Stacks over sheaves and epimorphic descent

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

A cover of a site can have overlaps that are not objects of that site. The sheaf category contains those overlaps, but a stack originally assigns categories only to site objects. We construct its values on arbitrary sheaves and prove the descent theorem that makes those new bases legitimate. The construction retains every morphism, including noninvertible ones.

The site may have no terminal object or fiber products, and its topology need not be subcanonical. In particular, the representable presheaf of a site object may need sheafification. Recovery of the original stack will use that sheafification explicitly.

We use covering sieves, the complete local section tests and sheafification from Sieves, local isomorphisms, and presheaf localization, and cartesian descent with all its arrows from Stackification by cartesian sections on covering sieves. Identity evaluation and both Yoneda variances are as in Points, fibres and universal representations. The result supplies arbitrary sheaf bases for Representing objects from local data.

1. Cartesian sections on a presheaf

Fix a small Grothendieck site \((C,J)\), and put \(E=\operatorname{Sh}(C,J)\). Work in fixed universes large enough for the stated categories and families. A presheaf below is set-valued in the small universe. Its element category has a small set of objects \((U,p)\), with \(p\in P(U)\), and arrows

Let \(\mathsf S:C^{op}\to\mathsf{Cat}\) be a stack in categories. Choose its restriction functors \(f^*\), their unit comparisons, and their coherent composition comparisons

For any presheaf \(P\), define \(\operatorname{Cart}(P,\mathsf S)\) as follows. An object consists of \(A_{U,p}\in\mathsf S(U)\) for every \((U,p)\), and isomorphisms

The identity comparison is the specified unit comparison. A morphism \(u:A\to B\) is a family of arbitrary fiber morphisms satisfying

These morphisms need not be isomorphisms. Their identities and compositions are componentwise; equation (1.2) is stable under both. A map \(v:P\to Q\) gives

Its comparisons and morphisms are the same indexed comparisons and morphisms. Thus pullback respects identity and composition of presheaf maps strictly.

For \(h_U=\operatorname{Hom}_C(-,U)\), evaluation at \((U,1_U)\) gives

Indeed, \(A\in\mathsf S(U)\) gives the family \(f^*A\), with comparisons \(a_{f,g}\). For any cartesian family, its comparison at \(f:(V,f)\to(U,1_U)\) identifies its component with \(f^*A_{U,1_U}\). These comparisons form a natural isomorphism of families by (1.1). Equation (1.2) says that a family morphism is determined by its component at \(1_U\), and any such component extends by restriction. These are the two inverse functors and their identifications. They commute with restriction in \(U\), using the original comparisons \(a_{f,g}\).

A stack on the sieve site means precisely that, for every covering sieve \(R\subseteq h_U\), the restriction

is an equivalence. Its full faithfulness is the sheaf condition for morphisms between fixed objects. Its essential surjectivity is effective object descent. We use both parts, with every base arrow of the sieve included.

2. Local isomorphisms do not change cartesian sections

A presheaf map \(v:P\to Q\) is a local isomorphism if it is locally surjective and locally injective in the exact sieve sense of the first prerequisite. Thus every \(q\in Q(U)\) lifts to \(P\) over a covering sieve, and two lifts of the same section become equal over a covering sieve. These conditions are equivalent to \(av:aP\to aQ\) being an isomorphism, where \(a\) denotes sheafification.

Theorem 2.1. For a local isomorphism \(v:P\to Q\), the pullback functor \(v^*\) in (1.3) is an equivalence.

Proof: full faithfulness. Let \(A,B\in\operatorname{Cart}(Q,\mathsf S)\), and suppose a morphism \(t:v^*A\to v^*B\) is given. For \(q\in Q(U)\), choose a covering sieve \(R_q\) of arrows \(f:V\to U\) on which \(q|_f\) has a lift \(p_f\in P(V)\). The component \(t_{V,p_f}\), transported through the comparisons of \(A,B\), is a morphism

This local morphism does not depend on the chosen lift. Two lifts with the same image become equal on a covering sieve; there their \(t\)-components are identified by equation (1.2). The sheaf condition for morphisms makes the two transported maps equal on \(V\). The same argument, applied to \(p_f|_g\) and \(p_{fg}\), makes the maps compatible under every arrow of \(R_q\). They glue to a unique \(t_{U,q}\).

