Draft lesson.

Sieves, local isomorphisms and presheaf localization

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Sheafification can identify sections that agree only locally and can create sections that exist only by gluing. Accordingly, a map need not be a bijection on its original section sets to become invertible. The relevant maps are exactly those which lift sections locally and identify sections only locally. Their pullback and pushout properties give concrete fraction descriptions of every morphism between associated sheaves.

Fix universes \(\mathcal U\in\mathcal V\), with \(\mathcal C\) a \(\mathcal U\)-small category and presheaves valued in \(\mathsf{Set}_{\mathcal U}\). No fiber products in \(\mathcal C\) are assumed. Presheaf limits and colimits exist pointwise; their factor maps are defined componentwise, and the presheaf equations follow from the uniqueness in those universal properties. Denominator categories are taken in \(\mathcal V\), which contains the category of these presheaves. Their Hom colimits will nevertheless be identified with \(\mathcal U\)-small sets. Choice is allowed.

Prerequisites are the set matching-family arguments in Twisted modules from coherent gluing, Lemma 7.1; the adjunction and finite-limit comparisons in Associated sheaves with general coefficients, Sections 4–5; Filtered stages and finite limits, Sections 1 and 4; and the incoming, outgoing and transformation interfaces of One-sided fractions and saturation, Section 5. We retain those constructions and specify their sieve and local-isomorphism maps below.

1. Covers without fiber products in the category

Write \(h_U(T)=\operatorname{Hom}_{\mathcal C}(T,U)\). A sieve \(R\subseteq h_U\) is closed under precomposition: if \(t:T\to U\) belongs to it, then so does \(tv\) for every \(v:V\to T\). For \(u:V\to U\), define the sieve \(u^*R\) by the condition \(v\in u^*R\) if and only if \(uv\in R\). This is a presheaf pullback; it exists even when no corresponding fiber product is represented in \(\mathcal C\).

A Grothendieck topology specifies covering sieves \(J(U)\). The maximal sieve covers, base change of a cover covers, and the following transitivity holds: if \(R\) covers and a sieve \(S\) has \(t^*S\) covering for every \(t\in R\), then \(S\) covers. These axioms imply upward closure and closure under finite intersections. For upward closure, pull the larger sieve back along a member of the smaller one; that pullback is maximal. For two covers \(R,S\), the pullback of \(R\cap S\) along a member \(t\in R\) is \(t^*S\), so transitivity proves that their intersection covers. The empty intersection is the maximal sieve. Empty covering sieves are allowed.

A matching family on \(R\) in a presheaf \(F\) is a natural map \(R\to F\): entries \(x_t\in F(T)\) commute with every further restriction. A sheaf is precisely a presheaf for which

is bijective for every covering sieve. The map restricts a section along all members. In particular an empty cover requires \(F(U)\) to be a singleton. None of these definitions needs an overlap object in \(\mathcal C\).

2. The retained matching construction and its local rules

The matching-family construction of Lemma 7.1 in the first prerequisite applies to sets before introducing the module operations. It forms

Here later sieves are smaller, and the transitions restrict the matching families. This is a small nonempty filtered poset: the maximal cover supplies an object and intersections give common later stages. A class restricts along \(u\) by pulling back its representing sieve and restricting its entries. The identities and composite restrictions agree on every entry, hence on the colimit. Thus plus is a functor, with unit given by all the restrictions of an original section.

Recall the exact set arguments. Equality of two classes means equality on a common covering subsieve. A plus-section locally comes from its own matching entries. If plus-sections are locally equal, choose comparison sieves on the members of that cover; transitivity and intersection combine them into a covering comparison of their original representatives. Hence plus is separated. For separated \(F\), a matching family of plus-sections has local representatives in \(F\); on the composite covering sieve any two representatives have equal plus-images, so separation makes them equal already in \(F\). They define a matching family whose class glues the prescribed sections. Plus-separation gives uniqueness. This proves that \(F^{++}\) is a sheaf. Gluing the images of matching families in any sheaf \(E\) gives the unique extension of a map \(F\to E\), first through plus and then through plus again. All equations commute with restrictions. These are exactly the matching arguments already used in the prerequisite; they require no module operations or base fiber products.

