Section 6.9 (0071): Sheaves of algebraic structures

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 4.14.11. 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:

$F$ is faithful,
$\mathcal{C}$ has limits and $F$ commutes with them, and
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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The Stacks project

6.9 Sheaves of algebraic structures

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