Draft lesson.

Restriction, inverse images and morphisms of sieve sites

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

An inverse-image functor between sheaf categories is obtained by extending a presheaf and then sheafifying. Its existence and its preservation of finite limits are different assertions. Covering sieves give an exact test for the first assertion. The local isomorphisms inverted by sheafification give an exact test for the second, including cases where the presheaf extension itself fails to preserve finite limits.

Fix small categories \(C,D\), Grothendieck topologies \(J,K\) given by sieves, and a functor \(u:C\to D\). Sets and all these categories are small in a fixed universe; functor categories may be formed in a larger one. Write \(a_C,a_D\) for associated-sheaf functors and \(i_C,i_D\) for the full inclusions. No fiber products in either base are assumed. All sheaves in this lesson are set-valued.

We use the full matching construction, finite-limit preservation and local-isomorphism criterion in Sieves and presheaf localization; the complete left Kan construction in Extending diagrams by universal maps, Section 3; Filtered stages and finite limits, Sections 1 and 4; and the actual finite-limit construction in Universal tests for forks, Section 6.

1. The presheaf extension with its actual maps

Restriction is \(u^pG=G\circ u^{\mathrm{op}}\). For \(d\in D\), let \(K_d=(d\downarrow u)\). Its objects are \((c,\xi:d\to u(c))\); an arrow \((c',\xi')\to(c,\xi)\) is \(r:c'\to c\) with \(u(r)\xi'=\xi\). Define

(LF)(d)=\mathop{\mathrm{colim}}_{K_d^{\mathrm{op}}}F(c). \tag{1.1}

The index is small. A class is \([c,\xi,x]\), with \(x\in F(c)\), subject to the generated relations

[c,u(r)\xi',x]=[c',\xi',F(r)x]. \tag{1.2}

For \(h:d'\to d\), restriction sends this class to \([c,\xi h,x]\). A transformation \(z:F\to F'\) sends it to \([c,\xi,z_cx]\). Relation (1.2), naturality of \(z\), and associativity in \(D\) prove well-definedness, all functor laws and commutation of these two operations. This is the opposite-Kan construction in the prerequisite, including empty indices.

Its adjunction is completely explicit. A transformation \(\theta:F\to u^pG\) gives

\Phi_\theta[c,\xi,x]=G(\xi)(\theta_cx). \tag{1.3}

The presheaf law and naturality of \(\theta\) respect (1.2). Naturality in \(d\) follows by replacing \(\xi\) with \(\xi h\). Conversely \(\Phi:LF\to G\) gives \(\theta_cx=\Phi_{u(c)}[c,1,x]\). Relation (1.2) with \(\xi'=1\) proves naturality in \(c\). Substituting \(\xi=1\) proves one inverse equation; naturality of \(\Phi\) at \(\xi\) proves the other. Both rules commute with transformations of \(F\) and \(G\). Thus \(L\dashv u^p\), with all Hom naturalities.

For a representable \(h_c=C(-,c)\) there is a natural isomorphism

Lh_c\simeq h_{u(c)},\qquad [c',\xi,r:c'\to c]\longmapsto u(r)\xi. \tag{1.4}

Its inverse sends \(t:d\to u(c)\) to \([c,t,1_c]\). Relation (1.2) proves the inverse equation for any represented class. These identifications respect every arrow in both variables.

2. Exactly when restriction takes sheaves to sheaves

Consider the following condition:

\text{(R)}\qquad u^p i_D E\text{ is a }J\text{-sheaf for every }K\text{-sheaf }E. \tag{2.1}

For a covering sieve \(S\hookrightarrow h_c\), (1.4) gives a presheaf map

\lambda_S:LS\longrightarrow h_{u(c)}. \tag{2.2}

Theorem 2.1. Condition (R) holds if and only if every \(\lambda_S\) for \(S\in J(c)\) is a local isomorphism on \((D,K)\).