For an arbitrary arrow \(f:(V,q|_f)\to(U,q)\), equation (1.2) for these glued maps can be checked on the pullback of a lifting sieve. There it is the equation for the given \(t\); hence it holds globally. The resulting family is a morphism \(A\to B\) whose pullback is \(t\), since that equality can likewise be tested on a lifting sieve. Uniqueness follows from the same local test. This proves bijectivity on all Hom sets, not only on isomorphisms.

Proof: constructing objects. Take \(D\in\operatorname{Cart}(P,\mathsf S)\). For \(q\in Q(U)\), choose \(R_q\) and lifts \(p_f\) as above, and use \(D_{V,p_f}\) over each \(f:V\to U\) in \(R_q\). For a base arrow \(g:W\to V\), the two lifts \(p_f|_g\) and \(p_{fg}\) have the same image. On their equality sieve the comparisons of \(D\) give a specified isomorphism

These local maps agree after common restriction, by the composition equation (1.1) for \(D\). The sheaf condition for morphisms glues them uniquely. Their inverse maps glue too, so the glued maps are invertible. The two composites for a further base arrow agree on a covering sieve where the finitely many lifts involved are equal. The sheaf condition therefore makes the composition equation hold on all of its domain. The identity equation is verified in the same way. Thus the objects and glued comparisons form a cartesian section on \(R_q\).

Effective descent in (1.5) gives \(A_{U,q}\in\mathsf S(U)\) together with its specified local identifications. Choose such a descended object and identifications for every \((U,q)\). To give a comparison for \(f:V\to U\), compare \(f^*A_{U,q}\) and \(A_{V,q|_f}\) locally on the pullback lifting sieve. After a further covering refinement their chosen lifts agree. The specified descent identifications and the comparisons of \(D\) then give local isomorphisms between these two objects. They agree on further restrictions and glue to \(c_f\).

The unit and composition equations for the \(c_f\) hold after covering refinements where the finitely many relevant lifts agree. Separation of the corresponding Hom sheaf makes them hold globally. Hence \(A\) is an object of \(\operatorname{Cart}(Q,\mathsf S)\).

Finally, for \((U,p)\), compare \(A_{U,v(p)}\) with \(D_{U,p}\) on the chosen lifting sieve. Local injectivity makes the chosen lifts equal to restrictions of \(p\) after refinement. Their specified local comparisons therefore glue to an isomorphism. The same local check proves equation (1.2), giving \(v^*A\simeq D\). This proves essential surjectivity and completes the equivalence. All chosen comparisons are unique relative to their specified local restrictions; no assertion that arbitrary isomorphisms of the objects are unique is being made.

Empty covering sieves cause no exception. The stack axiom on such a sieve says that the fiber category is equivalent to the terminal category. Both objects and the unique compatible morphisms then descend from the empty data. \(\square\)

3. Extension to sheaves and epimorphic descent

For \(Q\in E\), set

Use (1.3) for every morphism of base sheaves. The unit \(h_U\to ah_U\) is a local isomorphism, so Theorem 2.1 and (1.4) give equivalences

with the coherent restriction identifications just described. We use \(ah_U\), not an assumption that \(h_U\) is already a sheaf. An object over the terminal sheaf means an object of \(\operatorname{Cart}(1,\mathsf S)\); it does not require a terminal object of \(C\).

For a family \((Q_i\to Q)_{i\in I}\), jointly epic means that its induced map \(\coprod_i Q_i\to Q\) is an epimorphism in \(E\). The sheaf local-lifting criterion gives the following useful formulation: for every \(q\in Q(U)\), the arrows \(f:V\to U\) on which \(q|_f\) lifts to some \(Q_i(V)\) form a covering sieve. A local section of the sheaf coproduct belongs locally to a summand, by the sheafification local-section rule, so the formulation is equivalent to joint epimorphy.

Theorem 3.1. The extension \(\overline{\mathsf S}\) is a stack for jointly epic families of sheaves. In particular,

is an equivalence. All products defining the overlaps in this formula are products in \(E\); no intersection object in \(C\) is required.

Proof. Write a descent datum as objects \(A_i\in\overline{\mathsf S}(Q_i)\) and isomorphisms on \(Q_i\times_Q Q_j\), with direction from the \(j\)-object to the \(i\)-object, satisfying the diagonal and triple-overlap equations. The morphisms of descent data are arbitrary compatible families of morphisms.

