A presheaf on a site is a sheaf if its value on any object of the site is given by its compatible values on any covering of that object.
See also
There are several equivalent ways to characterize sheaves. We start with the general but explicit componentwise definition and then discuss more general abstract equivalent reformulations. Finally we give special discussion applicable in various common special cases.
The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the site has pullbacks).
Let $(C,J)$ be a site in the form of a small category $C$ equipped with a coverage $J$.
A presheaf $A \in PSh(C)$ is a sheaf with respect to $J$ if
for every covering family $\{p_i : U_i \to U\}_{i \in I}$ in $J$
and for every compatible family of elements, given by tuples $(s_i \in A(U_i))_{i \in I}$ such that for all $j,k \in I$ and all morphisms $U_j \stackrel{f}{\leftarrow} K \stackrel{g}{\to} U_k$ in $C$ with $p_j \circ f = p_k \circ g$ we have $A(f)(s_j) = A(g)(s_k) \in A(K)$
then
If in the above definition there is at most one such $s$, we say that $A$ is a separated presheaf with respect to $J$.
In this form the definition appears for instance in (Johnstone, def. C2.1.2).-
We now reformulate the above component-wise definition in general abstract terms.
Write
for the Yoneda embedding.
Given a covering family $\{f_i : U_i \to U\}$ in $J$, its sieve is the presheaf $S(\{U_i\})$ defined as the coequalizer
in $PSh(C)$.
Here the coproduct on the left is over the pullbacks
in $PSh(C)$, and the two morphisms between the coproducts are those induced componentwise by the two projections $p_j, p_k$ in this pullback diagram.
Using that limits and colimits in a category of presheaves are computed objectwise, we find that the sieve $S(\{U_i\})$ defined this way is the presheaf that sends any $K \in C$ to the set of morphisms $K \to U$ in $C$ that factor through one of the $f_i$.
For every covering family there is a canonical morphism
that is induced by the universal property of the coequalizer from the morphisms $j(f_i) : j(U_i) \to j(U)$ and $j(U_j) \times_{j(U)} j(U_k) \to J(U)$.
A sheaf on $(C,J)$ is a presheaf $A \in PSh(C)$ that is a local object with respect to all $i_{\{U_i\}}$: an object such that for all covering families $\{f_i : U_i \to U\}$ in $J$ we have that the hom-functor $PSh_C(-,A)$ sends the canonical morphisms $i_{\{U_i\}} : S(\{U_i\}) \to j(U)$ to isomorphisms.
Equivalently, using the Yoneda lemma and the fact that the hom-functor $PSh_C(-,A)$ sends colimits to limits, this says that the diagram
is an equalizer diagram for each covering family.
This is also called the descent condition for descent along the covering family.
For many examples of sites that appear in practice – but by far not for all – it happens that the pullback presheaves $j(U_j) \times_{j(U)} \times j(U_k)$ are themselves again representable, hence that the pullback $U_j \times_U U_k$ exists already in $C$, even before passing to the Yoneda embedding.
In this special case we may apply the Yoneda lemma once more to deduce
Then the sheaf condition is that all diagrams
The condition that $PSh_C(S(\{U_i\}), A)$ is an isomorphism is equivalent to the condition that the set $A(U)$ is isomorphic to the set of matching families $(s_i \in A(U_i))$ as it appears in the above component-wise definition.
We may express the set of natural transformations $PSh_C(j(U_j) \times_{j(U)} j(U_k), A)$ (as described there) by the end
Using this in the expression of the equalizer
as a subset of the product set on the left manifestly yields the componenwise definition above.
A morphism of sheaves is just a morphism of the underlying presheaves. So the category of sheaves $Sh_J(C)$ is the full subcategory of the category of presheaves on the sheaves:
We discuss equivalent characterizations of sheaves that are applicable if the underlying site enjoys certain special properties.
An important special case of sheaves is those over a (0,1)-site such as a category of open subsets $Op(X)$ of a topological space $X$. We consider some equivalent ways of characterizing sheaves among presheaves in such a situation.
(The following was mentioned in Peter LeFanu Lumsdaine’s comment here).
Suppose $Op = Op(X)$ is the category of open subsets of some topological space regarded as a site with the canonical coverage where $\{U_i \hookrightarrow U\}$ is covering if the union $\cup_i U_i \simeq U$ in $Op$.
Then a presheaf $\mathcal{F}$ on $Op$ is a sheaf precisely if for every complete full subcategory $\mathcal{U} \hookrightarrow Op$, $\mathcal{F}$ takes the colimit in $Op$ over $\mathcal{U} \hookrightarrow Op$ to a limit:
A complete full subcategory $\mathcal{U} \hookrightarrow Op$ is a collection $\{U_i \hookrightarrow X\}$ of open subsets that is closed under forming intersections of subsets. The colimit
is the union $U \coloneqq \cup_{i \in I} U_i$ of all these open subsets. Notice that by construction the component maps $\{U_i \hookrightarrow U\}$ of the colimit are a covering family of $U$.
