nLab
sheaf

Idea

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

• motivation for sheaves, cohomology and higher stacks.

Definition

There are several equivalent ways to think about sheaves. We start with the explicit componentwise definition and then discuss more general abstract reformulations.

The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the site has pullbacks).

Definition

Let $(C,J)$ be a site in the form of a small category $C$ equipped with a coverage $J$.

A presheaf $A\in \mathrm{PSh}(C)$ is a sheaf with respect to $J$ is

• 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 $(s_i\in A(U_i))$ such that for all morphisms $U_i\stackrel{f}{\leftarrow }K\stackrel{g}{\to }{U}_j$ such that $A(f)(s_i)=A(g)(s_j)$ for all $i,j\in I$

• there is a unique element $s\in A(U)$ such that $A(p_i)(s)=s_i$ for all $i\in I$.

Remark

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 in Johnstone, def. C2.1.2

We now reformulate the above component-wise definition in general abstract terms.

Write

$$j:C\hookrightarrow \mathrm{PSh}(C)$$j : C \hookrightarrow PSh(C)

for the Yoneda embedding.

Definition

Given a covering family $\{f_i:U_i\to U\}$ in $J$, say its sieve is the presheaf $S(\{U_i\})$ defined as the coequalizer

$$\coprod _{i,j}j(U_i){\times }_{j(u)}j(U_j)\overrightarrow{\to }\coprod _{i}j(U_i)\to S(\{U_i\})$$\coprod_{i,j} j(U_i) \times_{j(u)} j(U_j) \stackrel{\overset{}{\to}}{\to} \coprod_i j(U_i) \to S(\{U_i\})

in $\mathrm{PSh}(C)$.

Here the coproduct on the left is over the pullbacks

$$\begin{array}{ccc}j(U_i){\times }_{j(U)}j(U_j)& \stackrel{p_i}{\to }& j(U_i)\\ {}^{p_j}\downarrow & & {\downarrow }^{j(f_i)}\\ j(U_j)& \stackrel{j(f_j)}{\to }& j(U)\end{array}$$\array{ j(U_i) \times_{j(U)} j(U_j) &\stackrel{p_i}{\to}& j(U_i) \\ {}^{\mathllap{p_j}}\downarrow && \downarrow^{\mathrlap{j(f_i)}} \\ j(U_j) &\stackrel{j(f_j)}{\to}& j(U) }

in $\mathrm{PSh}(C)$, and the two morphisms between the coproducts are those induced componentwise by the two porjections $p_i,p_j$ in this pullback diagram.

Remark

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 $f_i$

Observation

For every covering family there is a canonical morphism

$$i_{\{U_i\}}:S(\{U_i\})\to j(U)$$i_{\{U_i\}} : S(\{U_i\}) \to j(U)

that is induced by the universal property of the coequalizer from the morphisms $j(f_i):j(U_i)\to j(U)$ and $j(U_i){\times }_{j(U)}j(U_j)\to J(U)$.

Definition

A sheaf on $(C,J)$ is a presheaf $A\in \mathrm{PSh}(C)$ 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 ${\mathrm{PSh}}_C(-,A)$ sends the canonical morphisms $i_{\{U_i\}}:S(\{U_i\})\to j(U)$ to isomorphisms.

$${\mathrm{PSh}}_C(i_{\{U_i\}},A):{\mathrm{PSh}}_C(j(U),A)\stackrel{\simeq }{\to }{\mathrm{PSh}}_C(S(\{U_i\}),A).$$PSh_C(i_{\{U_i\}}, A) : PSh_C(j(U), A) \stackrel{\simeq}{\to} PSh_C(S(\{U_i\}), A) \,.

Equivalently, using the Yoneda lemma and the fact that the hom-functor ${\mathrm{PSh}}_C(-,A)$ sends colimits to limits, this say that the diagram

$$A(U)\to \prod _{i}A(U_i)\overrightarrow{\to }\prod _{i,j}{\mathrm{PSh}}_C(j(U_i){\times }_{j(U)}j(U_j))$$A(U) \to \prod_i A(U_i) \stackrel{\to}{\to} \prod_{i,j} PSh_C(j(U_i) \times_{j(U)} j(U_j))

is an equalizer diagram for each covering family.

