Rapid tour through sheaves

Let $X$ be a topological space. We have a category $\operatorname{Opens}(X)$ whose objects are open subsets and $$\operatorname{Hom}(U,V) = \begin{cases} U \to V & U \subseteq V \\ \emptyset & U \not \subseteq V \end{cases}$$

Definition. Let $\mathcal C$ be a category. A pre-sheaf valued in $\mathcal C$ $\mathcal F$ on $X$ is a functor $$\mathcal F : \operatorname{Opens}(X)^{op} \to \mathcal C$$ A morphisms of pre-sheaves is a natural transformation of the functors.

We can unpack this as follows:

• For every open $U$ we have an object $\mathcal F(U)$ of $\mathcal C$. We can this sections of $\mathcal F$ over $U$ and will also write this as $\Gamma(U,\mathcal F)$.
• If $U \subseteq V$, we have a morphism $$r_{VU} : \mathcal F(V) \to \mathcal F(U)$$ with $r_{UU} = 1_{\mathcal F(U)}$. We usually call this map restriction from $V$ to $U$ and is also often denoted by $$r_{VU}(s) = s|_U$$
• If $U \subseteq V \subseteq W$, the diagram

commutes.

For a morphism $f : \mathcal F \to \mathcal G$ we have morphisms $$f(U) : \mathcal F(U) \to \mathcal G(U)$$ with the commutative diagrams

Generally, when we just say pre-sheaf without specifying the values, we will mean sets or abelian groups.

Examples

• Take an object $C$ from $\mathcal C$, then we have the constant pre-sheaf $\underline{C}$ $$\underline{C}(U) = C \\ r_{VU} = 1_C.$$

• Let $C_x$ be a set of objects of $\mathcal C$ indexed by $X$. We have the pre-sheaf $$U \mapsto \prod_{x \in U} C_x$$ where $$r_{VU} : \prod_{x \in V} C_x \to \prod_{x \in U} C_x$$ is the projection onto the components with $x \in U$.

• We have the pre-sheaf of continuous functions $$\mathcal O(U) = \lbrace f : X \to Y \mid f \text{ continuous} \rbrace$$ with target $Y$. If $Y$ is a topological ring, then $\mathcal O$ is a pre-sheaf of rings.

• Fix $x \in X$ and $C \in \mathcal C$. Assume that $\mathcal C$ has a terminal object $\ast$. Then we have $$i_{x \ast} \underline{C} (U) = \begin{cases} C & x \in U \\ \ast & x \not \in U. \end{cases}$$ This works, in particular, if $\mathcal C$ is an abelian category.

Definition. A pre-sheaf $\mathcal F$ is a sheaf if given an open cover $U_\alpha$ of $V$ and sections $s_\alpha \in \mathcal F(U_\alpha)$ satisfying $$s_{\alpha}|_{U_\alpha \cap U_\beta} = s_{\beta}|_{U_\alpha \cap U_\beta}$$ for all $\alpha, \beta$, there exists a unique $s \in \mathcal F(V)$ with $$s|_{U_\alpha} = s_\alpha$$ for all $\alpha$.

Morphisms of sheaves are morphisms of the underlying pre-sheaves.

We have a forgetful functor $$\operatorname{Sh}(X,\mathcal C) \to \operatorname{PreSh}(X,\mathcal C)$$ which often has a left adjoint. To give a (somewhat) explicit description, we need an auxiliary notion.

Definition. Let $x \in X$ and $\mathcal F$ a pre-sheaf on $X$ with values in $\mathcal C$. Then the stalk of $\mathcal F$ at $x$ is $$\mathcal F_x = \operatorname{colim}_{x \in U} \mathcal F(U)$$

In general, this need not exist. For common targets, like $\operatorname{Ab}$, this is not an issue.

Assume that $\mathcal C$ has small colimits and limits. Let $$\widetilde{\mathcal F}(U) \subset \prod_{x \in U} \mathcal F_x$$ where $(s_x)$ is a section of $\widetilde{\mathcal F}$ over $U$ if there exists a cover $V_\alpha$ of $U$ and sections $s_\alpha \in \mathcal F(V_\alpha)$ with $$(s_\alpha)_y = s_y$$ for all $y \in V_\alpha$. We call $\widetilde{F}$ the sheafification of $\mathcal F$.