Consequently \(a\) is left adjoint to the full inclusion of sheaves. We use its actual unit \(\eta_F:F\to aF\). It has the following two rules, including when covers are empty:

1. Every section of \(aF\) is represented locally by sections of \(F\).
2. Two sections of \(F(U)\) have the same image exactly when their equality sieve is covering.

The first rule applies the plus representative rule twice and then transitivity. The second uses the common-refinement equality rule twice, with transitivity for the converse local implication. These rules also prove uniqueness and both-variable naturality in the sheafification adjunction. For a sheaf the unit is an isomorphism.

For later use \(a\) preserves finite limits. For each sieve, \(\operatorname{Nat}(R,-)\) preserves all limits, by taking compatible families of maps. The filtered colimit in (2.1) commutes with finite limits of sets by the complete comparison theorem in the third prerequisite. The resulting comparison maps respect sieve pullback and all matching entries. Plus and then double plus therefore preserve finite presheaf limits. Limits of sheaves are their presheaf limits, since (1.1) is preserved by a limit. Sheaf colimits are obtained by sheafifying presheaf colimits, by the adjunction; in particular \(a\) preserves these colimits with their specified coprojections.

3. Local section tests and the exact class inverted

For \(f:F\to G\), call it locally surjective when, for every \(y\in G(U)\), the sieve

is covering. Call it locally injective when the equality sieve of \(x,x'\in F(U)\) is covering whenever \(f(x)=f(x')\). Both displayed collections are sieves by restriction. A local isomorphism satisfies both conditions. Denote their class by \(W\).

The local rules show that

Here are all the implications. If \(af\) is injective on sections, equality of two original images forces equality of their images in \(aF\), and rule 2 gives local injectivity. Conversely, represent two sections of \(aF\) locally by original sections. If their \(af\)-images agree, rule 2 for \(G\) makes their original \(f\)-images equal on a further cover. Local injectivity then makes the original sections equal on yet another cover. Transitivity and separation of \(aF\) give equality of the two initial sections. Thus local injectivity is exactly sectionwise injectivity of \(af\).

Similarly local surjectivity is equivalent to local section-lifting for \(af\). In the forward direction, represent a section of \(aG\) locally by \(y\), and apply (3.1). In the reverse direction, start with \(\eta_G(y)\), locally lift it to \(aF\), represent those lifts by original \(x\)'s, and use rule 2 for \(G\) to make \(f(x)=y\) on a further cover. If both local tests hold, the lifts in \(aF\) are unique after restriction because \(af\) is injective. They are consequently a matching family and glue to a global lift of any section of \(aG\). Hence \(af\) is bijective on sections and has the natural inverse given by those unique lifts. Conversely an isomorphism has the lifting and injectivity properties just proved. This establishes (3.2).

These tests also justify the names local monomorphism and local epimorphism. A monomorphism of sheaves is sectionwise injective: test two sections using the maps from \(ah_U\), by Yoneda and the adjunction. The converse is pointwise cancellation. A locally surjective sheaf map is epic because two maps equal after it are equal on local lifts, hence equal by separation.

For completeness, every epic sheaf map \(q:A\to B\) is locally surjective. Let \(I\subseteq B\) be its presheaf image. Finite-limit preservation makes \(aI\to B\) monic, and the factorization of \(q\) makes it epic. Form the presheaf pushout of two copies of \(B\) over \(aI\), then sheafify. The two maps from \(B\) to this sheaf agree on \(aI\), so are equal by epimorphy. For \(b\in B(U)\), their equal images imply, by rule 2, equality of its two copies locally in the presheaf pushout. A pushout of sets along a subset identifies those copies exactly on that subset. Thus \(b\) locally belongs to \(aI\); its unique preimages glue because \(aI\) is a subsheaf. Therefore \(aI=B\). Rule 1 for \(I\) now expresses every section of \(B\) locally as an image of a section of \(A\). Applied to \(af\), this proves the categorical epi assertion as well. Local tests do not assert sectionwise surjectivity of the original presheaf map.

4. Both systems of fractions

By (3.2), \(W\) contains isomorphisms, is closed under composition, and has the two-out-of-three property. The unit \(\eta_F\) belongs to \(W\), since its two local rules give the two tests. Finite-limit preservation makes \(W\) stable under presheaf pullback; colimit preservation makes it stable under presheaf pushout. Indeed the corresponding sheaf square is the pullback or pushout of an isomorphism, with its canonical projection or coprojection an isomorphism.