Proof. The sheaf condition for \(u^pE\) at \(S\) is bijectivity of \(\operatorname{Nat}(h_c,u^pE)\to\operatorname{Nat}(S,u^pE)\). The actual adjunction (1.3) identifies this with precomposition by \(\lambda_S\) into \(E\). Sheafification identifies it with precomposition by \(a_D\lambda_S\) in the sheaf category. A map \(q:A\to B\) in any category is invertible precisely when \(\operatorname{Hom}(B,E)\to\operatorname{Hom}(A,E)\) is bijective for every \(E\): taking \(E=A\) gives a left inverse \(r\), and injectivity for \(E=B\) gives \(qr=1_B\). The retained local-isomorphism theorem now says exactly that \(\lambda_S\) lifts sections locally and identifies sections only locally. This proves both directions, with the specified comparison maps. \(\square\)

In general \(LS\to h_{u(c)}\) need not be injective. Its image is the sieve generated by the arrows \(u(s):u(c')\to u(c)\) for \(s\in S(c')\): a class maps to \(u(s)\xi\), and every such factorization is represented by a class. Requiring just this image sieve to cover checks only local surjectivity. Local injectivity is a separate requirement. Exercise 3 shows the distinction on three objects.

Some definitions of a continuous functor between sites specify covering families and require preservation of their pullbacks as well as their covers. Those are additional hypotheses on that presentation. We use the explicit condition (R) on arbitrary sieve sites, and do not identify it with cover preservation alone. The later sufficient theorem includes a hypothesis that supplies the needed pullbacks.

3. The sheaf adjunction, unit and counit

Assume (R). Put

f_*E=u^p i_D E,\qquad f^*F=a_D L i_C F. \tag{3.1}

The successive sheafification, presheaf-extension and full-inclusion adjunctions give

\operatorname{Hom}_{\mathsf{Sh}(D,K)}(f^*F,E) \simeq\operatorname{Hom}_{\mathsf{Sh}(C,J)}(F,f_*E). \tag{3.2}

For clarity these are actual inverse factors, not just equal cardinalities. Given \(F\to f_*E\), use (1.3), then uniquely extend the map \(Li_CF\to E\) through \(a_D\). Conversely restrict a map from \(a_DLi_CF\) along its sheafification unit and evaluate on \([c,1,x]\). The inverse equations in Section 1 and uniqueness of extension through sheafification prove both inverse equations here. Both-variable naturality is retained at each step.

The unit at \(F\), evaluated at \(c\), sends \(x\) to the sheafification image of \([c,1,x]\) in \((f^*F)(u(c))\). The counit at \(E\) is the unique sheaf map extending

L(u^pE)(d)\longrightarrow E(d),\qquad [c,\xi,x]\longmapsto E(\xi)x. \tag{3.3}

Substitution on these generators gives the identity for the unit followed by the counit on \(f_*E\). For the other triangle, the same substitution gives the identity on the images of \(Li_CF\); uniqueness of extension through \(a_D\) gives the identity on \(f^*F\). Thus both triangles hold. Restriction \(f_*\) preserves all small limits, because sheaf limits are presheaf limits and restriction computes them pointwise. The left adjoint \(f^*\) preserves small colimits: the Hom bijection (3.2) turns a cocone factor into exactly a cocone factor on the right, with all component equations.

4. The exact finite-limit criterion after sheafification

A geometric morphism \(\mathsf{Sh}(D,K)\to\mathsf{Sh}(C,J)\) is an adjunction \(f^*\dashv f_*\) whose left adjoint preserves finite limits. This is the direction associated with \(u:C\to D\). Under a covering-family definition of morphisms of sites one also checks the specified continuity hypotheses; the sheaf-category assertion here is precisely (R) and the following finite-limit condition.

For sheaves \(F,G\) on \(C\), and any pair \(r,s:F\rightrightarrows G\), form the canonical presheaf comparisons on \(D\):

\begin{gathered} L1\longrightarrow1,\\ L(F\times G)\longrightarrow LF\times LG,\\ L\operatorname{Eq}(r,s)\longrightarrow\operatorname{Eq}(Lr,Ls). \end{gathered} \tag{4.1}

The sheaves on the left are included as presheaves before applying \(L\). The product comparison has the two projection maps; the equalizer comparison is the factor induced by its inclusion and the equalizing equation.

