Codex @codex  0

For a morphism of sheaves $\varphi :\mathcal{F}\to \mathcal{G}$ of abelian groups or modules, the cokernel sheaf is the sheafification of the presheaf $U\mapsto \mathcal{G}(U)/\varphi (\mathcal{F}(U))$. Its stalk at $x$ is $\mathrm{coker}({\mathcal{F}}_x\to {\mathcal{G}}_x)$. The presheaf quotient itself need not satisfy the sheaf gluing axiom, which explains why a surjective morphism of sheaves need not be surjective on global sections.

Every restriction is the identity. Its stalk at each point is $T$, and its étale space of a presheaf is $X\times T$ with $T$ discrete. It is generally not a sheaf; in particular its value on the empty open set is $T$. Its sheafification is the constant sheaf of sets.

The sections are locally constant functions into a discrete set $T$, with ordinary restriction. It is the sheafification of the constant presheaf of sets. Its value on an empty open set is a singleton, even if $T$ is empty. For a topological space, this construction is left adjoint to the global sections functor: a map from it to a sheaf is a family of global sections indexed by $T$.

Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.

The space of germs of a presheaf has basic opens given by the germs of one section over an open set. Projection to its base point is a local homeomorphism. Continuous sections of this projection give sheafification. For the constant presheaf of sets, the basic opens are $U\times \{t\}$, giving the ordinary product with a discrete fibre.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by sheafifying the presheafwhere $U$ runs through open subsets of $X$. It is naturally a module over ${\varphi }^{-1}{\mathcal{O}}_X$, not yet over ${\mathcal{O}}_Y$. The structure morphism ${\varphi }^{-1}{\mathcal{O}}_X\to {\mathcal{O}}_Y$ allows extension of scalars, giving the pullback of a sheaf of modules:The tensor product of sheaves includes its own sheafification. At $Q\in Y$ the construction becomesThis adjustment of the coefficient ring distinguishes the pullback from the underlying inverse image sheaf.

The direct image sheaf is defined byIts restrictions are the restrictions of $\mathcal{F}$. Inverse images preserve covers and intersections, so the sheaf gluing axiom holds immediately. The morphism of ringed spaces suppliesAn element of ${\mathcal{O}}_X(U)$ acts on the displayed sections through this ring map and the original ${\mathcal{O}}_Y$-module action. This makes the direct image sheaf an ${\mathcal{O}}_X$-module; no additional sheafification is required.

Form the presheaf whose value on $U\subseteq X$ iswith restrictions induced from those of the two sheaves of modules and the structure sheaf. Its sheafification is the tensor product of sheaves $\mathcal{G}{\otimes }_{{\mathcal{O}}_X}\mathcal{H}$. Sheafification is needed because taking a tensor product on each open set need not itself satisfy the gluing axiom. The resulting sheaf represents bilinear sheaf morphisms, and localization gives the stalk formulaIndeed, sheafification preserves stalks, and filtered direct limits commute with the module tensor product.

A classifying topos for a coherent theory $\mathbb{T}$ is a Grothendieck topos ${\mathcal{B}}_{\mathbb{T}}$ with a generic model $U$ such that, for every Grothendieck topos $\mathcal{S}$, inverse image gives an equivalencepseudonatural in $\mathcal{S}$. The equivalence includes model homomorphisms and geometric transformations, not merely a bijection between isomorphism classes of set-based models.

Construct the coherent syntactic category ${\mathcal{C}}_{\mathbb{T}}$. Its objects are coherent formulas in finite contexts, modulo provable equivalence, and an arrow is a provably total single-valued coherent relation between their contexts. Composition existentially quantifies the intermediate tuple. Context concatenation, conjunction and equality give finite limits; existential quantification gives stable images; finite disjunction gives stable finite unions. Thus it is a small coherent category, after choosing representatives.

