{f_i : U_i \to U\}$ in $C$ the morphism
is surjective.
For $X \in PSh(C)$ any presheaf, write
for the adjunct of the restriction map
which in turn is the adjunct of the component map of the functor
A presheaf $X : C^{op} \to Set$ on a concrete site is a concrete presheaf if for each $U \in C$ the map $\tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))$ is injective.
A concrete sheaf is a presheaf that is both concrete and a sheaf.
So a concrete presheaf $X$ is a subobject of the presheaf $U \mapsto Hom_{Set}(Hom_C(*,U), X(*))$.
Write $Conc(Sh(C)) \hookrightarrow Sh(C)$ for the full subcategory of the category of sheaves on concrete sheaves.
A more abstract perspective on the previous definition is obtained by noticing the following.
The category of sheaves on a concrete site is a local topos.
Taking $U=*$ 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 $E$ over any base topos $S$:
Let
be a local geometric morphism. Since then by definition $S \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} E$ is a subtopos the morphisms $V = \Gamma^{-1}(isos(S)) \subset Mor E$ that are inverted by $\Gamma$ are the local isomorphisms with respect to which the objects of $S$ are sheaves/$V$-local objects in $E$.
The concrete sheaves are the objects of $E$ that are the $V$-separated objects.
For $E = Sh(C) \stackrel{\Gamma = Hom(*,-)}{\to} Set$ the category of sheaves on a concrete site, this is equivalent to the previous definition.
Since $C$ is concrete, in the global sections geometric morphism $(Disc,\Gamma)\colon Sh(C) \to Set$, the direct image $\Gamma$ is evaluation on the point: $X\mapsto X(*)$. The further right adjoint $Codisc \colon Set\to Sh(C)$, sends a set $A$ to the functor $U\mapsto Hom_{Set}(Hom_C(*,U),A)$. Moreover, this right adjoint $Codisc$ is fully faithful and thus embeds $Set$ as a subtopos of $Sh(C)$.
We observe that $(\Gamma \dashv Codisc) : Set \to Sh(C)$ is the localization of $Sh(C)$ at the set $\{Disc \Gamma U \to U | U \in C\}$ of counits of the adjunction $(Disc \dashv \Gamma)$ on representables: because if for $X \in Sh(C)$ we have that
is an isomorphism, then clearly $X = Codisc(X(*))$.
On the other hand, comparison with the previous definition shows that this is a monomorphism precisely if $X$ is a concrete sheaf. But this is also the definition of a separated object.
So the concrete sheaves on $C$ are precisely the separated objects for this Lawvere-Tierney topology on $Sh(C)$ that corresponds to the subtopos $Codisc : S \hookrightarrow Sh(C)$.
Let $\Gamma : E \to S$ be a local topos. From the definition of concrete sheaves as separated presheaves it follows immediately that
The category of concrete sheaves $Conc_\Gamma(E)$ forms a reflective subcategory of $E$
which is a quasitopos.
The left adjoint $Conc$ is concretization which sends a sheaf $X$ to the image sheaf
Let $(Disc \dashv \Gamma \dashv coDisc) : \mathcal{E} \to \mathcal{S}$ be a Grothendieck topos that is a local topos over $\mathcal{S}$ and let $X \in \mathcal{E}$ be a concrete object, equivalently an object such that the $(\Gamma \dashv coDisc)$-counit $X \to coDisc \Gamma U$ is a monomorphism.
We discuss properties of the over-topos $\mathcal{E}/X$.
Notice that
is the canonical topos point of $\mathcal{E}$.
For every global element $(x \in \Gamma(X)) : * \to X$ (for every $X \in \mathcal{E}$) there is a topos point of the form
This is discussed in detail at over-topos – points.
(relative concretization)
Let $X \in \mathcal{E}$ be concrete. Then the image under the $coDisc/X \circ \Gamma/X$-monad of any object $(A \to X) \in \mathcal{E}/X$ is an object $(\tilde A \to X)$ with $\tilde A$ being concrete.
This $\tilde A$ is the finest concrete sheaf structure on $\Gamma A$ that extends $\Gamma A \to \Gamma X$ to a morphism of concrete sheaves.
By definition of the slice geometric morphism we have that $coDisc/X \circ \Gamma/X (A \stackrel{f}{\to} X)$ is the pullback $\tilde A \to X$ in
where the bottom morphism is the $(\Gamma \dashv coDisc)$-unit. Since this is a monomorphism by assumption on $X$ it follows that $\tilde A\to coDisc \Gamma A$ is a monomorphism. Since $coDisc$ is a full and faithful functor by assumption on $\mathcal{E}$ and $\Gamma$ is a right adjoint it follows that the adjunct $\Gamma \tilde A \to \Gamma coDisc\Gamma A \stackrel{\simeq}{\to} \Gamma A$ is a monomorphism, as is its image $coDisc \Gamma \tilde A\to coDisc \Gamma A$ under the right adjoint $coDisc$.
Then by the universal property of the unit we have a commuting diagram
where the bottom and the right morphisms are monomorphisms. Therefore also the diagonal morphism, the $(\Gamma \dashv coDisc)$-unit on $\tilde A$, is a monomorphism, and hence $\tilde A$ is concrete.
concrete sheaf
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.