Fix \(q\in Q(U)\). Choose a covering sieve \(R_q\) as in the local-lifting formulation, together with an index \(i(f)\) and a lift \(q_f\in Q_{i(f)}(V)\) for every \(f:V\to U\) in that sieve. Over \(f\), take the object

For \(g:W\to V\), the pair \((q_{fg},q_f|_g)\) is a section of \(Q_{i(fg)}\times_Q Q_{i(f)}\) over \(W\). Use the cartesian comparison in \(A_{i(f)}\), followed by the overlap isomorphism from the \(i(f)\)-object to the \(i(fg)\)-object at that pair. This compares the displayed object restricted to \(W\) with the object chosen for \(fg\). Naturality of the overlap maps and their triple equation give (1.1). The diagonal equation gives the unit condition. This is descent data on the covering sieve \(R_q\), and therefore descends to \(A_{U,q}\in\mathsf S(U)\) with its local identifications.

For a base arrow \(f:V\to U\), compare \(f^*A_{U,q}\) with \(A_{V,q|_f}\) over the common refinement of their lifting sieves. At a local pair of chosen lifts, use the supplied overlap isomorphism. These comparisons agree under further restrictions by the triple equation. They glue uniquely as morphisms in \(\mathsf S\), as do their inverses. Their unit and composition equations hold on the same local refinements and hence globally. Thus the \(A_{U,q}\) form \(A\in\operatorname{Cart}(Q,\mathsf S)\).

For a section \(q_i\in Q_i(U)\), compare this construction for its image \(q\) with \((A_i)_{U,q_i}\). On each chosen lifting domain there is a pair \((q_f,q_i|_f)\) in the appropriate overlap sheaf. Its specified overlap map gives the comparison. These comparisons glue, are cartesian in \((U,q_i)\), and reproduce the original overlap maps by their triple equation. Hence the pullback of \(A\) realizes the given descent datum.

For full faithfulness, suppose maps between two descended objects are given on every \(Q_i\), compatibly on overlaps. At \((U,q)\), pull the maps back to a lifting sieve. Overlap compatibility makes them a matching family of maps between the two fixed objects of \(\mathsf S(U)\). The Hom sheaf glues them uniquely. Their base-arrow compatibility is local, so the glued maps form a cartesian family. Its pullbacks are the prescribed maps, and it is unique. This proves the required Hom bijection and the equivalence.

If the family is empty, joint epimorphy forces \(Q\) to be initial. The same proof uses empty lifting sieves at any test section of \(Q\). Effective empty descent and the Hom sheaf condition give the unique object and maps, as required. \(\square\)

Thus this extension supports both object descent and morphism descent over arbitrary sheaf bases. Its value at the terminal sheaf, its pullbacks to overlaps, and its effective gluing are all supplied by the original stack axioms on \(C\).

4. Functors, sheaf values and internal Yoneda

A cartesian functor \(F:\mathsf S\to\mathsf T\) extends by applying \(F_U\) to every \(A_{U,p}\). Its restriction comparison identifies \(f^*F_U(A_{U,p})\) with \(F_V(f^*A_{U,p})\), after which \(F_V(c_f)\) supplies (1.1). The coherence axiom of \(F\) and equation (1.1) give the new composition equation. Apply \(F_U\) to a family morphism for its action on arrows. A cartesian transformation between two such functors extends componentwise; its compatibility with restriction is exactly equation (1.2). Identity and composition comparisons extend by these same components, so all their equations remain the original pointwise equations.

There is also a natural identification

Use the same objects and inverse cartesian comparisons; a component arrow in the opposite fiber is an arrow in the reverse direction in the original fiber. Inverting the comparisons turns (1.1) and (1.2) into the corresponding equations on the other side. These operations are inverse, and commute with base change. In particular, taking an opposite retains effective descent.

To interpret sheaf-valued functors precisely, let \(\mathsf H(U)=\operatorname{Sh}(C/U)\), with restriction given by composition of arrows into \(U\). The topology on \(C/U\) is inherited from \(C\). For a presheaf \(P\), give \(C/P\), its element category, the topology in which a sieve covers \((U,p)\) exactly when its underlying arrows form a covering sieve on \(U\).