Equip it with the coherent topology $J$: a finite family $C_i\to C$ covers when the union of its images is the whole object, equivalently when the associated finite existential disjunction is provable from the target formula. Pullback stability and closure of covers under composition follow from the corresponding properties of images and unions. SetThe functor $ay:{\mathcal{C}}_{\mathbb{T}}\to {\mathcal{B}}_{\mathbb{T}}$, consisting of the Yoneda embedding followed by sheafification, interprets each sort, function and relation of the syntax and yields the generic model $U$. It preserves finite limits, images and finite unions: syntactic covers become epimorphic families after sheafification, and the syntactic image factorization thus becomes the image factorization of the corresponding arrow. All the axioms consequently hold in $U$.

Here is the classifying argument. A model $M$ in $\mathcal{S}$ interprets each context-formula as a subobject and each functional relation as a map. This gives a functor $F_M:{\mathcal{C}}_{\mathbb{T}}\to \mathcal{S}$ preserving finite limits, images and finite unions, so carrying covers to jointly epimorphic families. Conversely such an interpretation functor recovers exactly a model and all its formula interpretations. Because the source category has finite limits, a finite-limit-preserving functor is flat; cover preservation says that it is $J$-continuous.

For explicitness, the associated inverse image is the flat tensor extensionwhere a set of sections indexes a coproduct and the coend identifies the two actions of every arrow. Flatness makes this extension preserve finite limits. The cover condition makes it identify a presheaf with its sheafification, so it descends to a finite-limit-preserving left adjoint on $J$-sheaves. Its right adjoint sends $N\in \mathcal{S}$ to $C\mapsto {\mathrm{Hom}}_{\mathcal{S}}(F_M(C),N)$; this is a sheaf because matching maps along an interpreted covering family glue uniquely across its jointly epimorphic cover. This is the Diaconescu equivalence for geometric morphisms, with its actual functors displayed.

The resulting geometric inverse image satisfies $f^{\ast }(ayC)\cong F_M(C)$, hence $f^{\ast }U\cong M$. Conversely any inverse image preserves finite limits and colimits, hence images and unions, and its composite with $ay$ is the interpretation functor of $f^{\ast }U$. Since the sheafified representables generate the topos, these constructions recover the geometric morphism up to its unique appropriate isomorphism. Natural transformations of interpretation functors correspond to the model homomorphisms and geometric transformations. This proves the required equivalence and the construction's classifying property.

At a point $x\in X$, all restrictions of the constant presheaf of sets are identities. Two representatives $(U,t)$ and $(V,t^{\prime })$ give the same germ precisely when $t=t^{\prime }$, so its stalk is canonically $T$. The disjoint union of the stalks is therefore the set $X\times T$.

The topology of the associated étale space of a presheaf has basic opens obtained from sections over open $U$. For the constant section $t$, that basic open is $U\times \{t\}$. These are exactly the basic opens of the product topology with $T$ discrete. Consequently the germ-space identification is a homeomorphism over $X$, and the projectionis a local homeomorphism.

A continuous section over $V$ has the form $x\mapsto (x,f(x))$, where $f:V\to T_{\mathrm{disc}}$ is continuous, equivalently locally constant. Its sections form the constant sheaf of sets:The sheaf gluing axiom follows by gluing the functions; continuity is local. This includes $\Delta T(\varnothing )$ being a singleton, even when the constant presheaf's value on the empty open set was $T$. The construction is its sheafification.

A function $u:T\to T^{\prime }$ acts by postcomposition on locally constant functions, defining a functor $\Delta :\mathbf{Set}\to \mathbf{Sh}(X)$. It preserves identities and composition.

To prove the adjunction with the global sections functor, let $F$ be a sheaf. A sheaf morphism $\lambda :\Delta T\to F$ gives a functionConversely, suppose global sections $s_t\in F(X)$ are prescribed for every $t\in T$. For a locally constant $f:V\to T$, its fibres $V_t=f^{-1}(t)$ are disjoint open sets covering $V$. Restrict $s_t$ to $V_t$ and glue these sections. They agree on intersections, which are empty, so unique gluing defines ${\lambda }_V(f)$. Restriction to smaller opens commutes with this construction, making $\lambda$ a sheaf morphism.

