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Representing objects from local data

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

A universal construction is often easy to find locally and difficult to describe globally. Local representing objects need not have identical descriptions on overlaps. What makes gluing possible is stronger: their universal properties determine the comparison maps uniquely. Uniqueness supplies the cocycle law automatically.

We assume categories, the Yoneda lemma and sheaves on a site. For the categorical prerequisite, read The Yoneda lemma. For sheaves of modules on topological spaces, read Sheaves of modules on a ringed space, Section 2, Theorem 2.1. Basic references are [Stacks], [D'Agnolo–Polesello] and [Schapira, Sheaves].

1. A universal property includes its identification

Let \(\mathsf S\) be a stack in categories on a small site. We work with categories and sets in fixed universes so that all the stated Hom sets and covers are small in the relevant universe. A stack has two gluing properties. Morphisms between fixed local objects form sheaves. A family of objects with compatible invertible transition maps glues to an object. The morphisms of its fiber categories may be noninvertible.

For objects \(A,B\) over \(U\), write \(\mathcal Hom_{\mathsf S}(A,B)\) for the sheaf on the slice site that sends \(V\to U\) to \(\operatorname{Hom}_{\mathsf S(V)}(A|_V,B|_V)\).

Consider a functor of prestacks

\[ \mathcal F:\mathsf S^{\mathrm{op}}\longrightarrow\mathsf{Sh}, \]

where \(\mathsf{Sh}(U)\) is the category of sheaves of sets over \(U\). This means that \(\mathcal F_U\) is a contravariant functor on \(\mathsf S(U)\), with specified coherent isomorphisms commuting with restriction. A representation over \(U\) is a pair \((A,\alpha)\) with an isomorphism of such functors

\[ \alpha:\mathcal Hom_{\mathsf S}(-,A)\simeq\mathcal F|_U. \tag{1.1} \]

It specifies what the universal property does to every test object and how it behaves on every smaller region.

Lemma 1.1. Two representations \((A,\alpha)\), \((B,\beta)\) over the same region have a unique isomorphism \(u:A\to B\) satisfying \(\beta\circ\mathcal Hom(-,u)=\alpha\).

Proof. Yoneda identifies the natural transformation \(\beta^{-1}\alpha\) with a unique map \(u:A\to B\). On a slice without a represented terminal object, apply ordinary Yoneda on each test object; restriction compatibility identifies the resulting local maps, which glue because morphisms in \(\mathsf S\) are sheaves. Applying the same argument to \(\alpha^{-1}\beta\) gives an inverse. Their composites correspond to identity transformations, so they are identity maps by Yoneda. \(\square\)

The identification \(\alpha\) matters. If one remembers only that \(A\) and \(B\) are isomorphic, their automorphisms leave the transition maps undetermined.

2. Local representability is global

Theorem 2.1. If \(\mathcal F\) has a representation after a covering of the terminal sheaf, then it has a global representation. This representation is unique up to the unique isomorphism preserving (1.1).

Proof. Write the covering as \((U_i\to1)\), with local representations \((A_i,\alpha_i)\). Products in the sheaf topos provide the overlaps even if the original site does not represent them. Over \(U_i\times U_j\), Lemma 1.1 gives the unique map \(u_{ij}:A_j\to A_i\) preserving their identifications with \(\mathcal F\).

On a triple overlap, \(u_{ij}u_{jk}\) and \(u_{ik}\) both preserve that identification. Lemma 1.1 makes them equal. The same uniqueness gives identity maps on the diagonal. Thus \((A_i,u_{ij})\) is descent data. Effectiveness in \(\mathsf S\) produces a global object \(A\) and isomorphisms \(A|_{U_i}\simeq A_i\) realizing these transitions.

For any test object \(T\) over any region \(V\), the isomorphisms

\[ \mathcal Hom(T,A)|_{U_i\times V} \simeq\mathcal F(T)|_{U_i\times V} \]

agree on pairwise overlaps, by the definition of \(u_{ij}\). Morphisms of sheaves glue uniquely, so they give an isomorphism on \(V\). Naturality in \(T\) and compatibility with restriction are equalities of sheaf maps. They can be checked on the same covering, where they hold. This gives (1.1) globally. Lemma 1.1 supplies the final uniqueness assertion. \(\square\)

The topos formulation can equivalently be read using the site of test objects mapping to a sheaf \(U\). A stack extends to that site by compatible families. A family gives its local components on each test object; its coherence gives their transition maps. This explains the use of an overlap sheaf that is not itself a site object.

The theorem is about objects inside a stack. It does not assert that a sheaf of sets locally represented by objects of an arbitrary base category is globally represented by an object of that base category. The effectiveness hypothesis is precisely what allows the representing object to glue.

There is a dual version for covariant functors \(\mathsf S\to\mathsf{Sh}\) represented as \(\mathcal Hom(A,-)\). Apply Theorem 2.1 to the opposite stack. That opposite is again a stack: reverse morphisms, and replace descent isomorphisms by their inverses. The same local gluing constructions verify its two stack axioms.