Lemma 4.1. The prestack \(\mathsf H\) is a stack, and there are natural equivalences

For a sheaf \(Q\), this identifies \(\overline{\mathsf H}(Q)\) with the slice category \(E/Q\).

Proof: the displayed equivalence. Given a cartesian family \(B\) of sheaves, define

For \(f:(V,p|_f)\to(U,p)\), first restrict a section of \(B_{U,p}\) from \(1_U\) to \(f\), and then apply the cartesian identification at \(1_V\). Equation (1.1) gives the restriction laws. A matching family over a covering sieve of \((U,p)\) is, under those identifications, a matching family in the sheaf \(B_{U,p}\) on \(C/U\). It glues uniquely, so \(G\) is a sheaf on \(C/P\).

Conversely, for such a \(G\), define the sheaf \(B_{U,p}\) by

The sheaf condition is the one for \(G\); comparison along a base arrow is the identity on these expressions. The two constructions are inverse up to the original cartesian comparisons: at \(f\), that comparison identifies \(B_{U,p}(f)\) with \(B_{V,p|_f}(1_V)\). A map of families gives a map of the \(G\)'s by evaluation, and a map of the \(G\)'s gives all these component maps. These operations are inverse by the same identifications. This proves full faithfulness as well as the assertion on objects.

Proof: descent for \(\mathsf H\). Let \(R\subseteq h_U\) be a covering sieve. By (4.2), its cartesian descent category is \(\operatorname{Sh}(C/R)\). If \(G\) is a sheaf there, extend it to a presheaf \(B\) on \(C/U\) by the rule

Restrictions pull back these families. We verify the sheaf condition. Suppose a covering sieve \(T\) on \(V\) gives compatible sections \(s_t\in B(gt)\). For \(r:Z\to V\) with \(gr\in R\), the sieve \(r^*T\) covers \(Z\). For \(k:W\to Z\) in this sieve, \(rk\in T\); the section \(s_{rk}\) has a component at \(1_W\), because \(grk\in R\). These components form a matching family in \(G\) over \((Z,gr)\). Glue them to \(x_r\in G(Z,gr)\).

The \(x_r\) commute with restriction: test the two proposed restrictions on the pullback of \(T\), where both are the corresponding components of the original \(s_t\), and use separation of \(G\). Thus they give an element of \(B(g)\). Its restriction to \(t\in T\) is \(s_t\), since each component is checked on a covering pullback and then by separation. Uniqueness follows by the same test for every \(r\). Hence \(B\) is a sheaf.

If \(g\in R\), its pullback sieve is maximal, so evaluation at \(1_V\) identifies \(B(g)\) with \(G(V,g)\). Conversely a sheaf on \(C/U\) is identified with its matching families over the covering sieve \(g^*R\) for every \(g\). Therefore restriction and (4.3) are inverse equivalences, also on morphisms by applying a map to each matching entry. In view of (4.2), this is precisely the stack condition for \(\mathsf H\).

Proof: the slice identification. The fiber construction of Points, fibres and universal representations, Section 6 sends a map \(A\to Q\) to its sets of sections lying over \(q\in Q(U)\). It restricts to sheaves. Indeed, matching local sections in a fiber glue uniquely in \(A\), and their images agree with \(q\) by separation of \(Q\). Conversely, if \(G\) is a sheaf on \(C/Q\), its total presheaf is

A matching family first glues its labels in the sheaf \(Q\), then glues its fiber entries in \(G\). Thus \(A\) is a sheaf and has its projection to \(Q\). The inverse constructions and maps are the stated fiber construction, so \(\operatorname{Sh}(C/Q)\simeq E/Q\). \(\square\)

Consequently a cartesian functor \(\mathcal F:\mathsf S^{op}\to\mathsf H\) extends, with all its functor and restriction data, to

For \(A,B\in\overline{\mathsf S}(Q)\), their internal Hom sheaf on \(C/Q\) is explicitly

with restriction by \(f^*\) and conjugation by the cartesian comparisons. The Hom sheaf condition in \(\mathsf S\) proves that it is a sheaf. A compatible family of its sections over a base sheaf \(W\to Q\) is exactly a family morphism \(A|_W\to B|_W\). Thus (4.5) is the internal Hom with its required meaning on every base sheaf, not only on represented tests.