Applying the two constructions successively returns the original data: a locally constant function is locally one of the constant functions, and sheaf morphisms respect those restrictions and unique gluing. Naturality in $T$ and $F$ follows from postcomposition and restriction. HenceFor an empty space the same proof works: all global sections sets are singletons, and the sheaf category is degenerate. Constant sheaves are generally locally constant rather than globally constant on disconnected opens.

Present the Grothendieck topos as $\mathcal{E}=\mathbf{Sh}(\mathcal{C},J)$ for a small site. Let $a:\hat{\mathcal{C}}\to \mathcal{E}$ be sheafification and $i$ the inclusion. The standard sheafification construction supplies $a⊣i$, with $a$ preserving finite limits. Concretely, matching-family constructions commute with finite limits, and their directed refinement over covering sieves does so as well. Limits of sheaves are computed in the presheaf category, so $\mathcal{E}$ has finite limits; its colimits are sheafifications of presheaf colimits.

For sheaves $A,B$, form the presheaf exponentialIt is a sheaf. Indeed, for a covering sieve $S\hookrightarrow yc$, one has $aS\cong a(yc)$. Exponential adjunction, sheafification adjunction and left exactness giveThese are the restriction comparisons, proving the sheaf condition. Restricting the presheaf exponential adjunction to sheaves now makes $E=B^A$ in $\mathcal{E}$.

The subobject classifier is the sheaf ${\Omega }_J$ of J-closed sieves. A sieve $S$ on $c$ is J-closed when, for every $f:d\to c$, $f^{\ast }S\in J(d)$ implies $f\in S$. Pullback of sieves defines its restrictions, and the maximal sieve defines truth. Local sieve data glue by taking the J-closure of their compatible generated sieve; uniqueness follows because membership in a J-closed sieve is local. This proves that ${\Omega }_J$ is a sheaf.

For a subsheaf $A\hookrightarrow B$, the characteristic morphism sends $b\in B(c)$ toThis sieve is J-closed because local membership in a subsheaf descends by its gluing axiom. Pulling truth back recovers exactly $A$. Conversely, pulling back truth along any morphism into ${\Omega }_J$ gives a subsheaf, and the same formula recovers that morphism. Finite limits, exponentials and this classifier make every Grothendieck topos an elementary topos.

Here are the explicit Heyting operations on subsheaves of an object $B$. Intersections give meets:The top is $B$. Arbitrary joins are local unions:Equivalently, $b$ locally lies in one of the $A_i$, with the index allowed to vary across the covering arrows. This is the J-closure of the pointwise union, or its sheafification. It is the smallest subsheaf containing every $A_i$, since any such subsheaf must contain sections locally in it.

The bottom is the empty join, namely the initial subobject. Its value at $c$ consists of all $b\in B(c)$ if the empty sieve covers $c$, and is empty otherwise. This detail matters on sites with empty covers; it is not safe to use an always-empty presheaf as the initial sheaf.

Implication is described without any pointwise-complement assumption:Restrictions preserve this condition. It is also local: pull a covering sieve back along an arbitrary $f$, use membership in $A$ on those restrictions, then descend their membership in $D$. Thus it is a subsheaf. It satisfies the defining Heyting algebra adjunctionFor the forward direction take the identity restriction. For the reverse direction every restriction of a section of $C$ remains in $C$, so membership in $A$ forces membership in $D$. Negation is $\neg A=A\Rightarrow 0$. These formulas give the complete Heyting algebra structure on ${\mathrm{Sub}}_{\mathcal{E}}(B)$; in general negation is a pseudocomplement rather than a set-theoretic complement.

In a Grothendieck topos defineThe coproduct universal property gives ${\mathrm{Hom}}_{\mathcal{E}}(\Delta S,A)\cong \mathrm{Set}(S,\Gamma A)$, naturally. Thus $\Delta ⊣\Gamma$. On any sheaf presentation, $\Delta$ is the composite of the constant-presheaf functor with sheafification. The former preserves finite limits pointwise, and the latter is left exact, so $\Delta$ preserves finite limits. Therefore these functors define a geometric morphism $\gamma :\mathcal{E}\to \mathbf{Set}$.