Inspection then shows that the limit $\underset{\leftarrow}{\lim}_{i \in I} \mathcal{F}(U_i)$ is the corresponding set of matching families (use the description of limits in terms of products and equalizers ). Hence the statement follows with def. 1.
The above prop. 2 shows that often sheaves are characterized as contravariant functors that take some colimits to limits. This is true in full generality for the following case
Let $\mathcal{T}$ be be a topos, regarded as a large site when equipped with the canonical topology. Then a presheaf (with values in small sets) on $\mathcal{T}$ is a sheaf precisely if it sends all colimits to limits.
We now describe the derivation and the detailed description of various aspects of sheaves, the descent condition for sheaves and sheafification, relating it to all the related notions
We start by assuming that a geometric embedding into a presheaf category is given and derive the consequences.
So let $S$ be a small category and write $PSh(S) = PSh_S = [S^{op}, Set]$ for the corresponding topos of presheaves.
Assume then that another topos $Sh(S) = Sh_S$ is given together with a geometric embedding
i.e. with a full and faithful functor
and a left exact functor
Such that both form a pair of adjoint functors
with $f^*$ left adjoint to $f_*$.
Write $W$ for the category
consisting of all those morphisms in $PSh(S)$ that are sent to isomorphisms under $f^*$.
From the discussion at geometric embedding we know that $Sh(S)$ is equivalent to the full subcategory of $PSh(S)$ on all $W$-local objects.
Recall that an object $A \in PSh(S)$ is called a $W$-local object if for all $p : Y \to X$ in $W$ the morphism
is an isomorphism. This we call the descent condition on presheaves (saying that a presheaf “descends” along $p$ from $Y$ “down to” $X$). Our task is therefore to identify the category $W$, show how it determines and is determed by a Grothendieck topology on $S$ – equipping $S$ with the structure of a site – and characterize the $W$-local objects. These are (up to equivalence of categories) the objects of $Sh$, i.e. the sheaves with respect to the given Grothendieck topology.
A morphism $Y \to X$ is in $W$ if and only if for every representable presheaf $U$ and every morphism $U\to X$ the pullback $Y \times_X U \to U$ is in $W$
Since $W$ is stable under pullback (as described at geometric embedding: simply because $f^*$ preserves finite limits) it is clear that $Y \times_X U \to U$ is in $W$ if $Y \to X$ is.
To get the other direction, use the co-Yoneda lemma to write $X$ as a colimit of representables over the comma category $(Y/const_X)$ (with $Y$ the Yoneda embedding):
Then pull back $Y \to colim_{U_i \to X} U$ over the entire colimiting cone, so that over each component we have
Using that in $PSh(S)$ colimits are stable under base change we get
But since $X \simeq colim_i U_i$ the right hand is $X \times_X Y$, which is just $Y$. So $Y = colim_i (Y \times_X U_i)$ and we find that $Y \to X$ is a morphism of colimits. But under $f^*$ the two respective diagrams become isomorphic, since $Y \times_X U_i \to U_i$ is in $W$. That means that the corresponding morphism of colimits $f^*(Y \to X)$ (since $f^*$ preserves colimits) is an isomorphism, which finally means that $Y \to X$ is in $W$.
A presheaf $A \in PSh(S)$ is a local object with respect to all of $W$ already if it is local with respect to those morphisms in $W$ whose codomain is representable
Rewriting the morphism $Y \to X$ in $W$ in terms of colimits as in the above proof
we find that $A(X) \to A(Y)$ equals
If $A$ is local with respect to morphisms $W$ with representable codomain, then by the above if $Y \to X$ is in $W$ all the morphisms in the limit here are isomorphisms, hence
Every morphism $Y \to X$ in $W \subset PSh(S)$ factors as an epimorphism followed by a monomorphism in $PSh(S)$ with both being morphisms in $W$.
Use factorization through image and coimage, use exactness of $f^*$ to deduce that the factorization exists not only in $PSh(S)$ but even in $W$.
More in detail, given $Y \to X$ we get the diagram
Because $f^*$ is exact, the pullbacks and pushouts in this diagram remain such under $f^*$. But since $f^*(Y \to X)$ is an isomorphism by assumption, the all these are pullbacks and pushouts along isomorphisms in $Sh(S)$, so all morphisms in the above diagram map to isomorphisms in $Sh(S)$, hence the entire diagram in $PSh(S)$ is in $W$.
Since the morphism $Y \sqcup_{Y \times_X Y} Y \to X$ out of the coimage is at the same time the equalizing morphism into the image $lim(X \stackrel{\to}{\to} X \sqcup_Y X)$, it is a monomorphism.