Proposition 4.2 (Internal identity evaluation). Cartesian natural transformations from \(\mathcal Hom(-,A)\) to a contravariant sheaf-valued functor \(\overline{\mathcal F}\) correspond to global sections of \(\overline{\mathcal F}(A)\) over \(Q\). In particular,

Proof. Evaluate a transformation on the identity section of \(\mathcal Hom(A,A)\). Conversely, a section \(s\) of \(\overline{\mathcal F}(A)\) assigns, to a local arrow \(f:T\to A\), the section \(\overline{\mathcal F}(f)(s)\). The functor composition law gives naturality in \(T\), and cartesian restriction compatibility gives a map of the Hom sheaves on every smaller base. Identity evaluation returns \(s\). For the other composite, naturality of the original transformation along each local \(f\) identifies its value at \(f\) with the displayed formula. Sheaf maps are determined by these sections on all tests, so the two transformations are equal. The construction commutes with base change. For the particular target \(\mathcal Hom(-,B)\), a global section is exactly a family morphism \(A\to B\) by (4.5), proving (4.6). The covariant statement follows from (4.1). \(\square\)

Theorem 3.1 and Proposition 4.2 now give the precise setting for local representing objects: their uniquely specified comparison maps live over actual overlap sheaves, obey the triple equation, and descend effectively. Lemma 4.1 gives the target sheaves and Proposition 4.2 gives uniqueness with its universal identification.

Corollary 4.3 (Linear structure and specified adjunctions). Let \(R\) be a sheaf of commutative rings, and suppose \(\mathsf S\) is an \(R\)-linear stack: its fiber Hom sets are \(R(U)\)-modules, composition is bilinear, and restriction is linear with the coefficient restriction \(R(U)\to R(V)\). Then \(\overline{\mathsf S}\) is an \(R\)-linear stack over base sheaves. Its internal Hom in (4.5) is a module over \(R_Q\), the pullback of \(R\) to \(E/Q\). Every cartesian \(R\)-linear functor extends to a cartesian \(R\)-linear functor.

Any specified adjunction of cartesian stack functors also extends, including its unit, counit, triangles and restriction comparisons. Such an adjunction includes cartesian unit and counit transformations. Separate adjoints to the fiber functors do not provide those data. The gluing of local stack adjoints, with their restriction condition, is proved in Dual and tensor products of locally module stacks, Lemma 2.1; the precise boundary is also explained in Functor stacks, Section 5.

Proof. On the test \((U,q)\) over \(Q\), the Hom set in (4.5) has its given \(R(U)\)-module structure. Its restriction is \(u\mapsto c_f^B f^*(u)(c_f^A)^{-1}\). The restriction functor is linear with the restricted coefficient, and the two compositions are linear by bilinearity. Therefore these module operations commute with restriction. Their axioms hold on every test, so (4.5) is a sheaf of \(R_Q\)-modules. All base-sheaf pullbacks in (1.3) simply relabel these same operations.

For the global fiber category over \(Q\), write \(R(Q)=\operatorname{Hom}_E(Q,R)\), with pointwise ring operations. If \(s\in R(Q)\) and \(u,v:A\to B\) are cartesian family morphisms, define

Since \(s_U(q)|_f=s_V(q|_f)\), equation (1.2) for \(su\) follows from

The same calculation with sums, additive inverses and zero proves that all the operations stay inside the family Hom set. The module axioms, the scalar unit and bilinearity of composition are componentwise. This gives the stated linear structure both on global Hom sets and on their internal Hom sheaves.

If \(F:\mathsf S\to\mathsf T\) is cartesian and \(R\)-linear, its extension applies \(F_U\) to each component. Hence \(F_U(s_U(q)u_{U,q})=s_U(q)F_U(u_{U,q})\), and it preserves sums and zero there. The family equalities prove linearity on every base, with the same coefficient restrictions.

Finally let \(F\dashv G\) be a specified cartesian stack adjunction, with unit \(\eta\) and counit \(\epsilon\). Using the identity and composite comparisons of Section 4, its extended unit and counit have components

Their cartesian equations are exactly the original restriction equations for \(\eta,\epsilon\), after the functor restriction comparisons. Their naturality is the original componentwise naturality. Each triangle composite at \((U,q)\) is the original triangle at \(A_{U,q}\) or \(B_{U,q}\), so is the identity. Family morphisms are equal when their components are equal; thus both triangles hold over every \(Q\). Relabelling along \(W\to Q\) preserves all these formulas and comparisons, proving the asserted stack adjunction. This argument uses the supplied cartesian adjunction; it does not choose unrelated fiber adjoints. \(\square\)

The Hom module in this corollary is the coefficient structure used when local tensor–Hom corepresentations are compared over arbitrary overlap sheaves.