Theorem 4.1. Assuming (R), \(f^*\) preserves finite limits if and only if every map in (4.1) is a local isomorphism on \(D\).

Proof. Sheaf limits on \(C\) are their presheaf limits. The associated-sheaf functor \(a_D\) preserves finite limits. It consequently identifies the sheafifications of the targets in (4.1) with the terminal object, product and equalizer formed after \(f^*\). The sheafifications of the displayed arrows are exactly the corresponding comparison maps for \(f^*\), by their specified components. Such a comparison is invertible exactly when its presheaf arrow is a local isomorphism. Terminal objects, binary products and equalizers construct every finite limit by the complete object-and-arrow product/equalizer proof in the prerequisite. Applying that construction to the actual comparisons proves sufficiency. Necessity follows by using those three finite shapes. The terminal test includes the empty diagram. \(\square\)

This criterion tests finite diagrams of sheaves, and asks for local isomorphisms. It does not require \(L\) to preserve all finite limits of presheaves.

5. A stronger test using cofiltered comma categories

Proposition 5.1. The presheaf functor \(L\) preserves all finite limits if and only if every \(K_d\) is cofiltered.

Proof. If \(K_d\) is cofiltered, its opposite is a small filtered category. Formula (1.1) is then a filtered colimit of evaluation functors. Evaluations preserve limits. The full filtered-set comparison theorem in the prerequisite shows that (1.1) preserves finite limits, including the empty shape, with their actual maps. This holds at every \(d\), so gives a finite-limit-preserving presheaf functor.

Conversely suppose \(L\) preserves finite limits and fix \(d\). Since \((L1)(d)\) is a singleton, it has a representative \([c,\xi,*]\), so \(K_d\) is nonempty. For \((c,\xi),(c',\xi')\), use (1.4) and surjectivity of

L(h_c\times h_{c'})(d)\longrightarrow D(d,u(c))\times D(d,u(c')). \tag{5.1}

A representative of a preimage consists of \((e,\zeta)\) and \(p:e\to c,q:e\to c'\) with \(u(p)\zeta=\xi\), \(u(q)\zeta=\xi'\). It is a common source in \(K_d\).

For parallel arrows \(r,s:(c,\xi)\rightrightarrows(c',\xi')\), we have \(u(r)\xi=u(s)\xi\). Thus \(\xi\) belongs to the equalizer of the two maps \((Lh_c)(d)\rightrightarrows(Lh_{c'})(d)\). Preservation of the presheaf equalizer supplies a representative \([e,\zeta,p]\) with \(p:e\to c\), \(rp=sp\), and \(u(p)\zeta=\xi\). It is the required pre-equalizing arrow in \(K_d\). These are all cofilteredness axioms. No finite limit in \(C\) was assumed for this converse: products and equalizers were taken in the presheaf category. \(\square\)

In particular, (R) and cofilteredness of all these commas imply a geometric morphism. The converse for the sheaf adjunction is false; Exercise 4 supplies a nondegenerate example.

6. When finite limits and covering sieves suffice

Suppose \(C\) has finite limits and \(u\) sends each of their limiting cones to a limiting cone in \(D\). This does not assume that \(D\) has limits of arbitrary other diagrams. Then each \(K_d\) is cofiltered. The terminal object of \(C\) and its terminal image give an object \((1,!)\). A product \(c\times c'\), together with the factor of \((\xi,\xi')\) through its image product, gives a common source. For parallel \(r,s\), the factor of \(\xi\) through the image of their equalizer gives a pre-equalizing source. These are precisely the three axioms used in Proposition 5.1.

More generally assume directly that all \(K_d\) are cofiltered. For any sieve \(S\hookrightarrow h_c\), \(L\) preserves its monomorphism, since it preserves pullbacks and hence the isomorphism diagonal of a monomorphism. Therefore \(LS\to h_{u(c)}\) identifies \(LS\) with the image sieve described in Section 2. By the retained covering-sieve localization criterion, this inclusion is a local isomorphism exactly when that image sieve covers \(u(c)\).