The monomorphisms in $PSh(S)$ which are in $W$ are called dense monomorphisms.
Every monomorphism $Y \to X$ with $X$ representable is of the form
for $U = \sqcup_{\alpha} U_\alpha$ a disjoint union of representables
This is a direct consequence of the standard fact that subfunctors are in bijection with sieves.
If a presheaf $A$ is local with respect to all dense monomorphisms, then it is already local with respect to all morphisms $Y \to X$ of the form
with the left vertical morphism a dense monomorphism
(and with $U = \sqcup_\alpha U_\alpha$ the disjoint union (of representable presheaves) over a covering family of objects.)
The morphisms in $W$ with representable codomain
of the form $colim (U \times_X U \stackrel{\to}{\to} U) \to X$ as above are covers:
of the form $colim (W \stackrel{\to}{\to} U) \to X$ (with $W$ a cover of $U \times_X U$) as above are hypercovers
of the representable $X$.
A presheaf $A$ is $W$-local, i.e. a sheaf, already if it is local (satisfies descent) with respect to all covers, i.e. all dense monomorphisms with codomain a representable.
Urs: the above shows this almost. I am not sure yet how to see the remaining bit directly, without making recourse to the full machinery leading up to section VII, 4, corollary 7 in Sheaves in Geometry and Logic.
So we finally conclude:
We have:
Systems $W$ of weak equivalences defined by choice of geometric embedding $f : Sh(S) \to PSh(S)$ are in canonical bijection with choice of Grothendieck topology.
A presheaf $A$ is $W$-local, i.e. local with respect to all local isomorphisms, if and only if it is local already with respect to all dense monomorphism, i.e. if and only if it satisfies sheaf condition for all covering sieves.
From the assumption that $f : Sh(S) \to PSh(S)$ is a geometric embedding follows at once the following explicit description of the sheafification functor $f^* : PSh(S) \to Sh(S)$.
For $A \in PSh(S)$ a presheaf, its sheafification $\bar A := f_* f^* A$ is the presheaf given by
By the discussion at geometric embedding the category $Sh(S)$ is equivalent to the localization $PSh(S)[W^{-1}]$, which in turn is the category with the same objects as $PSh(S)$ and with morphisms given by spans out of hypercovers in $W$
So we have
and deduce
by Yoneda that $\bar A(U) = PSh_S(U, \bar A)$;
by the hom-adjunction this is $\cdots \simeq Sh_S(\bar U, \bar A)$;
by the equivalence just mentioned this is $\cdots \simeq PSh_S[W^{-1}](U,A)$.
For checking the sheaf condition the dense monomorphisms, i.e. the ordinary covers are already sufficient. But for sheafification one really needs the local isomorphisms, i.e. the hypercovers. If one takes the colimit in the sheafification prescription above only over covers, one obtains instead of sheafification the plus-construction.
For $A \in PSh(S)$ a presheaf, the plus-construction on $A$ is the presheaf
where the colimit is over all dense monomorphisms (instead of over all local isomorphisms as for sheafification $\bar A$).
In general $A^+$ is not yet a sheaf. It is however in general closer to being a sheaf than $A$ is, namely it is a separated presheaf.
But applying the plus-construction twice yields the desired sheaf
This is essentially due to the fact that in the context of ordinary sheaves discussed here, all hypercovers are already of the form
for $W \to U \times_X U$ a cover. For higher stacks the hypercover is in general a longer simplicial object of covers and accordingly if one restricts to covers instead of using hypercovers one will need to use the plus-construction more and more often.
The archetypical example of sheaves are sheaves of functions:
for $X$ a topological space, $\mathbb{C}$ a topological space and $O(X)$ the site of open subsets of $X$, the assignment $U \mapsto C(U,\mathbb{C})$ of continuous functions from $U$ to $\mathbb{C}$ for every open subset $U \subset X$ is a sheaf on $O(X)$.
for $X$ a complex manifold and $\mathbb{C}$ a complex manifold, the assignment $U \mapsto C_{hol}{X,\mathbb{C}}$ of holomorphic functions in a sheaf.
presheaf / separated presheaf / sheaf / cosheaf
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
Section C2 in
The book by Kashiwara and Schapira discusses sheaves with motivation from homological algebra, abelian sheaf cohomology and homotopy theory, leading over in the last chapter to the notion of stack.
A quick pedagogical introduction with an eye towards the generalization to (∞,1)-sheaves is in
Classics of sheaf theory on topological spaces are
Roger Godement, Topologie algébrique et théorie des faisceaux, Hermann, 1958, 283 p. gBooks
Recently, an improvement in understanding the interplay of derived functors (inverse image and proper direct image) in sheaf theory on topological spaces has been exhibited in