Presheaves, sheaves, and the sheaf axioms (Tag D-RCCFY)
A presheaf \(\mcf\) of abelian groups on a space \(X\) assigns a group \(\mcf(U)\) to each open \(U\) and a restriction \(\res{U}{V}: \mcf(U) \to \mcf(V)\) to each inclusion \(V \subseteq U\), functorially, with \(\mcf(\emptyset) = 0\). Equivalently, \(\mcf\) is a contravariant functor from the category of open subsets of \(X\), with inclusions as morphisms, to abelian groups. More generally, a presheaf with values in a category \(\mcc\) (sets, rings, modules) is a contravariant functor from open subsets of \(X\) to \(\mcc\), and sheaves with values in \(\mcc\) are defined by the same two axioms. Elements of \(\mcf(U)\) are sections of \(\mcf\) over \(U\), and elements of \(\mcf(X)\) are global sections.
(Sheaf)
A presheaf is a sheaf if for every open \(U\) and every open cover \(\ts{U_i}\) of \(U\):
identity: a section \(s \in \mcf(U)\) with \(\ro{s}{U_i} = 0\) for all \(i\) is \(0\);
gluing: sections \(s_i \in \mcf(U_i)\) agreeing on every overlap \(U_i \intersect U_j\) come from a section of \(\mcf(U)\).
A presheaf satisfying the identity axiom is a separated presheaf.
For a separated presheaf \(\mcf\), the map \(\mcf\to\mcf^+\) to its sheafification is injective on sections; for an arbitrary presheaf, its kernel on \(\mcf(U)\) consists of the sections whose restrictions to the members of some open cover of \(U\) vanish.
(The equalizer form)
Both axioms at once say that \[ \mcf(U) \to \prod_i \mcf(U_i) \rightrightarrows \prod_{i,j} \mcf(U_i \intersect U_j) \] is an equalizer, the two maps being restriction from \(U_i\) and from \(U_j\) to the overlap. Injectivity of the first map is identity; that its image is exactly the equalizer is gluing. This equalizer condition, with fibre products \(U_i\times_UU_j\) in place of intersections, is the definition of a sheaf on a site (D-GROTHTOP).
Presheaves that are not sheaves (Tag FE-AM5Z8)
For a fixed abelian group \(A\), the constant presheaf with \(P(U)=A\) for \(U\ne\emptyset\), \(P(\emptyset)=0\), and identity restrictions between nonempty open sets fails the gluing axiom. On a disconnected \(U = U_1 \sqcup U_2\) the sections \(a \in A\) on \(U_1\) and \(b \neq a\) on \(U_2\) agree on the empty overlap, and no element of \(P(U)=A\) restricts to both. Its sheafification is the sheaf of locally constant \(A\)-valued functions.
(Bounded functions)
Let \(C\) be the sheaf of continuous real functions on \(\RR\) and \(B\subseteq C\) the presheaf of bounded continuous functions. \(B\) satisfies the identity axiom, being a subpresheaf of a sheaf, and fails gluing: the restrictions of \(f(x)=x\) to the intervals \((n-1,n+1)\) are bounded and agree on overlaps, while \(f\) is unbounded. The presheaf quotient \(U\mapsto C(U)/B(U)\) fails the identity axiom: the class of \(f(x)=x\) on \(\RR\) is nonzero, and its restriction to each interval \((n-1,n+1)\) is zero.
(Tensor presheaf)
On \(X = \PP^1_k\) take \(\mcf = \OO(1)\) and \(\mcg = \OO(-1)\). The presheaf \(P(U) = \mcf(U) \tensor_{\OO_X(U)} \mcg(U)\) has \(P(X) = k^2 \tensor_k 0 = 0\). On each standard chart \(U_i = D_+(x_i)\) both sheaves are trivial, so \(P(U_i) \cong \OO(U_i)\), and the sections \(x_0 \tensor x_0^{-1}\) on \(U_0\) and \(x_1 \tensor x_1^{-1}\) on \(U_1\) agree on \(U_0 \cap U_1\), since both equal the unit section there. They do not glue to a section of \(P(X) = 0\), so \(P\) is not a sheaf; its sheafification is \(\OO(1) \tensor \OO(-1) \cong \OO_X\), with \(H^0 = k\).
Given \(\varphi : \mcf \to \mcg\) of sheaves, the presheaf \(U \mapsto \mcg(U)/\varphi(\mcf(U))\) need not be a sheaf, and the sheaf cokernel is its sheafification (D-A7LCT); the presheaf \(U \mapsto \ker \varphi_U\) is a sheaf. A sequence of sheaves is exact if and only if it is exact on every stalk; a surjection of sheaves need not be surjective on sections over every open set, as \(\exp\colon\OO\to\OO^\times\) on \(\CC\sm\ts{0}\) shows.