The incoming Ore square for \(f:F\to G\) and \(s:K\to G\) in \(W\) is their actual presheaf pullback. Its projection \(F\times_GK\to F\) belongs to \(W\). For cancellation, if \(t f=t g\) and \(t\in W\), then \(af=ag\). The equalizer inclusion \(e:\operatorname{Eq}(f,g)\to F\) is in \(W\), since its sheafification is the equalizer of the equal maps \(af,ag\), hence the identity on \(aF\). It satisfies \(fe=ge\).

Dually, the outgoing Ore square is a presheaf pushout along a member of \(W\). If \(fs=gs\) and \(s\in W\), then \(af=ag\). The coequalizer \(q\) of \(f,g\) belongs to \(W\): after sheafification it is the coequalizer of an equal pair, canonically the identity of their target. It satisfies \(qf=qg\). Thus both fraction systems have their actual square and cancellation maps. Their construction takes place in the presheaf category, regardless of limits or colimits in \(\mathcal C\).

5. Roofs, equality and both Hom formulas

An incoming roof \(F\xleftarrow{w}H\xrightarrow{v}G\), with \(w\in W\), represents

Every sheaf map \(\alpha:aF\to aG\) has this form: use

The projection \(w:H\to F\) is a base change of \(\eta_G\), and the other projection is \(v\). The pullback square and the unit identifications give exactly \(av=\alpha aw\).

Two incoming roofs have the same image if and only if they have a common refinement with both numerators equal. For the nontrivial direction, form \(K=H\times_FH'\), with its denominator \(z:K\to F\) in \(W\). The two numerators \(v\operatorname{pr}_1,v'\operatorname{pr}_2\) have equal sheafifications by (5.1). Their equalizer \(E\to K\) is in \(W\) by Section 4. It provides the required common refinement and strict numerator equality. Conversely any such refinement has invertible sheafification and leaves (5.1) unchanged.

To compose \(F\leftarrow H\to G\) and \(G\xleftarrow{s}K\to L\), use \(H\times_GK\). The projection to \(H\) is in \(W\); compose it with the first denominator and use the other projection followed by the second numerator. Its image under (5.1) is the composite of the two images, by the pullback equation. The equality criterion proves independence of representatives. Identities are identity roofs; injectivity of (5.1) then proves associativity, including choices of pullback objects.

Let \(I_F\) have objects \(w:H\to F\) in \(W\) and arrows the ordinary maps over \(F\). It is cofiltered: pullbacks give common sources and the equalizer argument gives equalization of parallel maps. Its opposite indexes the numerator sets contravariantly. Let \(D_G\) have objects \(t:G\to K\) in \(W\) and arrows maps under \(G\). Pushouts and coequalizers make it filtered. In the stated ambient universe the actual natural bijections are

The incoming transition precomposes a numerator by a refinement \(H_{w'}\to H_w\). The outgoing transition postcomposes it by \(K_t\to K_{t'}\). Its representative is \((at)^{-1}af\). Every \(\alpha\) has the outgoing representative \(f=\alpha\eta_F:F\to aG\), \(t=\eta_G:G\to aG\). For outgoing equality, push two targets out over \(G\), then coequalize their two maps from \(F\). The latter maps have equal sheafifications, so the coequalizer is in \(W\) and gives a common later numerator. This proves both directions of equality for that colimit as well. The Hom sets in the middle are \(\mathcal U\)-small, although the denominator indices were formed in \(\mathcal V\). All comparisons are induced by the specified roofs, so commute with maps in both variables.

6. The full localization and recovery of the topology

The roof category just constructed has the presheaves as objects. Its functor \(Q\) from presheaves uses denominator identities. Every member of \(W\) becomes invertible, with inverse given by the roof using that member as denominator and the identity as numerator. Equations (5.1)–(5.3) make its functor to sheaves fully faithful. It is essentially surjective, since the unit of a sheaf is an isomorphism. Thus

and the comparison with the original presheaf functor is precisely \(a\), with its unit. This is a left exact reflective localization.

