nLab
sheaf

Idea

A sheaf is a presheaf that satisfies descent.

A gentle, detailed introduction to the basic ideas of the notion “sheaf” is at

• motivation for sheaves, cohomology and higher stacks.

A presheaf can be regarded as an assignment of “sets of structures to spaces”, such that these structures can be pulled back along maps of spaces. A presheaf is a sheaf if this assignment satisfies descent: if $\pi :Y\to X$ is a cover of a space $X$ by a space $Y$, then the collection of structures assigned to $X$ is isomorphic to the collection of structures assigned to $Y$ which glue and hence descend from $Y$ down to $X$.

Generally, to any hypercover or local isomorphism $Y\to X$ and presheaf $A$ there is associated a descent set $\mathrm{Desc}(Y,A)$ whose objects are the elements of $A(Y)$ which glue. There is also a canonical morphism

$$\mathrm{Hom}(X,A)=A(X)\to \mathrm{Desc}(Y,G)=\mathrm{Hom}(Y,A).$$Hom(X,A) = A(X) \to Desc(Y,G) = Hom(Y,A) \,.

A presheaf is a sheaf if this canonical morphism is an isomorphism, i.e. which is a local object with respect to local isomorphisms (or already for dense monomorphisms).

Remarks

• The categorifications of sheaves are

  • stacks;
  • infinity-stacks.

Here the descent set ($0$-category) is replaced by a descent category or $\mathrm{\infty }$-category. Notably in the context of (infinity,1)-category theory the (infinity,1)-category of (infinity,1)-sheaves has literally the same definition as the category of sheaves itself as a geometric embedding into the corresponding collection of presheaves.

Definition

There are many different equivalent aspects of the definition of sheaf.

Let $S$ be a site.

A sheaf (of sets) is a presheaf (of sets) in $S$ that satisfies descent with respect to the corresponding Grothendieck topology in that it is a local object with respect to the local isomorphisms defined by the Grothendieck topology. For this it is sufficient to be a local object with respect to the dense monomorphisms into representables, as described in more detail below. A dense monomorphism into a representable is a covering sieve, namely a morphism of the form

$$\mathrm{colim}(U{\times }_{X}U\overrightarrow{\to }U)\to X$$colim( U \times_X U \stackrel{\to}{\to} U) \to X

where

$U={\coprod }_{\alpha }{U}_{\alpha }$ is the coproduct of a covering family $\{U_{\alpha }\}$ for $X$ of objects in $S$ as presheaves (the Yoneda embedding $S\to \mathrm{PSh}(S)$ is implicit throughout).

So a presheaf $A$ is local with respect to this morphism if there is a bijection of sets

$$\mathrm{Hom}(X,A)\simeq \mathrm{Hom}(\mathrm{colim}(U{\times }_{X}U\overrightarrow{\to }U)).$$Hom(X,A) \simeq Hom(colim( U \times_X U \stackrel{\to}{\to} U) ) \,.

By the defining universal property of the colimit (coproduct and coequalizer in this case) it follows that the hom-set on the right consists of

• morphisms $U\to A$, wich by the coproduct property are the same as collections of morphisms $\{U_{\alpha }\to A\}$, which by the Yoneda lemma are the same as collections of elements $\{s_{\alpha }\in A(U_{\alpha })\}$;

• such that the two pullbacks to $U\times XU={\coprod }_{\alpha ,\beta }{U}_{\alpha }{\times }_X{U}_{\beta }$ coincide; which means in terms of the $\{s_{\alpha }\}$ that these glue on double overlaps in that $s_{\alpha }{\mid }_{U_{\alpha ,\beta }}=s_{\beta }{\mid }_{U_{\alpha ,\beta }}$.

This is the sheaf condition as one often sees it stated in the literature, especially for the case that $S$ is a category of open subsets of some topological space.

The category of sheaves $\mathrm{Sh}(S)$ is the full subcategory of $\mathrm{PSh}(S)=[S^{\mathrm{op}},\mathrm{Set}]$ on presheaves that are sheaves. The category of sheaves is a topos and the full and faithful functor

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

is a geometric embedding of topoi. The left exact left adjoint $f^{*}:\mathrm{PSh}(S)\to \mathrm{Sh}(S)$ is sheafification.

Conversely, every category of sheaves, hence every Grothendieck topology arises this way. So Gorthendieck topologies and their corresponding categories of sheaves can alternatively be defined as geometric embeddings into categories of presheaves.

The notion of sheaf and of sheafification makes sense for presheaves with values in some classes of categories $A$ other than sets, notably for $A$ a Grothendieck category. Such abelian sheaves give rise to abelian sheaf cohomology.

Remarks

• The definition of category of sheaves as geometric embedding into presheaf categories is the one that generalizes straightforward to an (infinity,1)-categorical context where it yields a notion of infinity-stack in terms of (infinity,1)-category of (infinity,1)-sheaves.

Explicit description

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

In terms of geometric embedding

Here 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 morpphism $Y\to X$ 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

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

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 $W$

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

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.