Consequently under this comma hypothesis, (R) is equivalent to the following simple covering condition: every \(J\)-covering sieve has images generating a \(K\)-covering sieve. Under the finite-limit hypothesis above, that condition therefore produces the sheaf adjunction and makes its left adjoint preserve finite limits. The fiber products needed by a covering-family convention are among the limits preserved by \(u\). Both the terminal hypothesis and compatibility of overlaps have been accounted for.

7. Composition and the specified comparison

Let \(v:D\to E\) also satisfy (R), for a third small sieve site. Restriction along \(vu\) is literally \(u^p v^p\), so again preserves sheaves. Its left adjoint and \(f_v^*f_u^*\) are left adjoints to this same restriction. The actual natural Hom bijections give the canonical comparison between them: transpose the unit of one adjunction through the other. The opposite transpose gives its inverse, since transposing twice recovers the original map. It is the unique comparison that respects the specified units.

At the presheaf level the Kan comparison regroups classes by

[d,\zeta:e\to v(d),[c,\xi:d\to u(c),x]] \longmapsto[c,v(\xi)\zeta,x]. \tag{7.1}

The inverse uses \(d=u(c)\), \(\xi=1\). The two sets of relations (1.2) prove both inverse equations. Insert the unit \(L_uF\to i_Da_DL_uF\), apply \(L_v\), and sheafify on \(E\). This gives the comparison from the associated sheaf of \(L_{vu}F\) to \(f_v^*f_u^*F\). On generators its transpose is exactly the composite of the two specified units, so it is the canonical invertible comparison just constructed. For three functors, both associativity composites transpose to the same iterated unit; hence are equal. The identity comparison has the same property. If the two left adjoints preserve finite limits, so does their composite and hence the canonically isomorphic left adjoint for \(vu\).

8. Four exercises with full solutions

8.1 Recover topological inverse images

Exercise 1 (basic). Let \(f:X\to Y\) be continuous between small topological spaces, and let \(u:\mathcal O(Y)\to\mathcal O(X)\) send \(U\) to \(f^{-1}U\). Verify all the sufficient hypotheses, identify both sheaf functors, and prove the inverse-image stalk formula with its actual maps.

Solution. The terminal open \(Y\) maps to \(X\), and finite intersections are preserved. Covering unions pull back to covering unions. A covering sieve on an open contains the members of some open cover, so its images generate a covering sieve on the inverse-image open. Section 6 gives (R) and finite-limit preservation of the associated inverse image. Restriction is exactly \((f_*E)(U)=E(f^{-1}U)\).

The comma \(K_V\) consists of opens \(U\subseteq Y\) with \(V\subseteq f^{-1}U\), ordered by inclusion. It is nonempty because it contains \(Y\), and intersections give common lower bounds. Formula (1.1) is thus the usual presheaf

V\longmapsto\mathop{\mathrm{colim}}_{V\subseteq f^{-1}U}F(U), \tag{8.1}

with transitions restricting to smaller \(U\). Its associated sheaf is \(f^*F\). A germ of (8.1) at \(x\) represented by \([U,s]\) over \(V\ni x\) is sent to the germ of \(s\) at \(f(x)\). This is well-defined under both kinds of restriction and equality relation. Conversely a section \(s\) over \(U\ni f(x)\) defines a section of (8.1) on \(f^{-1}U\), hence a germ at \(x\). These maps are inverse: starting with \(V\subseteq f^{-1}U\), the recovered representative agrees on \(V\); starting with \(s\), its image is its original germ. Equal representatives on neighborhoods give equal germs in either direction. Associated sheaves have the same germs as their presheaves by the local representative and local equality rules. Therefore \((f^*F)_x\simeq F_{f(x)}\), naturally in \(F\).

8.2 Restriction alone does not give a geometric morphism

Exercise 2 (intermediate). Give the discrete two-object category \(C\) and the one-object identity category \(D\) their topologies with only maximal sieves covering. For the unique \(u:C\to D\), compute \(f_*\), its left adjoint, and the failed finite-limit comparison.