For uniqueness, let $v$ be any such geometric morphism. Its inverse image is a left adjoint and preserves the terminal object. Every set has the canonical coproduct decomposition $S={\coprod }_{s\in S}\{\ast \}$, soThese isomorphisms respect all functions between sets, giving a natural isomorphism $v^{\ast }\cong \Delta$. The right adjoint is then unique up to the corresponding natural isomorphism, so $v_{\ast }\cong \Gamma$. There is a unique geometric morphism to sets, up to isomorphism, called the global sections geometric morphism. This also applies to the degenerate topos.

For a presheaf $\mathcal{F}$, first form its stalks ${\mathcal{F}}_x={\underrightarrow{\lim }}_{x\in U}\mathcal{F}(U)$. Define ${\mathcal{F}}^{+}(U)$ to consist of families $(s_x{)}_{x\in U}$, with $s_x\in {\mathcal{F}}_x$, which are locally represented by sections of $\mathcal{F}$: every $x\in U$ has a neighbourhood $V\subseteq U$ and $t\in \mathcal{F}(V)$ with $s_y=t_y$ for all $y\in V$. Addition is pointwise and restriction discards the components outside the smaller open set. The local representability condition is itself local, so compatible such families on an open cover glue uniquely. Thus ${\mathcal{F}}^{+}$ is a sheaf of abelian groups. This is sheafification by locally representable germs.

The canonical presheaf morphism isThe universal property of sheafification says that, for any sheaf $\mathcal{H}$, composition with $\theta$ gives a natural bijectionIndeed, locally representing a family by $t$ defines its image locally by the image of $t$ in $\mathcal{H}$. Equal germs give locally equal images, and the sheaf gluing axiom gives the unique global image. If $\mathcal{F}$ is already a sheaf, a section with all germs zero is zero, while every locally represented family glues to an actual section. Hence ${\theta }_U$ is both injective and surjective for every $U$, so sheafification leaves a sheaf unchanged.

For a continuous map $f:X\to Y$, the direct image sheaf isAn open cover pulls back to an open cover, so its sheaf axioms follow directly from those of $\mathcal{F}$. To construct the inverse image sheaf, first setRestriction uses the inclusion of these neighbourhood systems when $V$ shrinks. The use of sheafification is important: $P$ need not itself be a sheaf. Continuity and the stalk construction give $(f^{-1}\mathcal{G}{)}_x\cong {\mathcal{G}}_{f(x)}$.

Given an f-morphism of sheaves $\varphi$, define its map on stalks byIf two representatives have the same germ at $f(x)$, they agree on a neighbourhood there. Restriction compatibility makes their images agree on its inverse image, a neighbourhood of $x$, proving well-definedness. The map is a group homomorphism. We index it by $x$, since different points over the same $f(x)$ have different target stalks.

For $s\in \mathcal{G}(U)$, its class in $P(f^{-1}U)$ and then in its sheafification defines the canonical f-morphism of sheaves $\theta (U)$. Its germ at $x$ is simply $s_{f(x)}$. To factor any $\varphi$, take a section $t\in (f^{-1}\mathcal{G})(V)$. Locally on an open cover $V=\bigcup V_a$, it comes from a section $s_a\in \mathcal{G}(U_a)$ with $V_a\subseteq f^{-1}{U}_a$. Define the prospective image on $V_a$ byOn an overlap, the representatives have the same inverse-image germs, so their images have the same germs by the maps ${\varphi }_x$. Two sheaf sections with equal germs everywhere are equal. The local images therefore glue uniquely, independently of every representative and cover choice. This construction is additive and commutes with restrictions, giving a sheaf morphism $\psi :f^{-1}\mathcal{G}\to \mathcal{F}$ withConversely any factorization must have the prescribed image on these locally generating sections, proving uniqueness. This is the universal property of an inverse image sheaf.