5. Four exercises with complete solutions

Exercise 1 (Foundational: descend a noninvertible arrow). Let a site have one object and only its identity, with only the maximal covering sieve. Its sheaf category is \(\mathsf{Set}\). Let the stack have value an arbitrary small category \(A\). Describe descent along \(\{0,1\}\to\{*\}\) on objects and morphisms, and exhibit a descended noninvertible morphism.

Solution. A presheaf is a set, its element category is discrete, and \(\overline{\mathsf S}(Q)=A^Q\). Descent along the indicated epimorphism consists of two objects \(a_0,a_1\), identity diagonal comparisons, and isomorphisms \(u_{ij}:a_j\to a_i\) for \(i,j\in\{0,1\}\), with \(u_{ij}u_{jk}=u_{ik}\). These equations say that \(u_{10}=u_{01}^{-1}\); the datum is determined by \(a_0,a_1,u_{10}\). It is isomorphic to the constant datum on \(a_0\), with component maps \(1_{a_0}\) and \(u_{10}:a_0\to a_1\).

A morphism to a second datum \(b_0,b_1,v_{10}\) is a pair \(f_i:a_i\to b_i\) satisfying

Thus \(f_0\) determines \(f_1=v_{10}f_0u_{10}^{-1}\), and every \(f_0\) is allowed. Evaluation at index zero is an inverse equivalence to constant descent data, on all Hom sets. Take \(A\) to have objects \(0,1\) and a single nonidentity arrow \(0\to1\). Between the two constant data, use that arrow at both indices. It descends to \(0\to1\), which is not invertible. Replacing the fibers by their cores would discard this morphism. \(\square\)

Exercise 2 (Intermediate: a nonsubcanonical site). Let \(C\) have one object with endomorphism monoid \(M=\{1,e\}\), \(e^2=e\). Declare the sieve \(R=\{e\}\) and the maximal sieve covering. Compute sheaves and the sheafification of the representable presheaf. For any small category \(A\), take the constant stack with identity restrictions. Compute its extension over every sheaf \(Q\).

Solution. The only sieves are \(\varnothing,R,M\). Pullback along \(1\) fixes them, and \(e^*R=M\). These rules verify stability of the two declared covering sieves. For transitivity with outer sieve \(R\), the only test is pullback along \(e\); the empty sieve does not pass that test, while \(R,M\) do. The maximal outer sieve imposes the usual identity test. Thus this is a Grothendieck topology.

A presheaf is a right \(M\)-set, equivalently a set \(X\) with an idempotent map \(t(x)=x e\). Matching sections over \(R\) are the fixed points of \(t\), and the restriction map is

It is always surjective. It is injective exactly when \(t=1_X\): injectivity applied to \(t(x)=t(t(x))\) gives \(x=t(x)\). Hence the sheaves are sets with trivial \(e\)-action and \(E\simeq\mathsf{Set}\).

Sheafification sends \(X\) to \(\operatorname{Fix}(t)=\operatorname{im}(t)\), with the displayed unit and trivial action. Every equivariant map to a trivial-action set satisfies \(f(x)=f(t(x))\), so factors uniquely through that unit, proving the adjunction. The representable is the regular right \(M\)-set \(\{1,e\}\), on which \(t(1)=e=t(e)\). Its sheafification is a singleton. It is not already a sheaf, so the topology is not subcanonical.

For the constant category \(A\), a cartesian section on \(R\) is an object \(a\) with an invertible comparison \(c_e:a\to a\) satisfying \(c_e^2=c_e\). Invertibility forces \(c_e=1_a\). Its morphisms are all morphisms of \(A\). Consequently \(A\to\operatorname{Cart}(R,A)\) is an equivalence, so this constant prestack is a stack.