Proposition. The assignment $$\mathcal F \mapsto \widetilde{\mathcal F}$$ is a functor $$\operatorname{PreSh}(X,\mathcal C) \to \operatorname{Sh}(X,\mathcal C)$$ which is left adjoint to the forgetful functor $$\operatorname{Sh}(X,\mathcal C) \to \operatorname{PreSh}(X,\mathcal C)$$

Proof. (Expand to view)

Note that $$U \mapsto \prod_{x \in U} C_x$$ is a sheaf so we only need to check that the condition: "there exists a cover $V_\alpha$ of $U$ and sections $s_\alpha \in \mathcal F(V_\alpha)$ with $$(s_\alpha)_y = s_y$$ for all $y \in V_\alpha$" cuts out a subsheaf. We leave the reader to convince themselves of this. Next, one checks that if $\mathcal G$ is a sheaf then any map $\phi: \mathcal F \to \mathcal G$ admits a unique factorization

Assume that $\mathcal A$ is an abelian category and sheafification exists. Then we have a an abelian category structure on $\operatorname{PreSh}(X,\mathcal A)$ where $$0 \to \mathcal F \to \mathcal G \to \mathcal H \to 0$$ if exact if each $$0 \to \mathcal F(U) \to \mathcal G(U) \to \mathcal H(U) \to 0$$ is exact.

Given a map $\phi : \mathcal F \to \mathcal G$, then $\ker \phi$ is a sheaf in general but $\operatorname{cok} \phi$ is not.

We transfer the abelian category structure to $\operatorname{Sh}(X, \mathcal A)$ using sheafification. Precisely, in $\operatorname{Sh}(X, \mathcal A)$ $$\operatorname{cok} \phi (U) := \widetilde{\operatorname{cok} \phi(U)}$$

We lose exactness on open sets in general but still have a simple criteria for exactness in $\operatorname{Sh}(X, \mathcal A)$.

Proposition. A sequence of sheaves $$\mathcal F \to \mathcal G \to \mathcal H$$ is exact if and only if for each $x \in X$ the sequence of stalks $$\mathcal F_x \to \mathcal G_x \to \mathcal H_x$$ is exact.

Proof. (Expand to view)

We note that $$\widetilde{\mathcal F}_x = \mathcal F_x$$ for all $x \in X$. One checks that exactness reduces to this. We leave the details to the reader.

If $\mathcal A$ has enough injectives then so does $\operatorname{Sh}(X,\mathcal A)$. For each $x \in X$, we choose some injection $$\mathcal F_x \to \mathcal I(x)$$ to get $$\widetilde{F} \to \prod \mathcal F_x \to \prod \mathcal I(x)$$ which still has no kernel.

One then checks that

Lemma. The sheaf $\prod \mathcal I(x)$ is injective in $\operatorname{Sh}(X,\mathcal A)$.

As a consequence, we can define right derived functors. One of particular interest is global sections.

The functor \begin{aligned} \Gamma(X,-) : \operatorname{Sh}(X,\mathcal A) & \to \mathcal A \\ \mathcal F & \mapsto \mathcal F(X) \end{aligned} is left-exact but not right exact in general.

Definition. The i-th (sheaf) cohomology of $\mathcal F$ is the i-th right derived functor of global sections $$H^i(X,\mathcal F) := \mathbf{R}^i\Gamma(X,\mathcal F)$$

One very useful computational tool is the following.

Proposition. Assume we have an exact sequence of sheaves $$0 \to \mathcal F \to \mathcal G \to \mathcal H \to 0$$ then we get a long exact sequence of cohomologies $$0 \to H^0(X,\mathcal F) \to H^0(X,\mathcal G) \to H^0(X,\mathcal H) \to H^1(X,\mathcal F) \to \cdots$$

Proof. (Expand to view)

We have seen that right derived functors are homological and we know that $$0 \to \mathcal F \to \mathcal G \to \mathcal H \to 0$$ provides a triangle in $D(\operatorname{Sh}(X,\mathcal A))$.

Global sections is special case of a more general functor between categories of sheaves coming from continuous maps of topological spaces.

Definition. Given a continuous map $f: X \to Y$, we have the pushforward $$f_\ast : \operatorname{Sh}(X,\mathcal C) \to \operatorname{Sh}(X,\mathcal C)$$ where $$f_\ast \mathcal F(U) := F(f^{-1}U)$$

Note that for the map $f: X \to \ast$ to the one-point space we have $$f_\ast \mathcal F(\ast) = \Gamma(X,\mathcal F)$$ So the derived pushforward $\mathbf{R}f_\ast$ is generalization of cohomology which can be heuristically viewed as “cohomology along the fibers of $f$”.

Definition. Given a sheaf $\mathcal G$ on $Y$, we also have the pre-sheaf on $X$ $$U \mapsto \operatorname{colim}_{f(U) \subseteq V} \mathcal G(V)$$ The inverse image sheaf under $f$ $f^{-1}\mathcal G$ is the sheafification of this pre-sheaf.

Proposition. We have an adjunction $f^{-1} \vdash f_\ast$.