This is also called the descent condition for descent along the covering family.

Remark

For many examples of sites that appear in practice – but by far not for all – it happens that the pullback presheaves $j(U_i){\times }_{j(U)}\times j(U_j)$ are themselves again representable, hence that the pullback $U_i{\times }_U{U}_j$ 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

$${\mathrm{PSh}}_C(j(U_i){\times }_{j(U)}j(U_j),A)\simeq A(U_i{\times }_U{U}_j).$$PSh_C(j(U_i) \times_{j(U)} j(U_j), A) \simeq A(U_i \times_U U_j) \,.

Then the sheaf condition is that all diagrams

$$A(U)\to \prod _{i}A(U_i)\overrightarrow{\to }\prod _{i,j}A(U_i{\times }_U{U}_j)$$A(U) \to \prod_i A(U_i) \stackrel{\to}{\to} \prod_{i,j} A(U_i \times_U U_j)

are equalizer diagrams.

Proposition

The condition that ${\mathrm{PSh}}_C(S(\{U_i\}),A)$ is an isomorphism is equivlent to the condition that the set $A(U)$ is isomorphic to the set of compatible families $(s_i\in A(U_i))$ as it appears in the above component-wise definition.

Proof

We may express the set of natural transformations ${\mathrm{PSh}}_C(j(U_i){\times }_{j(U)}j(U_j),A)$ (as described there) by the end

$${\mathrm{PSh}}_C(j(U_i){\times }_{j(U)}j(U_j),A)\simeq {\int }^{K\in C}\mathrm{Set}(C(K,U_i){\times }_{C(K,U)}C(K,U_j),A(K)).$$PSh_C(j(U_i) \times_{j(U)} j(U_j), A) \simeq \int^{K \in C} Set( C(K,U_i) \times_{C(K,U)} C(K,U_j) , A(K)) \,.

Using this in the expression of the equalizer

$$\prod _{i}A(U_i)\simeq \prod _i{\int }^{K\in C}\mathrm{Set}(C(K,U_i),A(K))\overrightarrow{\to }\prod _{i,j}{\int }^{K\in C}\mathrm{Set}(C(K,U_i){\times }_{C(K,U)}C(K,U_j),A(K))$$\prod_i A(U_i) \simeq \prod_i \int^{K \in C} Set( C(K,U_i), A(K)) \stackrel{\to}{\to} \prod_{i,j} \int^{K \in C} Set( C(K,U_i) \times_{C(K,U)} C(K,U_j) , A(K))

as a subset of the product set on the left manifestly yields the componenwise definition above.

Definition

A morphism of sheaves is just a morphism of the underlying presheaves. So the category of sheaves ${\mathrm{Sh}}_J(C)$ is the full subcategory of the category of presheaves on the sheaves:

$${\mathrm{Sh}}_J(C)\hookrightarrow \mathrm{PSh}(C)$$Sh_J(C) \hookrightarrow PSh(C)

Sheaves and localization

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

• geometric embedding

  • localization

  • homotopy category

• coverage

  • Grothendieck topology

  • Lawvere-Tierney topology

• local isomorphism

  • sieve

    • cover

    • hypercover

  • dense monomorphism

  • local epimorphism

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 $\mathrm{PSh}(S)={\mathrm{PSh}}_S=[S^{\mathrm{op}},\mathrm{Set}]$ for the corresponding topos of presheaves.

Assume then that another topos $\mathrm{Sh}(S)={\mathrm{Sh}}_S$ is given together with a geometric embedding

$$f:\mathrm{Sh}(S)\to \mathrm{PSh}(S)$$f : Sh(S) \to PSh(S)

i.e. with a full and faithful functor

$$f_{*}:\mathrm{Sh}(S)\to \mathrm{PSh}(S)$$f_* : Sh(S) \to PSh(S)

and a left exact functor

$$f^{*}:\mathrm{PSh}(S)\to \mathrm{Sh}(S)$$f^* : PSh(S) \to Sh(S)

Such that both form a pair of adjoint functors

$$f^{*}⊣f_{*}$$f^* \dashv f_*

with $f^{*}$ left adjoint to $f_{*}$.