A sheaf \(Q\) has trivial \(e\)-action. Its element category is a disjoint union, indexed by \(q\in Q\), of copies of the one-object monoid category. On each copy the same idempotent comparison equation forces \(c_e=1\). Therefore

At \(ah_*=\{*\}\) it recovers \(A\), even though \(h_*\) was not a sheaf and its two elements have become equal. \(\square\)

Exercise 3 (Advanced: an epimorphism is not a local isomorphism). Explain why local surjectivity alone cannot replace local isomorphism in Theorem 2.1. Use the one-point site of Exercise 1 and the surjection \(\{0,1\}\to\{*\}\), and reconcile the failure with Theorem 3.1.

Solution. In the trivial topology a covering sieve is maximal, so local equality is actual equality. The surjection is locally surjective but not locally injective: its two different source elements have the same image. If \(A\) is the discrete category with two objects \(a,b\), pullback is the diagonal

The pair \((a,b)\) is not isomorphic to any diagonal pair, so this functor is not an equivalence. Theorem 3.1 instead compares \(A\) with objects of \(A\times A\) supplied with overlap isomorphisms and their cocycle. The pair \((a,b)\) admits no comparison between its two entries and is not descent data. In Exercise 1 that additional datum is exactly what made all legitimate pairs effective. Thus epimorphic descent and invariance under a local isomorphism are distinct assertions. \(\square\)

Exercise 4 (Synthesis: morphism gluing does not make objects effective). Give a prestack on the circle whose morphisms are sheaves, and a contravariant sheaf-valued functor that is locally represented but has no global representation. Prove the obstruction rather than assume a classification of line bundles.

Solution. Write \(X=\mathbb R/\mathbb Z\), with projection \(\pi\), and let \(R\) be the sheaf of continuous real functions. On an open \(U\), let \(\mathsf P(U)\) have one object, called \(R_U\), with endomorphisms \(R(U)\) under multiplication. Restrictions restrict functions. The Hom presheaf is the sheaf \(R\), so this is a prestack with morphism descent.

Define a sheaf of \(R\)-modules by

Scalar multiplication is by the pullback of a function on \(U\). Matching functions glue on the open cover of \(\pi^{-1}U\), and their displayed equations are local; hence this is a module sheaf. On an arc \(U\) with a lift \(I\subset\mathbb R\) on which \(\pi\) is a homeomorphism, \(\pi^{-1}U\) is the disjoint union of \(I+n\). The function equal to \((-1)^n\) on \(I+n\) is a basis of \(L|_U\). Thus \(L\) is locally free of rank one.

There is no global basis. Such a basis would be a continuous nowhere-zero \(f:\mathbb R\to\mathbb R\) with \(f(1)=-f(0)\). If \(f(0)=0\) it already vanishes. Otherwise replace \(f\) by \(-f\) if necessary, so that \(f(0)>0>f(1)\), and set \(c=\sup\{t\in[0,1]:f(t)>0\}\). Continuity at the endpoints gives \(0<c<1\). If \(f(c)>0\), continuity gives a positive value to the right of \(c\), contradicting the upper bound. If \(f(c)<0\), continuity gives a neighborhood of \(c\) with no positive values, contradicting positive points arbitrarily close to the supremum from below. Thus \(f(c)=0\). A rank-one local module basis must be nowhere zero in the local descriptions, so in every case this section cannot be a basis.

Let \(\mathcal F_U(R_U)=L|_U\); let an endomorphism \(r\) act by multiplication by \(r\). Commutativity of \(R\) makes this a contravariant functor, compatible with restriction. On each trivializing arc the chosen basis gives

A global representation could only use the sole object \(R_X\). Its identification would be an isomorphism of sheaves \(\phi:R\to L\) natural under every local scalar endomorphism. Naturality gives \(\phi(r)=r\phi(1)\) on every open set. Hence \(\phi\) is the module map determined by a global basis \(\phi(1)\), which has just been ruled out. The local bases instead give invertible scalar transition maps satisfying the cocycle equation. If that object descent were effective in \(\mathsf P\), it would give a global identification with \(L\), the same contradiction. Thus morphism descent holds but object descent fails, and the stack hypothesis in local representability is necessary. \(\square\)

Further reading

The Stacks Project, Stacks, Lemma 8.4.2, states the descent-category criterion. Its section on Representable sheaves explains the role of sheafified representables on a general site. The cartesian-section and epimorphic-descent arguments above use the full category-valued stack axioms.