Let X be a topological space. We have a category Opens(X) whose objects are open subsets and Hom(U,V)={U→V∅U⊆VU⊆V
Definition. Let C be a category. A pre-sheaf valued in CF on X is a functor F:Opens(X)op→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 F(U) of C. We can this sections of F over U and will also write this as Γ(U,F).
If U⊆V, we have a morphism rVU:F(V)→F(U) with rUU=1F(U). We usually call this map restriction from V to U and is also often denoted by rVU(s)=s∣U
If U⊆V⊆W, the diagram
commutes.
For a morphism f:F→G we have morphisms f(U):F(U)→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 C, then we have the constant pre-sheaf CC(U)=CrVU=1C.
Let Cx be a set of objects of C indexed by X. We have the pre-sheaf U↦x∈U∏Cx where rVU:x∈V∏Cx→x∈U∏Cx is the projection onto the components with x∈U.
We have the pre-sheaf of continuous functions O(U)={f:X→Y∣f continuous} with target Y. If Y is a topological ring, then O is a pre-sheaf of rings.
Fix x∈X and C∈C. Assume that C has a terminal object ∗. Then we have ix∗C(U)={C∗x∈Ux∈U. This works, in particular, if C is an abelian category.
Definition. A pre-sheaf F is a sheaf if given an open cover Uα of V and sections sα∈F(Uα) satisfying sα∣Uα∩Uβ=sβ∣Uα∩Uβ for all α,β, there exists a unique s∈F(V) with s∣Uα=sα for all α.
Morphisms of sheaves are morphisms of the underlying pre-sheaves.
We have a forgetful functor Sh(X,C)→PreSh(X,C) which often has a left adjoint. To give a (somewhat) explicit description, we need an auxiliary notion.
Definition. Let x∈X and F a pre-sheaf on X with values in C. Then the stalk of F at x is Fx=colimx∈UF(U)
In general, this need not exist. For common targets, like Ab, this is not an issue.
Assume that C has small colimits and limits. Let F(U)⊂x∈U∏Fx where (sx) is a section of F over U if there exists a cover Vα of U and sections sα∈F(Vα) with (sα)y=sy for all y∈Vα. We call F the sheafification of F.
Proposition. The assignment F↦F is a functor PreSh(X,C)→Sh(X,C) which is left adjoint to the forgetful functor Sh(X,C)→PreSh(X,C)
Proof. (Expand to view)
Note that U↦x∈U∏Cx is a sheaf so we only need to check that the condition: "there exists a cover Vα of U and sections sα∈F(Vα) with (sα)y=sy for all y∈Vα" cuts out a subsheaf. We leave the reader to convince themselves of this. Next, one checks that if G is a sheaf then any map ϕ:F→G admits a unique factorization
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Assume that A is an abelian category and sheafification exists. Then we have a an abelian category structure on PreSh(X,A) where 0→F→G→H→0 if exact if each 0→F(U)→G(U)→H(U)→0 is exact.
Given a map ϕ:F→G, then kerϕ is a sheaf in general but cokϕ is not.
We transfer the abelian category structure to Sh(X,A) using sheafification. Precisely, in Sh(X,A)cokϕ(U):=cokϕ(U)
We lose exactness on open sets in general but still have a simple criteria for exactness in Sh(X,A).
Proposition. A sequence of sheaves F→G→H is exact if and only if for each x∈X the sequence of stalks Fx→Gx→Hx is exact.
Proof. (Expand to view)
We note that Fx=Fx for all x∈X. One checks that exactness reduces to this. We leave the details to the reader. ■
If A has enough injectives then so does Sh(X,A). For each x∈X, we choose some injection Fx→I(x) to get F→∏Fx→∏I(x) which still has no kernel.
One then checks that
Lemma. The sheaf ∏I(x) is injective in Sh(X,A).
As a consequence, we can define right derived functors. One of particular interest is global sections.
The functor Γ(X,−):Sh(X,A)F→A↦F(X) is left-exact but not right exact in general.
Definition. The i-th (sheaf) cohomology of F is the i-th right derived functor of global sections Hi(X,F):=RiΓ(X,F)
One very useful computational tool is the following.
Proposition. Assume we have an exact sequence of sheaves 0→F→G→H→0 then we get a long exact sequence of cohomologies 0→H0(X,F)→H0(X,G)→H0(X,H)→H1(X,F)→⋯
Proof. (Expand to view)
We have seen that right derived functors are homological and we know that 0→F→G→H→0 provides a triangle in D(Sh(X,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→Y, we have the pushforwardf∗:Sh(X,C)→Sh(X,C) where f∗F(U):=F(f−1U)
Note that for the map f:X→∗ to the one-point space we have f∗F(∗)=Γ(X,F) So the derived pushforward Rf∗ is generalization of cohomology which can be heuristically viewed as “cohomology along the fibers of f”.
Definition. Given a sheaf G on Y, we also have the pre-sheaf on XU↦colimf(U)⊆VG(V) The inverse image sheaf under ff−1G is the sheafification of this pre-sheaf.