Write $W$ for the category

$$\mathrm{Core}(\mathrm{PSh}(S))\hookrightarrow W\hookrightarrow \mathrm{PSh}(S)$$Core(PSh(S)) \hookrightarrow W \hookrightarrow PSh(S)

consisting of all those morphisms in $\mathrm{PSh}(S)$ that are sent to isomorphisms under $f^{*}$.

$$W=(f^{*}{)}^{-1}(\mathrm{Core}({\mathrm{Sh}}_S)).$$W = (f^*)^{-1}(Core(Sh_S)) \,.

From the discussion at geometric embedding we know that $\mathrm{Sh}(S)$ is equivalent to the full subcategory of $\mathrm{PSh}(S)$ on all $W$-local objects.

Recall that an object $A\in \mathrm{PSh}(S)$ is called a $W$-local object if for all $p:Y\to X$ in $W$ the morphism

$$p^{*}:{\mathrm{PSh}}_S(X,A)\to {\mathrm{PSh}}_S(Y,A)$$p^* : PSh_S(X,A) \to PSh_S(Y,A)

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 $\mathrm{Sh}$, i.e. the sheaves with respect to the given Grothendieck topology.

Lemma

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 $W$

$$\begin{array}{ccc}Y{\times }_{X}U& \to & Y\\ {\downarrow }^{\in W}& & {\downarrow }^{\iff \in W}\\ U& \to & X\end{array}.$$\array{ Y \times_X U &\to& Y \\ \downarrow^{\in W} && \downarrow^{\Leftrightarrow \in W} \\ U &\to& X } \,.

Proof

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 $(Y/{\mathrm{const}}_X)$ (with $Y$ the Yoneda embedding):

$$X\simeq {\mathrm{colim}}_{U_i\to X}{U}_i.$$X \simeq colim_{U_i \to X} U_i \,.

Then pull back $Y\to {\mathrm{colim}}_{U_i\to X}U$ over the entire colimiting cone, so that over each component we have

$$\begin{array}{ccc}Y{\times }_X{U}_i& \to & Y\\ \downarrow & & \downarrow \\ U_i& \to & X\end{array}.$$\array{ Y \times_X U_i &\to& Y \\ \downarrow && \downarrow \\ U_i &\to& X } \,.

Using that in $\mathrm{PSh}(S)$ colimits are stable under base change we get

$${\mathrm{colim}}_i(Y{\times }_X{U}_i)\simeq ({\mathrm{colim}}_i{U}_i){\times }_{X}Y.$$colim_i (Y \times_X U_i) \simeq (colim_i U_i) \times_X Y \,.

But since $X\simeq {\mathrm{colim}}_i{U}_i$ the right hand is $X{\times }_{X}Y$, which is just $Y$. So $Y={\mathrm{colim}}_i(Y{\times }_X{U}_i)$ 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 $W$.

Lemma

A presheaf $A\in \mathrm{PSh}(S)$ 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

Proof

Rewriting the morphism $Y\to X$ in $W$ in terms of colimits as in the above proof

$$\begin{array}{ccc}{\mathrm{colim}}_{U\to X}{U}_i{\times }_{X}Y& \stackrel{\simeq }{\to }& Y\\ \downarrow & & \downarrow \\ {\mathrm{colim}}_{U\to X}U& \stackrel{\simeq }{\to }& X\end{array}$$\array{ colim_{U \to X} U_i \times_X Y &\stackrel{\simeq}{\to}& Y \\ \downarrow && \downarrow \\ colim_{U \to X} U &\stackrel{\simeq}{\to}& X }

we find that $A(X)\to A(Y)$ equals

$${\lim }_{U\to X}(A(U)\to A(U{\times }_{X}Y)).$$lim_{U \to X} (A(U) \to A(U \times_X Y)) \,.

