The Stacks project

6.9 Sheaves of algebraic structures

Let us clarify the definition of sheaves of certain types of structures. First, let us reformulate the sheaf condition. Namely, suppose that $\mathcal{F}$ is a presheaf of sets on the topological space $X$. The sheaf condition can be reformulated as follows. Let $U = \bigcup _{i\in I} U_ i$ be an open covering. Consider the diagram

\[ \xymatrix{ \mathcal{F}(U) \ar[r] & \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) } \]

Here the left map is defined by the rule $s \mapsto \prod _{i \in I} s|_{U_ i}$. The two maps on the right are the maps

\[ \prod \nolimits _ i s_ i \mapsto \prod \nolimits _{(i_0, i_1)} s_{i_0}|_{U_{i_0} \cap U_{i_1}} \text{ resp. } \prod \nolimits _ i s_ i \mapsto \prod \nolimits _{(i_0, i_1)} s_{i_1}|_{U_{i_0} \cap U_{i_1}}. \]

The sheaf condition exactly says that the left arrow is the equalizer of the right two. This generalizes immediately to the case of presheaves with values in a category as long as the category has products.

Definition 6.9.1. Let $X$ be a topological space. Let $\mathcal{C}$ be a category with products. A presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $X$ is a sheaf if for every open covering the diagram

\[ \xymatrix{ \mathcal{F}(U) \ar[r] & \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) } \]

is an equalizer diagram in the category $\mathcal{C}$.

Suppose that $\mathcal{C}$ is a category and that $F : \mathcal{C} \to \textit{Sets}$ is a faithful functor. A good example to keep in mind is the case where $\mathcal{C}$ is the category of abelian groups and $F$ is the forgetful functor. Consider a presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $X$. We would like to reformulate the condition above in terms of the underlying presheaf of sets (Definition 6.5.2). Note that the underlying presheaf of sets is a sheaf of sets if and only if all the diagrams

\[ \xymatrix{ F(\mathcal{F}(U)) \ar[r] & \prod \nolimits _{i\in I} F(\mathcal{F}(U_ i)) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{(i_0, i_1) \in I \times I} F(\mathcal{F}(U_{i_0} \cap U_{i_1})) } \]

of sets – after applying the forgetful functor $F$ – are equalizer diagrams! Thus we would like $\mathcal{C}$ to have products and equalizers and we would like $F$ to commute with them. This is equivalent to the condition that $\mathcal{C}$ has limits and that $F$ commutes with them, see Categories, Lemma 4.14.10. But this is not yet good enough (see Example 6.9.4); we also need $F$ to reflect isomorphisms. This property means that given a morphism $f : A \to A'$ in $\mathcal{C}$, then $f$ is an isomorphism if (and only if) $F(f)$ is a bijection.

Lemma 6.9.2. Suppose the category $\mathcal{C}$ and the functor $F : \mathcal{C} \to \textit{Sets}$ have the following properties:

  1. $F$ is faithful,

  2. $\mathcal{C}$ has limits and $F$ commutes with them, and

  3. the functor $F$ reflects isomorphisms.

Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$. Then $\mathcal{F}$ is a sheaf if and only if the underlying presheaf of sets is a sheaf.

Proof. Assume that $\mathcal{F}$ is a sheaf. Then $\mathcal{F}(U)$ is the equalizer of the diagram above and by assumption we see $F(\mathcal{F}(U))$ is the equalizer of the corresponding diagram of sets. Hence $F(\mathcal{F})$ is a sheaf of sets.

Assume that $F(\mathcal{F})$ is a sheaf. Let $E \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be the equalizer of the two parallel arrows in Definition 6.9.1. We get a canonical morphism $\mathcal{F}(U) \to E$, simply because $\mathcal{F}$ is a presheaf. By assumption, the induced map $F(\mathcal{F}(U)) \to F(E)$ is an isomorphism, because $F(E)$ is the equalizer of the corresponding diagram of sets. Hence we see $\mathcal{F}(U) \to E$ is an isomorphism by condition (3) of the lemma. $\square$

The lemma in particular applies to sheaves of groups, rings, algebras over a fixed ring, modules over a fixed ring, vector spaces over a fixed field, etc. In other words, these are presheaves of groups, rings, modules over a fixed ring, vector spaces over a fixed field, etc such that the underlying presheaf of sets is a sheaf.

Example 6.9.3. Let $X$ be a topological space. For each open $U \subset X$ consider the $\mathbf{R}$-algebra $\mathcal{C}^{0}(U) = \{ f : U \to \mathbf{R} \mid f\text{ is continuous}\} $. There are obvious restriction mappings that turn this into a presheaf of $\mathbf{R}$-algebras over $X$. By Example 6.7.3 it is a sheaf of sets. Hence by the Lemma 6.9.2 it is a sheaf of $\mathbf{R}$-algebras over $X$.

Example 6.9.4. Consider the category of topological spaces $\textit{Top}$. There is a natural faithful functor $\textit{Top} \to \textit{Sets}$ which commutes with products and equalizers. But it does not reflect isomorphisms. And, in fact it turns out that the analogue of Lemma 6.9.2 is wrong. Namely, suppose $X = \mathbf{N}$ with the discrete topology. Let $A_ i$, for $i \in \mathbf{N}$ be a discrete topological space. For any subset $U \subset \mathbf{N}$ define $\mathcal{F}(U) = \prod _{i\in U} A_ i$ with the discrete topology. Then this is a presheaf of topological spaces whose underlying presheaf of sets is a sheaf, see Example 6.7.5. However, if each $A_ i$ has at least two elements, then this is not a sheaf of topological spaces according to Definition 6.9.1. The reader may check that putting the product topology on each $\mathcal{F}(U) = \prod _{i\in U} A_ i$ does lead to a sheaf of topological spaces over $X$.


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