This entry is about the properties and the characterization of the category $Sh(S)$ of (set-valued) sheaves on a (small) site $S$, which is a Grothendieck topos. Among other things it gives a definition and a characterization of the notion of sheaf itself, but for more details on sheaves themselves see there.
Let $(C,J)$ be a site: a (small) category equipped with a coverage.
The category of sheaves on $(C,J)$ is the full subcategory of the category of presheaves
on those presheaves which are sheaves with respect to $J$.
Every category of sheaves is a reflective subcategory
hence a subtopos of the presheaf topos. Moreover, every such subtopos arises in this way: there is a bijection between Grothendieck topologies on $C$ and equivalence classes of geometric embeddings into $PSh(C)$.
This appears for instance as (Johnstone, corollary 2.1.11). See also Lawvere-Tierney topology.
Details on the first statement are at sheafification. A full proof for the second statement is at (∞,1)-category of (∞,1)-sheaves (there proven in (∞,1)-category theory, but the proof is verbatim the same in category theory).
The category of sheaves is equivalent to the homotopy category of the category with weak equivalences $PSh(C)$ with the weak equivalences given by $W =$local isomorphisms
The converse is also true: for every left exact functor $L : PSh(S) \to PSh(S)$ (preserving finite limits) which is left adjoint to the inclusion of its image, there is a Grothendieck topology on $S$ such that the image of $L$ is the category of sheaves on $S$ with respect to that topology.
We spell out proofs of some of the above claims.
Let $C$ be a small category and
a reflective subcategory, hence a full subcategory with a left adjoint $L : [C^{op}, Set] \to \mathcal{E}$, such that moreover $L$ preserves finite limits.
Write $W := L^{-1}(isos) \subset Mor([C^{op}, Set])$ for the class of morphisms in $[C^{op}, Set]$ that are sent to isomorphisms by $L$.
A presheaf $A \in [C^{op}, Set]$ is in $\mathcal{E}$ (meaning: in the essential image of $i$) precisely if for all $f : X \to Y$ in $W$ the induced function
is a bijection.
If $A \simeq i \hat A$, then by the $(L \dashv i)$-adjunction isomorphism we have
But by assumption $L(f)$ is an isomorphism, so the claim is immediate.
Conversely, if for all $f$ the function $Hom(f,A)$ is a bijection, define $\hat A := L(A)$ and let $\epsilon_A : A \to i L(A)$ be the $(L \dashv i)$-unit.
By the assumption that $i$ is a full and faithful functor and basic properties of adjoint functors we habe that the counit
is a natural isomorphism. By the zig-zag law the composite
is the identity and therefore $L \epsilon$ is an isomorphism and so $\epsilon_A$ is in $W$, under our assumption on $A$.
Using this it follows that
is an isomorphism. Write $k_A : i L A \to A$ for the preimage of $id_A$ under this isomorphism, which is therefore a left inverse of $\epsilon_A$. This immediately implies that also $k_A$ is in $W$, and so we can enter the same argument with $k_A$ to find that it has a left inverse itself. But this means that $k_A$ is in fact an isomorphism and hence so is $\epsilon A$, which thus exhibits $A$ as being in the essential image of $i$.
A morphism $f : X \to Y$ is in $W$ precisely if for every morphism $z : j(c) \to Y$ with representable domain, the pullback $z^* f$ in
is in $W$.
Assume first that $f$ is in $W$. Since by assumption $L$ preserves finite limits, it follows that
is still a pullback diagram in $\mathcal{E}$ and hence that $L(z^* f)$ is the pullback of the isomorphism $L f$ and thus itself an isomorphism. Therefore $z^* f$ is in $W$.
Conversely, suppose that all these pullbacks are in $W$. Then use the “co-Yoneda lemma” to write the presheaf $Y$ as a colimit over all representables mapping into it
Forming the pullback along $f$, using that in a topos (such as our presheaf topos) colimits are preserved by pullbacks, we get
Since $L$ preserves all colimits and finite limits, we also get
Since by assumption now all $L(z_i^* f )$ are isomorphisms, also ${\lim_\to}_i L(z_i^* f)$ is an isomorphism and hence three sides of the above square are isomorphisms. Therefore also $L(f)$ is and hence $f$ is in $W$.
The collection of sieves in $W$, hence the collection of monomorphisms in $W$ whose codomain is a representable, constitute a Grothendieck topology on $C$.
We check the list of axioms, given at Grothendieck topology:
Pullbacks of covering sieves are covering :
First of all, the pullback of a sieve along a morphism of representables is still a sieve, because monomorphisms are (as discussed there) stable under pullback.
Next, since $L$ preserves finite limits, $L$ applied to the pullback sieve is the pullback of an isomorphism, hence is an isomorphism, hence the pullback sieve is in $W$.
The maximal sieve is covering. Clear: $L$ applied to an isomorphism is an isomorphism.
Two sieves cover precisely if their intersection covers. This is again due to the pullback-stability of elements of $W$, due to the preservation of finite limits by $L$.