The presheaf of bounded real functions on \(\RR\) satisfies the identity axiom and fails gluing: the restrictions of \(x\) to the intervals \((-n,n)\) are bounded and compatible, and they glue only to the unbounded function \(x\). The presheaf quotient of all real functions by the bounded ones fails the identity axiom: the class of \(x\) is nonzero on \(\RR\) and zero on each \((-n,n)\).
Stalks and germs (Tag D-0QSI0)
For a presheaf \(\mcf\) on \(X\) and a point \(p\), \[ \mcf_p \da \colim_{U \ni p} \mcf(U) , \] the colimit over open neighbourhoods of \(p\) ordered by reverse inclusion. An element of \(\mcf_p\) is a germ of a section at \(p\): a pair \((U, s)\) with \(s \in \mcf(U)\), where two pairs are identified when the sections agree on some smaller neighbourhood.
The open neighbourhoods of \(p\) form a directed set under reverse inclusion, so the colimit is filtered. Filtered colimits of abelian groups are exact, so \(\mcf\mapsto\mcf_p\) is an exact functor on sheaves of abelian groups. A sequence of sheaves of abelian groups on \(X\) is exact if and only if it is exact on the stalk at every point of \(X\); in particular, a morphism of sheaves is injective, surjective, or an isomorphism if and only if it is so on every stalk.
For the sheaf \(C^\infty\) of smooth functions on \(\RR\), the map \(C^\infty(\RR)\to C^\infty_0\) is not injective: a nonzero smooth function supported in \([1,2]\) has germ \(0\) at \(0\). The germ of \(f\) at \(0\) determines every derivative \(f^{(k)}(0)\), but the derivatives do not determine the germ: the function equal to \(e^{-1/x^2}\) for \(x>0\) and to \(0\) for \(x\le0\) has Taylor series \(0\) at \(0\) and a nonzero germ at \(0\).
Comma categories, and colimits as initial cocones (Tag D-COMMACAT)
Let \(S \colon \mathsf{A} \to \mathsf{C}\) and \(T \colon \mathsf{B} \to \mathsf{C}\) be functors. The comma category \(S \downarrow T\) has as objects the triples \((A, B, h)\) with \(A \in \mathsf{A}\), \(B \in \mathsf{B}\) and \(h \colon S(A) \to T(B)\) a morphism of \(\mathsf{C}\). A morphism \((A_0, B_0, h_0) \to (A_1, B_1, h_1)\) is a pair \((f, g)\) of morphisms \(f \colon A_0 \to A_1\) in \(\mathsf{A}\) and \(g \colon B_0 \to B_1\) in \(\mathsf{B}\) such that \(h_1 \circ S(f) = T(g) \circ h_0\):
Composition is componentwise.
(Slice categories)
Let \(\mathsf{pt}\) be the category with one object \(\ast\) and only its identity morphism, \(X \in \mathsf{A}\), and \(T_X \colon \mathsf{pt} \to \mathsf{A}\) the functor with \(T_X(\ast) = X\). Then \(\id_{\mathsf{A}} \downarrow T_X\) has objects \((A, \ast, h \colon A \to X)\) and morphisms \((f, \id_\ast)\) with \(h_1 \circ f = h_0\), so it is the slice category \(\mathsf{A}_{/X}\) of objects over \(X\).
(Cocones)
Let \(F \colon \mathsf{J} \to \mathsf{C}\) be a functor, and let \(\Delta \colon \mathsf{C} \to \operatorname{Fun}(\mathsf{J}, \mathsf{C})\) be the diagonal functor, sending \(N\) to the constant functor \(\Delta(N)\) with value \(N\) on objects and \(\id_N\) on morphisms. A cocone over \(F\) is an object \(N \in \mathsf{C}\) with a natural transformation \(\psi \colon F \to \Delta(N)\): morphisms \(\psi_X \colon F(X) \to N\) for \(X \in \mathsf{J}\) with \(\psi_Y \circ F(u) = \psi_X\) for every \(u \colon X \to Y\) in \(\mathsf{J}\). The category of cocones over \(F\) is the comma category \(\ast_F \downarrow \Delta\), also written \(F \downarrow \Delta\), where \(\ast_F \colon \mathsf{pt} \to \operatorname{Fun}(\mathsf{J}, \mathsf{C})\) picks out \(F\); its objects are the cocones \((N, \psi)\), and a morphism \((N, \psi) \to (N', \psi')\) is a morphism \(g \colon N \to N'\) with \(g \circ \psi_X = \psi'_X\) for all \(X\).