In terms of sieves

One formalization of the descent set is obtained from representing a cover $Y$ by the corresponding presheaf $S_Y:C^{\mathrm{op}}\to \mathrm{Set}$ – a sieve – and defining the descent set as

$$\mathrm{Desc}(Y,G):=[S^{\mathrm{op}},\mathrm{Set}](S_Y,G).$$Desc(Y,G) := [S^{op}, Set](S_Y, G) \,.

So: given a site $(C,J)$, a sheaf over that site is a presheaf

$$G:C^{\mathrm{op}}\to \mathrm{Set}$$G: C^{op} \to Set

such that whenever $i:F\hookrightarrow \mathrm{hom}(-,c)$ is a covering sieve, the map

$$G(c)\stackrel{\mathrm{Yoneda}}{\cong }{\mathrm{Set}}^{C^{\mathrm{op}}}(\mathrm{hom}(-,c),G)\stackrel{{\mathrm{Set}}^{C^{\mathrm{op}}}(i,G)}{\to }{\mathrm{Set}}^{C^{\mathrm{op}}}(F,G)$$G(c) \stackrel{Yoneda}{\cong} Set^{C^{op}}(\hom(-, c), G) \stackrel{Set^{C^{op}}(i, G)}{\to} Set^{C^{op}}(F, G)

is an isomorphism.

In terms of orientals

When generalizing sheaves to stacks and then to infinity-stacks the Hom-set ${\mathrm{Set}}^{C^{\mathrm{op}}}(F,G)$ in the above definition, representing the descent set (0-category), needs to be generalized to a corresponding category or infinity-category of infinity-functors whose objects are the $\mathrm{\infty }$-functors from the cover/sieve $F$ to the presheaf $G$, whose morphisms are homotopies between these, whose higher morphisms are the higher homotopies. The technical problem is to formalize these $\mathrm{\infty }$-categories of $\mathrm{\infty }$-functors. This is one of the central issues of higher category theory.

One partial solution to the problem has been given by Ross Street, who defined descent strict omega-categories $\mathrm{Desc}(Y,G)$ for $\omega$-category-valued presheaves $G:C^{\mathrm{op}}\to \omega \mathrm{Cat}$ in terms of orientals. The $\omega$-category $\mathrm{Desc}(Y,G)$ can be regarded as the $\omega$-category of lax $\mathrm{\infty }$-functors from the sieve associated with $Y$ to $G$.

In the case that the $\omega$-categories in questions happen to be just 0-categories this reduces to the definitin of descent 0-categories for presheaves.

Of course the machinery of orientals is overkill for just defining sheaves, but it is instructive to understand sheaves in this language, since then the generalization to stacks and infinity-stacks is straightforward.

So let $\pi :Y\to X$ be a morphism of objects of $C$ such that the pullback of $\pi$ along itself

$$\begin{array}{ccc}{Y}^{[2]}& \stackrel{{\pi }_1}{\to }& Y\\ {\downarrow }^{{\pi }_2}& & {\downarrow }^{\pi }\\ Y& \stackrel{\pi }{\to }& X\end{array}$$\array{ Y^{[2]} &\stackrel{\pi_1}{\to}& Y \\ \downarrow^{\pi_2} && \downarrow^\pi \\ Y &\stackrel{\pi}{\to}& X }

exists. Let $G:C^{\mathrm{op}}\to \mathrm{Set}\hookrightarrow \omega \mathrm{Cat}$ be a strict omega-category valued presheaf which happens to take values just in $0$-categories. Then an object (an element) in the descent $\omega$-category $\mathrm{Desc}(Y,G)$ (which is a set here) is a tuple consisting of

• an element $a\in G(Y)$;

• a morphism ${\pi }_1^{*}a\stackrel{g}{\to }{\pi }_2^{*}a$ in $G(Y^{[2]})$, which is by assumption necessarily an identity morphism: ${\pi }_1^{*}a\stackrel{=}{\to }{\pi }_2^{*}a$.

The point to notice is that the morphism ${\pi }_1^{*}a\stackrel{g}{\to }{\pi }_2^{*}a$ is the image of the first oriental $G_1=\{a\to b\}$ in $G(Y^{[2]})$.

This definition can be formalized as an end

$$\mathrm{Desc}(Y,G):={\int }_{[n]\in \Delta }\mathrm{hom}(O([n]),G(Y^n)),$$Desc(Y,G) := \int_{[n] \in \Delta} hom(O([n]), G(Y^n)) \,,

where

• $\Delta$ is the simplex category;

• $O:\Delta \to \omega \mathrm{Cat}$ are the orientals.

From this definition one obtains the canonical morphism

$$G(X)\to \mathrm{Desc}(Y,G)$$G(X) \to Desc(Y,G)

form the universal property of then end. The sheaf condition is, as before, that this canonical morphism is an isomorphism for all covers $Y\to X$.

From the above explicit characterization of $\mathrm{Desc}(Y,G)$ this is manifestly the familiar gluing condition for sheaves: the presheaf $G$ is a sheaf if the elements in $G(Y)$ which coincide (glue) on double intersections $Y^{[2]}$ correspond bijectively to the elements in $G(Y)$.

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.

References

The book by MacLane and Moerdijk has an emphasis on the topos-theoretic aspects of sheaves:

• S. MacLane, I. 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