If all pullbacks of a sieve along morphisms of a covering sieve are covering, then the original sieve was covering .
This is the same argument as in the second part of the proof of prop. 4.
$\mathcal{E}$ is a Grothendieck topos.
By prop. 5 and prop. 3 we are reduced to showing that an object $A$ is in $\mathcal{E}$ already if for all monomorphisms $f$ in $W$ the function $Hom(f,A)$ is a bijection.
(…)
Categories of sheaves are examples of categories that are toposes: they are the Grothendieck toposes characterized among all toposes as those satisfying Giraud's axioms.
Sheaf toposes are equivalently the subtoposes of presheaf toposes.
This appears for instance as (Johnstone, corollary 2.1.11).
Sheaf toposes are equivalently the accessible reflective subcategories of categories of presheaves.
See at reflective sub-(∞,1)-category.
For $(L \dashv i) : \mathcal{E} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}$ a category of sheaves and $(C,J)$ a site such that we have an equivalence of categories $\mathcal{E} \simeq Sh_J(C)$ we say that $C$ is a site of definition for the topos $\mathcal{E}$.
There are always different sites $(C,J)$ whose categories of sheaves are equivalent. First of all for fixed $C$ and given a coverage $J$, the category of sheaves depends only on the Grothendieck topology generated by $J$. But there may be site structures also on inequivalent categories $C$ that have equivalent categories of sheaves.
For $(C,J)$ a site with coverage $J$ and $D \to C$ any subcategory, the induced coverage $J_D$ on $D$ has as covering sieves the intersections of the covering sieves of $C$ with the morphisms in $D$.
Let $(C,J)$ be a site (possibly large). A subcategory $D \to C$ (not necessarily full) is called a dense sub-site with the induced coverage $J_D$ if
If $D$ is a full subcategory then the second condition is automatic.
(comparison lemma)
Let $(C,J)$ be a (possibly large) site with $C$ a locally small category and let $f : D \to C$ be a small dense sub-site. The pair of adjoint functors
with $f^*$ given by precomposition with $f$ and $f_*$ given by right Kan extension induces an equivalence of categories between the categories of sheaves
This appears as (Johnstone, theorm C2.2.3).
Let $X$ be a locale with frame $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the join of the $U_i$ is $U$). Let $bOp(X)$ be a basis for the topology of $X$: a complete join-semilattice such that every object of $Op(X)$ is the join of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.
For $C = TopManifold$ the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp${}_{top}$ is a dense sub-site: every paracompact manifold has a good open cover by open balls homeomorphic to a Cartesian space.
Similaryl for $C =$ Diff the category of smooth manifolds equipped with the good open cover coverage, the full subcategory CartSp${}_{smooth}$ is a dense sub-site.
Let $\mathcal{C}$ be a small site and let $Sh(\mathcal{C})$ be its category of sheaves. Let $f \colon X \to Y$ be a homomorphism of sheaves, hence a morphism in $Sh(\mathcal{C})$. Then:
$f$ is a monomorphism or isomorphism precisely if it is so globally in that for each object $U \in \mathcal{C}$ in the site, then the component $f_U \colon X(U) \to Y(U)$ is an injection or bijection of sets, respectively.
$f$ is an epimorphism precisely if it is so locally, in that: for all $U \in C$ there is a covering $\{p_i : U_i \to U\}_{i \in I}$ such that for all $i \in I$ and every element $y \in Y(U)$ the element $f(p_i)(y)$ is in the image of $f(U_i) : X(U_i) \to Y(U_i)$.
But if $\{x_i\}_{i \in I}$ is a set of points of a topos for $Sh(\mathcal{C})$ such that these are enough points (def.) then the morphism $f$ is epi/mono/iso precisely it is is so an all stalks, hence precisely if
is a surjection/injection/bijection of sets, respectively, for all $i \in I$.
Every sheaf topos satisfies the following exactness properties. it is an
category of sheaves
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
Peter Johnstone, sections A.4 and C.2 in Sketches of an Elephant
The Stacks Project, Sites and sheaves (pdf)
The characterization of sheaf toposes and Grothendieck topologies in terms of left exact reflective subcategories of a presheaf category is also in
where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.
The characterization of $Sh(S)$ as the homotopy category of $PSh(S)$ with respect to local isomorphisms is emphasized at the beginning of the text
Details are in
It’s a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed:
exercise 16.7 shows that sheafification inverts precisely the local isomorphisms, so that in particular every local isomorphism between sheaves is an isomorphism;
lemma 16.3.2 states that the unit of the adjunction $Id_{PSh(S)} \rightarrow i \circ \bar{(-)} : PSh(S) \to PSh(S)$ is componentwise a local isomorphism;
using this corollary 7.2.2 says that $Sh(S) \simeq Ho_{PSh(S)}$ with the homotopy category $Ho_{PSh(S)}$ formed using local isomorphisms as weak equivalences.