A colimit of \(F\) (D-5MX7E) is exactly an initial object of the category of cocones over \(F\). In particular a colimit is unique up to unique isomorphism.
For a presheaf \(\mathcal{F}\) on a space \(Y\) and a point \(p \in Y\), the stalk \(\mathcal{F}_p\) (D-0QSI0) is the colimit of the functor \(U \mapsto \mathcal{F}(U)\) on the category of open neighbourhoods of \(p\) with the opposite of inclusion, and the maps \(\mathcal{F}(U) \to \mathcal{F}_p\), \(s \mapsto s_p\), form the initial cocone.
Sheafification (Tag T-3VX80)
Every presheaf \(\mcf\) admits a morphism \(\theta : \mcf \to \mcf^+\) to a sheaf, universal among morphisms from \(\mcf\) to sheaves. The map \(\theta\) is an isomorphism on every stalk.
The construction is the reason the stalk statement is free: \(\mcf^+(U)\) is the set of functions \(s: U \to \coprod_{p \in U} \mcf_p\) with \(s(p) \in \mcf_p\) that are locally represented by sections of \(\mcf\). Nothing is added to a stalk, because a stalk is already a local object; what is added is the ability to assemble, and what is removed are the sections that were locally zero.
Sheafification is left adjoint to the inclusion of sheaves into presheaves, so it preserves colimits and the inclusion preserves limits. That is the abstract reason kernels of sheaf morphisms need no sheafification and cokernels do.
Kernel, image and cokernel sheaves, and which need sheafifying (Tag D-A7LCT)
Let \(\varphi : \mcf \to \mcg\) be a morphism of sheaves.
The kernel is \(U \mapsto \ker \varphi(U)\), which is already a sheaf.
The image is the sheafification of \(U \mapsto \im \varphi(U)\).
The cokernel is the sheafification of \(U \mapsto \mcg(U)/\im \varphi(U)\).
A section of \(\mcf\) lies in \(\ker\varphi(U)\) if and only if its restrictions to the members of an open cover of \(U\) do, so the kernel presheaf satisfies both sheaf axioms. A section \(t\in\mcg(U)\) lies in \(\im\varphi(U)\) when there exists \(s\in\mcf(U)\) with \(\varphi(s)=t\); local preimages on an open cover of \(U\) need not agree on overlaps, so they need not glue to a preimage on \(U\).
The image presheaf is a subpresheaf of the sheaf \(\mcg\), so it satisfies the identity axiom; it can fail gluing, and sheafification adds the sections of \(\mcg\) that lie in the image locally. The cokernel presheaf can fail both axioms; sheafification sends to \(0\) the sections that are locally zero.
(The image presheaf of the exponential map)
On \(X=\CC\sm\ts{0}\), let \(\varphi=\exp\colon\OO_X\to\OO_X^\times\). The section \(z\in\OO_X^\times(X)\) has a logarithm on every simply connected open subset of \(X\) but none on \(X\). So \(z\) is not in the image presheaf on \(X\), and its class in the cokernel presheaf on \(X\) is nonzero but locally zero. The image sheaf is \(\OO_X^\times\) and the cokernel sheaf is \(0\).
The map \(\mcf\to\mcf^+\) induces an isomorphism on every stalk. For a morphism \(\varphi\colon\mcf\to\mcg\) of sheaves, the presheaf kernel \(U\mapsto\ker\varphi_U\) is a sheaf; the presheaf cokernel \(U\mapsto\mcg(U)/\varphi(\mcf(U))\) need not be, and the sheaf cokernel is its sheafification.
Sheafification as a space over \(X\)
The espace étalé of \(\mcf\) is \(\coprod_{x\in X}\mcf_x\) with the topology for which the projection to \(X\) is a local homeomorphism and each \(s\in\mcf(U)\) gives an open section \(x\mapsto s_x\); \(\mcf^+(U)\) is the set of its continuous sections over \(U\).
The espace étalé, and sheafification as continuous sections (Tag D-VJFAP)
Let \(\mcf\) be a presheaf on \(X\). Set \[ \Et(\mcf) \da \disjoint_{p \in X} \mcf_p , \qquad \pi : \Et(\mcf) \to X , \] where \(\pi\) sends every germ in \(\mcf_p\) to \(p\).