The universal property includes functors and natural transformations. If \(T\) is a functor from presheaves to any category in an adequate ambient universe and inverts \(W\), send an incoming roof to \(T(v)T(w)^{-1}\). Refinements preserve this formula because their denominator equations make their \(T\)-images invertible. In a composition square \(vp=sq\), the identity \(T(q)T(p)^{-1}=T(s)^{-1}T(v)\) proves preservation of composition. Identity roofs give identities. Every roof is a numerator followed by an inverted denominator, so the extension is unique with its restriction fixed. A transformation natural on ordinary arrows is also natural on their inverses by multiplying its denominator naturality equation by the two inverse images. It is consequently natural on every roof, uniquely, with the same components. Hence restriction gives the full functor-category localization equivalence, not merely a rule on objects or isomorphism classes.

The topology itself is recovered from \(W\):

The inclusion is always locally injective. If \(R\) covers, the lifting sieve for \(t:T\to U\) is \(t^*R\), which covers. Conversely apply local surjectivity to \(\operatorname{id}_U\) to obtain exactly \(R\in J(U)\). A presheaf \(E\) is a sheaf if and only if precomposition gives a bijection \(\operatorname{Nat}(G,E)\to\operatorname{Nat}(F,E)\) for every \(f:F\to G\) in \(W\). For a sheaf this follows from the adjunction and (3.2). Conversely apply it to the inclusions in (6.2) and use Yoneda to get (1.1). No existence of enough points, or test on topological stalks, has been assumed.

7. Four exercises with full solutions

7.1 Both topologies on a one-object trivial category

Exercise 1 (basic). Let \(\mathcal C\) have one object and only its identity. Compute its two Grothendieck topologies, sheafification, local isomorphisms and the localized Hom sets. Give an explicit inverse roof for \(\varnothing\to\{*\}\) when it is inverted.

Solution. There are two sieves, empty and maximal. The topology containing only the maximal sieve imposes no condition, so sheaves are sets, \(a\) is identity, and \(W\) consists of bijections. The local tests reduce respectively to ordinary surjectivity and injectivity. Its localization therefore retains all the ordinary function sets.

The topology also declaring the empty sieve covering makes a sheaf a singleton set, since (1.1) for the empty sieve has a singleton target. Thus the sheaf category is equivalent to the terminal category and every set sheafifies to a singleton. Both local tests hold for every function: any required lifting or equality sieve is covering, even if empty. The localized Hom set between any two sets is a singleton. An incoming common refinement can always use the empty set. In particular the inverse of \(\varnothing\to\{*\}\) is the roof \(\{*\}\leftarrow\varnothing\to\varnothing\), with that map as denominator and the empty identity as numerator. Its two composite roofs agree with identities after the empty refinement.

7.2 A site with no overlap object

Exercise 2 (intermediate). Let the only nonidentity arrows be \(a\to u\) and \(b\to u\), and let \(R\subset h_u\) consist of these two arrows. Declare \(R\) and the maximal sieve covering on \(u\), and only the maximal sieves covering on \(a,b\). Verify the topology and compute sheafification, local isomorphisms and all localized Hom sets.

Solution. Pulling \(R\) back to \(a\) or \(b\) gives the maximal sieve. A sieve whose pullbacks along both arrows cover must contain both, hence contain \(R\); this verifies transitivity, and the other axioms follow immediately from maximality. There is no object with arrows to both \(a\) and \(b\), so their fiber product over \(u\) does not exist in this category.

A presheaf is a pair of restriction maps \(F_u\to F_a,F_b\). Its matching family on \(R\) is any pair in \(F_a\times F_b\), since there is no further arrow connecting the two members. Thus a sheaf has \(F_u\simeq F_a\times F_b\), with the projections. Sheaves are equivalent, on maps as well as objects, to pairs of sets. Sheafification leaves \(F_a,F_b\) unchanged and replaces \(F_u\) by \(F_a\times F_b\); its unit at \(u\) is the pair of restrictions. The adjunction inverse extends a map to a sheaf by its two component functions at \(a,b\), proving this is the actual associated sheaf.

Consequently \(f\in W\) exactly when \(f_a,f_b\) are bijections, and

For functions \(\alpha_a,\alpha_b\), the canonical incoming roof (5.2) has at \(u\) the pairs \((x,y)\in F_u\times G_u\) satisfying \(y|_a=\alpha_a(x|_a)\) and \(y|_b=\alpha_b(x|_b)\). At \(a,b\) its projections to \(F\) are bijective. Hence it is an allowed denominator even when the set of those pairs at \(u\) is empty. This verifies every map in (7.1) through an actual roof. The sieve inclusion \(R\hookrightarrow h_u\) is an example: it is a local isomorphism, but there is no ordinary inverse \(h_u\to R\), since Yoneda identifies such maps with \(R(u)=\varnothing\).