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

$$\cdots ={\mathrm{Id}}_{A(X)}.$$\cdots = Id_{A(X)} \,.

Lemma

Every morphism $Y\to X$ in $W\subset \mathrm{PSh}(S)$ factors as an epimorphism followed by a monomorphism in $\mathrm{PSh}(S)$ with both being morphisms in $W$.

Proof

Use factorization through image and coimage, use exactness of $f^{*}$ to deduce that the factorization exists not only in $\mathrm{PSh}(S)$ but even in $W$.

More in detail, given $Y\to X$ we get the diagram

$$\begin{array}{ccccc}Y{\times }_{X}Y& & \to & & Y\\ & & & \swarrow \\ \downarrow & & Y{\bigsqcup }_{Y{\times }_{X}Y}Y& & \downarrow \\ & \nearrow & & \searrow \\ Y& & \to & & X\end{array}.$$\array{ Y \times_X Y &&\to&& Y \\ &&& \swarrow \\ \downarrow &&Y \sqcup_{Y \times_X Y} Y && \downarrow \\ & \nearrow && \searrow \\ Y && \to && X } \,.

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 $\mathrm{Sh}(S)$, so all morphisms in the above diagram map to isomorphisms in $\mathrm{Sh}(S)$, hence the entire diagram in $\mathrm{PSh}(S)$ is in $W$.

Since the morphism $Y{\bigsqcup }_{Y{\times }_{X}Y}Y\to X$ out of the coimage is at the same time the equalizing morphism into the image $\lim (X\overrightarrow{\to }X{\bigsqcup }_{Y}X)$, it is a monomorphism.

Definition

The monomorphisms in $\mathrm{PSh}(S)$ which are in $W$ are called dense monomorphisms.

Lemma

Every monomorphism $Y\to X$ with $X$ representable is of the form

$$Y=\mathrm{colim}(U{\times }_{X}U\to U)$$Y = colim ( U \times_X U \to U )

for $U={\bigsqcup }_{\alpha }{U}_{\alpha }$ a disjoint union of representables

Proof

This is a direct consequence of the standard fact that subfunctors are in bijection with sieves.

Corollary

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

$$\begin{array}{c}Y\\ \downarrow \\ X\end{array}=\mathrm{colim}\left(\begin{array}{ccc}W& \overrightarrow{\to }& U\\ {\downarrow }^{\mathrm{dense}\mathrm{mono}}& & {\downarrow }^{\mathrm{Id}}\\ U{\times }_{X}U& \overrightarrow{\to }& U\end{array}\right)$$\array{ Y \\ \downarrow \\ X } = colim \left( \array{ W &\stackrel{\to}{\to}& U \\ \;\;\downarrow^{dense mono} && \downarrow^{Id} \\ U \times_X U & \stackrel{\to}{\to}& U } \right)

with the left vertical morphism a dense monomorphism

(and with $U={\bigsqcup }_{\alpha }{U}_{\alpha }$ the disjoint union (of representable presheaves) over a covering family of objects.)

Definition

The morphisms in $W$ with representable codomain

• of the form $\mathrm{colim}(U{\times }_{X}U\overrightarrow{\to }U)\to X$ as above are covers:

• of the form $\mathrm{colim}(W\overrightarrow{\to }U)\to X$ (with $W$ a cover of $U{\times }_{X}U$) as above are hypercovers

of the representable $X$.

Proposition

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:

Corollaries

We have:

• Systems $W$ of weak equivalences defined by choice of geometric embedding $f:\mathrm{Sh}(S)\to \mathrm{PSh}(S)$ 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 $f:\mathrm{Sh}(S)\to \mathrm{PSh}(S)$ is a geometric embedding follows at once the following explicit description of the sheafification functor $f^{*}:\mathrm{PSh}(S)\to \mathrm{Sh}(S)$.