Each \(s \in \mcf(U)\) gives a set-theoretic section \(\bar s : U \to \Et(\mcf)\), \(p \mapsto s_p\), with \(\pi \circ \bar s = \id_U\). Give \(\Et(\mcf)\) the finest topology making every such \(\bar s\) continuous.
For that topology, \[ \mcf^+(U) = \ts{ \text{continuous } t : U \to \Et(\mcf) \st \pi \circ t = \id_U } . \]
The sets \(\bar s(U)\), for \(U\) open and \(s \in \mcf(U)\), form a basis of the topology, and each \(\bar s\) is a homeomorphism of \(U\) onto the open set \(\bar s(U)\), with inverse \(\pi|_{\bar s(U)}\). Hence \(\pi\) is a local homeomorphism.
(Sheaves as spaces over $X$)
In T-3VX80, \(\mcf^+(U)\) is the set of compatible families of germs; the proposition identifies these with the continuous sections of \(\pi\). The functor \(\mcf\mapsto(\Et(\mcf)\to X)\) restricts to an equivalence between sheaves on \(X\) and spaces over \(X\) whose projection is a local homeomorphism; the inverse sends such a space to its sheaf of continuous sections.
Since \(\Et(\mcf)\) is built from the stalks, \((\mcf^+)_p = \mcf_p\) for every \(p\). The map \(\mcf \to \mcf^+\) is an isomorphism if and only if \(\mcf\) is a sheaf.
For an abelian group \(A\), the constant presheaf \(U\mapsto A\) and the constant sheaf \(\ul{A}\) have the same espace étalé, \(X \times A\) with \(A\) discrete; its continuous sections over \(U\) are the locally constant functions \(U\to A\), which are the sections of \(\ul{A}\).
From this construction, \(\mcf\) and \(\mcf^+\) have the same stalks, and \(\mcf\to\mcf^+\) is an isomorphism exactly when \(\mcf\) is a sheaf.
Direct limits of sheaves, and the Noetherian hypothesis that removes the sheafification (Tag PR-IP6ZG)
For a direct system \(\ts{\mcf_i}\) of sheaves on \(X\) over a directed index set, \(\colim_i \mcf_i\) is the sheafification of \[ U \longmapsto \colim_i \mcf_i(U) . \] Inverse limits are defined the same way, and there the presheaf \(U \mapsto \lim_i \mcf_i(U)\) is already a sheaf.
If \(X\) is a Noetherian topological space, the presheaf \(U \mapsto \colim_i \mcf_i(U)\) is already a sheaf, and \[ \Gamma\qty{X ; \colim_i \mcf_i} = \colim_i \Gamma(X ; \mcf_i) . \]
On a Noetherian space every open set is quasicompact, so it suffices to verify the sheaf axioms against finite covers. Both axioms together say that \[ \mcf(U) \to \prod_{i=1}^n \mcf(U_i) \rightrightarrows \prod_{i,j} \mcf(U_i \intersect U_j) \] is an equalizer, and for a finite cover these are finite products, so the whole diagram is a finite limit. Filtered colimits of abelian groups are exact and commute with finite limits, so the colimit of the equalizer diagrams is again an equalizer. The displayed identity is that statement for the cover consisting of \(X\) itself, once the presheaf colimit is known to be the sheaf colimit.
Both the hypothesis and its absence are the point. \(\Gamma\) is only left exact in general, and a colimit is a right-exact operation, so there is no reason for them to commute; the Noetherian condition supplies quasicompactness, which converts the sheaf axioms into a finite limit, and filtered colimits commute with those.
Dropping the hypothesis breaks the conclusion. A section of the colimit is locally a section of some \(\mcf_i\), but the index \(i\) may depend on the open set, and with infinitely many opens in an irreducible cover there need be no single \(i\) working everywhere — so the glued section lives in the sheafification and in no \(\mcf_i\). Quasicompactness is exactly what bounds the number of indices to be reconciled.
The same argument, run on Čech complexes, gives the cohomological version: on a Noetherian space \(H^k(X ; \wait)\) commutes with filtered direct limits of sheaves.
The noetherian hypothesis is needed. On the discrete space \(X=\NN\), let \(\mcf_i\) be the sheaf with stalk \(\ZZ\) at the points \(n\le i\) and \(0\) elsewhere. Then \(\colim_i\mcf_i\) is the constant sheaf \(\ZZ\), with \(\Gamma(X,\ZZ)=\prod_n\ZZ\), while \(\colim_i\Gamma(X,\mcf_i)=\bigoplus_n\ZZ\).