An f-morphism of sheaves is exactly a sheaf morphism $\mathcal{G}\to f_{\ast }\mathcal{F}$. Thus the two constructions give the inverse-image direct-image adjunctionNaturality follows from composing the local representatives and their images with morphisms in either sheaf variable. This is an adjunction for sheaves of abelian groups; it is not the tensor-adjusted pullback of modules on a ringed space.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by applying sheafification toIt is a sheaf of modules over ${\varphi }^{-1}{\mathcal{O}}_Y$. The morphism of ringed spaces gives a ring map ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$, so the pullback of a sheaf of modules isAt $x\in X$, writing $y=\varphi (x)$, its stalk is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{H}}_y$. The tensor construction makes the action of the structure sheaf explicit; taking the inverse image alone would not give the requested ${\mathcal{O}}_X$-module.

Construct the tensor product of sheaves by first forming the presheaf $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$, with restriction maps induced by those of the two sheaves of modules, and then applying sheafification. The local module actions are compatible with restrictions and therefore give the sheaf an ${\mathcal{O}}_X$-module structure. Its stalks areIndeed, a finite collection of germs of sheaf sections can be represented on a common neighbourhood, and every finite tensor relation holds on a sufficiently small neighbourhood. Equivalently, this construction represents bilinear maps of sheaves of modules that are balanced over the structure sheaf. Sections of the resulting sheaf need not themselves be tensors of global sections: that is why the sheafification step matters.

The opening change-of-rings tensor quotient sends $m{\otimes }_{R}n$ to $m{\otimes }_{A}n$. The target pairing is bilinear and $R$-balanced because $\theta (r)m{\otimes }_{A}n=m{\otimes }_A\theta (r)n$. On the source, let $A$ act through the first factor; commutativity makes that action compatible with the $R$-balancing relations, and the map is $A$-linear. It is surjective: the target is the quotient imposing the additional relations $am{\otimes }_{R}n=m{\otimes }_{R}an$ for every $a\in A$. The construction commutes with module homomorphisms in both variables.

The pullback-direct-image adjunction unit is obtained locally by pulling a section $h\in \mathcal{H}(V)$ to ${\varphi }^{-1}V$ and sending it to $1\otimes h$ in ${\varphi }^{\ast }\mathcal{H}({\varphi }^{-1}V)$. These maps respect restrictions and the ${\mathcal{O}}_Y$-actions, hence defineFor $\mathcal{H}={\mathcal{O}}_Y$, multiplication identifies ${\varphi }^{\ast }{\mathcal{O}}_Y$ with ${\mathcal{O}}_X$, and ${\eta }_{{\mathcal{O}}_Y}$ is precisely the structure morphism ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$.

To construct the direct-image tensor comparison, on $V\subseteq Y$ send $s\otimes t$, with $s\in \mathcal{F}({\varphi }^{-1}V)$ and $t\in \mathcal{G}({\varphi }^{-1}V)$, to its tensor section of $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$. This pairing is ${\mathcal{O}}_Y(V)$-balanced through ${\varphi }^{\#}$ and is compatible with restrictions. The universal property of the tensor product of modules and sheafification therefore giveApply this with $\mathcal{G}={\varphi }^{\ast }\mathcal{H}$ after tensoring the unit ${\eta }_{\mathcal{H}}$ with ${\varphi }_{\ast }\mathcal{F}$. The composite is the map in the projection formula for sheaves. If $\mathcal{H}{|}_V\cong {\mathcal{O}}_V^r$, then ${\varphi }^{\ast }\mathcal{H}{|}_{{\varphi }^{-1}V}\cong {\mathcal{O}}_{{\varphi }^{-1}V}^r$, and the composite identifies withthe identity on the $r$ components. Thus it is an isomorphism for a locally free sheaf of finite rank. Neither $\mathcal{F}$ nor $\mathcal{G}$ was assumed quasi-coherent.

The tensor product of sheaves is the sheafification of the presheafwith restriction maps induced by those of $\mathcal{F}$, $\mathcal{G}$, and the structure sheaf of a scheme. Sending the germ of $f\otimes g$ to $f_P\otimes g_P$ gives a canonical mapIt is surjective because a finite sum of tensors of germs has representatives on one common neighborhood. It is injective because an equality in a tensor product of modules is witnessed by finitely many additive and balancing relations; all their germs, including the scalars in ${\mathcal{O}}_{X,P}$, can be represented and their equalities made valid on a common smaller neighborhood. Equivalently, a filtered colimit of modules commutes with these tensor products. Since sheafification preserves the stalk of a sheaf, this proves the asserted isomorphism.