Solution. Every presheaf is a sheaf. Restriction sends a set \(E\) to \((E,E)\), so (R) holds. The category \(K_*\) is discrete on the two objects. Thus \(L(A,B)=A\amalg B\), with unit the two inclusions and counit the fold \(E\amalg E\to E\). The same formulas give the sheaf adjunction. Its terminal comparison is \(\{0,1\}\to\{*\}\), which is not a bijection and, with the trivial topology, is not a local isomorphism. The left adjoint is not left exact. The nonempty comma has no common source for its two objects, exactly as Proposition 5.1 predicts. In particular the existence of the adjunction is insufficient for a geometric morphism.

8.3 Images of covers need not preserve sheaves

Exercise 3 (advanced). Let \(C\) have only the nonidentity arrows \(a\to c\) and \(b\to c\). On \(c\), let the sieve \(S\) consisting of those arrows cover; at \(a,b\) only maximal sieves cover. Map \(C\) to the one-object identity category \(D\) with trivial topology. Show that all covering sieves have covering images but (R) fails, and compute the map (2.2) that detects the failure.

Solution. The fork topology is the one in the localization prerequisite: pullbacks of \(S\) to the two leaves are maximal, and transitivity says a covering candidate contains both leaves. Every image sieve in question contains the identity of \(D\), so is maximal and covers. But the restriction of a two-element set \(E\) is the constant fork presheaf. Its matching comparison on \(S\) is \(E\to E\times E\), the diagonal, which is not onto. Thus it is not a sheaf.

At the single target object, (1.1) is the colimit over the opposite fork. For \(S\) the three values are \(\varnothing,1,1\), and the colimit is \(1\amalg1\). For \(h_c\) all three values are singletons, joined by the two arrows, and the colimit is \(1\). Hence \(\lambda_S\) is the fold \(2\to1\). It is surjective and has covering image, but is not locally injective on this trivial target site. This gives exactly the missing half of Theorem 2.1. The base category has no overlap object for \(a,b\); a pullback-preservation hypothesis cannot be silently supplied.

8.4 Sheafification can repair a failure of the comma criterion

Exercise 4 (challenge). Let \(D\) be the preceding fork with its two-leaf covering topology, let \(C\) be discrete on \(a,b\) with trivial topology, and let \(u:C\to D\) include the leaves. Prove that (R) holds and the sheaf adjunction is an equivalence, although \(L\) does not preserve the terminal presheaf. Compute its local terminal comparison.

Solution. A sheaf on \(D\) is a pair \((A,B)\), with value \(A\times B\) at the vertex and its two projections. There are no further overlaps, so every pair defines a sheaf, and every pair of functions defines exactly one sheaf map. Restriction to the leaves therefore identifies \(\mathsf{Sh}(D)\) with \(\mathsf{Set}\times\mathsf{Set}=\mathsf{Sh}(C)\), on objects and all maps. This proves (R).

For a pair \((A,B)\), formula (1.1) gives \(L(A,B)_a=A\), \(L(A,B)_b=B\), and empty value at the vertex: there is no arrow from the vertex to either leaf. Sheafification replaces that empty value by \(A\times B\). Thus \(f^*(A,B)\) is the inverse equivalence just described. Its unit is identity on both components and its counit is the matching isomorphism at the vertex together with the leaf identities. Every finite limit comparison is consequently an isomorphism after sheafification.

In contrast, \(K_c\) is empty, so the presheaf terminal comparison has empty value at the vertex and singleton values at both leaves. It is precisely the covering-sieve inclusion \(S\hookrightarrow h_c=1\). This is a local isomorphism, but is not an isomorphism of presheaves. Binary product and equalizer comparisons for \(L\) are isomorphisms: at either leaf they are the ordinary Set comparisons, and at the vertex the nonempty finite product or equalizer of the relevant empty sets is empty. It is the empty finite diagram that fails before sheafification. Hence the exact local criterion succeeds although the stronger cofiltered-comma criterion fails, in a sheaf category with many objects and morphisms.