Lemma (Sheafification)

For $A\in \mathrm{PSh}(S)$ a presheaf, its sheafification $\overline{A}:=f_{*}{f}^{*}A$ is the presheaf given by

$$\overline{A}:U\mapsto {\mathrm{colim}}_{(Y\to U)\in W}A(Y)$$\bar A : U \mapsto colim_{(Y \to U) \in W} A(Y)

Proof

By the discussion at geometric embedding the category $\mathrm{Sh}(S)$ is equivalent to the localization $\mathrm{PSh}(S)[W^{-1}]$, which in turn is the category with the same objects as $\mathrm{PSh}(S)$ and with morphisms given by spans out of hypercovers in $W$

$$\mathrm{PSh}(S)[W^{-1}](X,A)={\mathrm{colim}}_{(Y\to X)\in W}A(Y).$$PSh(S)[W^{-1}](X,A) = colim_{(Y \to X) \in W} A(Y) \,.

So we have

$$\begin{array}{cccc}\mathrm{Sh}(S)& & \stackrel{\stackrel{f_{*}}{\to }}{\stackrel{f^{*}}{\leftarrow }}& \mathrm{PSh}(S)\\ & {\searrow }_{\simeq }& {\Downarrow }^{\simeq }& \downarrow \\ & & \mathrm{PSh}(S)[W^{-1}].\end{array}$$\array { Sh(S) &&\stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}}& PSh(S) \\ & \searrow_{\simeq}&\Downarrow^{\simeq}& \downarrow \\ && PSh(S)[W^{-1}] \,. }

and deduce

• by Yoneda that $\overline{A}(U)={\mathrm{PSh}}_S(U,\overline{A})$;

• by the hom-adjunction this is $\cdots \simeq {\mathrm{Sh}}_S(\overline{U},\overline{A})$;

• by the equivalence just mentioned this is $\cdots \simeq {\mathrm{PSh}}_S[W^{-1}](U,A)$.

Remark: covers versus hypercovers

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.

Definition: plus-construction

For $A\in \mathrm{PSh}(S)$ a presheaf, the plus-construction on $A$ is the presheaf

$$A^{+}:X\mapsto {\mathrm{colim}}_{(Y\hookrightarrow X)\in W}A(Y)$$A^+ : X \mapsto colim_{(Y \hookrightarrow X) \in W } A(Y)

where the colimit is over all dense monomorphisms (instead of over all local isomorphisms as for sheafification $\overline{A}$).

Remark: plus-construction versus sheafification

In general $A^{+}$ is not yet a sheaf. It is howver in general closer to being a sheaf than $A$ is, in that it is a separated presheaf.

But applying the plus-construction twice yields the desired sheaf

$$(A^{+}{)}^{+}=\overline{A}.$$(A^+)^+ = \bar A \,.

This is essentially due to the fact that in the context of ordinary sheaves discussed here, all hypercovers are already of the form

$$\mathrm{colim}(W\overrightarrow{\to }U)$$colim(W \stackrel{\to}{\to} U)

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.

Examples

The archetypical example of sheaves are sheaves of functions:

• for $X$ a topological space, $ℂ$ a topological space and $O(X)$ the site of open subsets of $X$, the assignment $U\mapsto C(U,ℂ)$ of continuous functions from $U$ to $ℂ$ for every open subset $U\subset X$ is a sheaf on $O(X)$.

• for $X$ a complex manifold and $ℂ$ a complex manifold, the assignment $U\mapsto C_{\mathrm{hol}}X,ℂ$ of holomorphic functions in a sheaf.

Related concepts

• presheaf / sheaf / cosheaf

• 2-sheaf / stack

• (∞,1)-sheaf / ∞-stack

• abelian sheaf cohomology

  • soft sheaf

  • fine sheaf

  • flabby sheaf

• descent

  • cover

  • cohomological descent

  • descent morphism

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

References

Section C2 in

• Peter Johnstone, Sketches of an Elephant

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

The book by Kashiwara and Shapira discusses sheaves with motivation from homological algebra, abelian sheaf cohomology and homotopy theory, leading over in the last chapter to the notion of stack.

• Kashiwara, Shapira, Categories and Sheaves, Springer

A quick pedagogical introduction with an eye towrds the generalization to (∞,1)-sheaves is in

• Dan Dugger, Sheaves and Homotopy Theory (pdf)