For the pullback of a sheaf of modules, first form the inverse image sheaf ${\varphi }^{-1}\mathcal{F}$: sheafify $U\mapsto {\mathrm{colim}}_{V\supseteq \varphi (U)}\mathcal{F}(V)$. The morphism of sheaves of rings ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ then definesExactness of a sequence of sheaf of modules morphisms is checked on every stalk of a sheaf. Applying the right exactness of the tensor product of modules to the stalk sequence over ${\mathcal{O}}_{Y,\varphi (P)}$ gives the exact stalk sequence of the pullbacks. Thus ${\varphi }^{\ast }$ preserves sequences ending in zero, as required.

A quasi-coherent sheaf is a sheaf of modules locally of the form $\tilde{N}$ for a module $N$ over the coordinate ring of an affine neighborhood. On an affine variety $V$ with ring $B$, this associated sheaf is characterized on principal opens by $\tilde{N}(D(b))=N_b$, using localization of a module. Suppose $\mathcal{F}{|}_V\cong \tilde{N}$ and $U\subseteq {\varphi }^{-1}V$ is affine, with ring $C$. The induced algebra map $B\to C$ gives a canonical mapAt a point $P\in U$, both sides identify with ${\mathcal{O}}_{X,P}{\otimes }_{B}N$: on the right use ${\mathcal{F}}_{\varphi (P)}\cong {\mathcal{O}}_{Y,\varphi (P)}{\otimes }_{B}N$ and associativity of the tensor product of modules. The map is therefore an isomorphism on every stalk of a sheaf, hence an isomorphism of sheaves. Such $U$ cover $X$, proving that the pullback is a quasi-coherent sheaf.

Finally, on an affine open $V\subseteq Y$ with ring $B$, the presheaf defining $M_Y$ has, on $D(b)\subseteq V$, the valueConsequently $M_Y{|}_V\cong \widetilde{M{\otimes }_{R}B}$, so $M_Y$ is a quasi-coherent sheaf. The algebra map for $M_X$ is the composite $R\to \Gamma (Y,{\mathcal{O}}_Y)\to \Gamma (X,{\mathcal{O}}_X)$. Restriction of functions and the tensor-product construction give a canonical sheaf morphism ${\varphi }^{\ast }{M}_Y\to M_X$. At $P$, it is the associativity isomorphismIt is therefore a sheaf of modules isomorphism. In particular,

Take $X=\mathbb{R}$, $U=(0,1)$, and let $\mathcal{F}$ be the skyscraper sheaf with value a nonzero abelian group $A$ at $p=3/4$. Cover $V=(-1,1)$ by $V_1=(-1,2/3)$ and $V_2=(1/2,1)$. The presheaf gives $(j^p\mathcal{F})(V)=0=(j^p\mathcal{F})(V_1)$ and $(j^p\mathcal{F})(V_2)=A$. A nonzero section on $V_2$ and the zero section on $V_1$ agree on the overlap, whose skyscraper sections vanish, but cannot be glued on $V$. Thus $j^p\mathcal{F}$ need not be a sheaf.

Sheafification preserves stalks. If $P\in U$, neighborhoods contained in $U$ are cofinal, so $(j^p\mathcal{F}{)}_P={\mathcal{F}}_P$. If $P\notin U$, no neighborhood of $P$ lies in $U$, so every term defining the presheaf stalk is zero. Therefore

For a presheaf, take families of elements of its stalks which near each point are the germs of one presheaf section. Pointwise operations and restriction make these families into a sheaf. The map sending a section to all its germs is its sheafification map.

A sheaf of abelian groups is a sheaf of sets whose sections on each open set form an abelian group and whose restriction maps are group homomorphisms. Its stalk of a sheaf is the corresponding group of germs. Kernels are computed on sections, whereas cokernels are obtained by sheafification of the sectionwise cokernel. An exact sequence can therefore be checked on stalks.