7.3 Local inversion can lose both ordinary categorical tests

Exercise 3 (advanced). On the preceding site, let \(B_u=\{0,1\}\), \(B_a=B_b=\{*\}\), with the unique restrictions. Show that \(B\to1\) is inverted although it is not a monomorphism of presheaves. Show that \(R\to1=h_u\) is inverted although it is not an epimorphism of presheaves. Exhibit the equalizer that implements cancellation.

Solution. Both maps are bijections at \(a,b\), so both belong to \(W\) by Exercise 2. The sections zero and one define two distinct maps \(s_0,s_1:h_u\to B\). Their composites with \(B\to1\) are equal, which proves that map is not monic. Their restrictions to \(R\) are equal, which proves \(R\to h_u\) is not epic. These are categorical failures, not merely failures of a chosen presentation of sections.

The equalizer of \(s_0,s_1\) has empty value at \(u\) and singleton values at \(a,b\); it is precisely \(R\hookrightarrow h_u\). Thus the incoming cancellation in Section 4 is literally restriction to this equalizer. In the localized category the two maps \(s_0,s_1\) agree, while the equalizer inclusion and the collapse \(B\to1\) are both isomorphisms. Sheafifying \(B\) gives singleton values everywhere. Local injectivity and local lifting explain exactly how these ordinary failures disappear.

7.4 Tail sieves invert an endomorphism

Exercise 4 (advanced). Take the one-object category with endomorphisms \(\mathbb N\) under addition. Declare all nonempty sieves covering. Describe its sheaves, sheafification and the two local tests for a morphism of presheaves. Compute the associated sheaf of \(\mathbb N\) with its shift endomorphism.

Solution. A sieve is empty or a tail \(S_n=\{m\ge n\}\). Its pullback along \(r\) is \(S_{\max(n-r,0)}\), so nonempty sieves are stable under base change. Transitivity holds: an empty sieve has empty pullback along every member of a nonempty covering sieve and cannot satisfy its premise. A nonempty candidate is already covering. Thus this is a Grothendieck topology.

A presheaf is a set \(E\) with an endomorphism \(T\). A matching family on \(S_n\) is determined freely by its value at \(n\), with value at \(n+r\) equal to \(T^r\) of that entry. Under this identification the sheaf comparison is \(T^n:E\to E\). A sheaf is therefore exactly a set with invertible \(T\).

Its associated sheaf is the sequential colimit of \(E\xrightarrow{T}E\xrightarrow{T}\cdots\). Write a class as \([x,n]\), with \([x,n]=[Tx,n+1]\). The induced endomorphism \(\widetilde T[x,n]=[Tx,n]\) has inverse \([x,n]\mapsto[x,n+1]\). For an equivariant map \(f:E\to D\) with invertible target endomorphism \(S\), the unique extension is \([x,n]\mapsto S^{-n}f(x)\). The displayed relation proves well-definedness, and every class is an inverse iterate of an original one, proving uniqueness. This is the full sheafification adjunction with its unit \(x\mapsto[x,0]\).

For an equivariant map \(f:(E,T)\to(D,S)\), local surjectivity says that each \(y\in D\) has some \(S^ny\) in the image of \(f\). Local injectivity says that \(f(x)=f(x')\) implies \(T^nx=T^nx'\) for some \(n\). These are precisely the nonempty lifting and equality sieves, and (3.2) says that together they characterize bijectivity of the induced colimit map. No uniform exponent for all sections is required.

For \(E=\mathbb N\), \(T(x)=x+1\), the map \([x,n]\mapsto x-n\) is a bijection to \(\mathbb Z\): every integer has such a representative, and equality of differences is exactly common-stage equality. The induced endomorphism is translation by one. The unit is \(\mathbb N\hookrightarrow\mathbb Z\), whose local lifting test holds because every integer eventually becomes nonnegative. Its inverse in the localization is \(\mathbb Z\leftarrow\mathbb N\to\mathbb N\), with the unit as denominator and the identity as numerator.

Further reading

The Stacks Project treats sieve topologies, sheafification, and injective and surjective maps of sheaves. The local tests, size conventions, both fraction systems and explicit localization comparisons used here are supplied above and in the linked course prerequisites.