3. Tensoring an object without choosing a trivialization

Let \(R\) be a sheaf of commutative rings. Suppose \(\mathsf S\) is an \(R\)-linear stack locally equivalent to the stack of \(R\)-modules. Let \(M\) be an ordinary \(R\)-module sheaf and \(K\) an object of \(\mathsf S\). Consider the covariant functor

\[ L\longmapsto \mathcal Hom_R\bigl(M,\mathcal Hom_{\mathsf S}(K,L)\bigr). \tag{3.1} \]

Here the internal Hom on the right is an \(R\)-module sheaf. Linearity makes (3.1) a functor to sheaves, compatible with restriction.

On a region where \(\mathsf S\simeq\mathsf{Mod}(R)\), ordinary tensor–Hom adjunction corepresents (3.1) by \(M\otimes_R K\). This uses arbitrary module sheaves; \(M\) need not be finite, flat or locally free. The dual form of Theorem 2.1 glues the local corepresentations.

Corollary 3.1. There is an object \(M\otimes_R K\), characterized by natural isomorphisms

\[ \mathcal Hom_{\mathsf S}(M\otimes_R K,L) \simeq\mathcal Hom_R\bigl(M,\mathcal Hom_{\mathsf S}(K,L)\bigr). \tag{3.2} \]

It is functorial in both \(M\) and \(K\), and has the ordinary tensor unit and associativity isomorphisms.

Proof. Existence follows from the preceding local corepresentation and Theorem 2.1. A map \(M\to M'\) gives, by precomposition on the right of (3.2), a map from the functor corepresented by \(M'\otimes K\) to the one corepresented by \(M\otimes K\). Covariant Yoneda converts it to \(M\otimes K\to M'\otimes K\). A map \(K\to K'\) gives a map in the same direction on tensors by precomposition in the inner Hom. Uniqueness proves the functor laws and the compatibility of the two variables.

Locally the unit \(R\otimes K\simeq K\) and the associator \((M\otimes N)\otimes K\simeq M\otimes(N\otimes K)\) are those of ordinary module tensor products. They identify the same universal properties in (3.2), so Lemma 1.1 glues them uniquely. The unit coherence diagrams and the associativity pentagon commute locally; equality of morphisms in a stack is local, so they commute globally. \(\square\)

For example, take \(R=\mathbb Z\) and \(M=\mathbb Z/n\mathbb Z\). In a local module description, \(M\otimes K\) is \(K/nK\). Formula (3.2) glues these local quotients without a global choice of that description. If \(K\) is a rank-one object locally, tensoring with this \(M\) can produce torsion; the construction asserts no local freeness of its output.

4. Exercises with solutions

Exercise 4.1 (foundation). If \(\alpha\) is replaced by \(\alpha\circ\mathcal Hom(-,v)\), where \(v\) is an automorphism of \(A\), what is the comparison from this new representation to \((A,\alpha)\)?

Solution. It is \(v\), because \(\alpha\circ\mathcal Hom(-,v)\) is the new identification. Lemma 1.1 determines it uniquely. This example shows why uniqueness concerns an object together with its identification, rather than the object alone.

Exercise 4.2 (calculation). In the ordinary category of modules over a commutative ring \(R\), corepresent the functor \(L\mapsto\operatorname{Hom}_R(R/I,\operatorname{Hom}_R(K,L))\).

Solution. Tensor–Hom adjunction corepresents it by \((R/I)\otimes_R K=K/IK\). Directly, a map from \(R/I\) to \(\operatorname{Hom}_R(K,L)\) is determined by a map \(f:K\to L\) annihilated by \(I\). Since \((af)(k)=f(ak)\), this means \(f\) vanishes on \(IK\). Such maps are exactly the maps from \(K/IK\) to \(L\).

Exercise 4.3 (hypotheses). Let \(\mathsf P\) be a prestack whose morphisms are sheaves, but whose object descent is not effective. Does the proof of Theorem 2.1 still produce a global object? Identify the precise remaining statement.

Solution. It produces local representing objects and canonical transition maps obeying the cocycle law. Thus it produces an object of the descent category for the cover. It does not produce an object of \(\mathsf P\) over the terminal sheaf, since that is exactly the missing effectiveness condition. Gluing morphisms cannot supply a missing object.

Exercise 4.4 (synthesis). Suppose two different covers and two different collections of local corepresentations construct tensors for the same pair \((M,K)\). Compare the resulting global tensors and their functorial maps.

Solution. Both carry (3.2), so the dual of Lemma 1.1 gives a unique comparison preserving it. It can be checked on a common refinement, where it is the ordinary tensor comparison. For a map in either variable, both composites with this comparison implement the same natural transformation of (3.1); Yoneda makes them equal. Hence the two tensor bifunctors are canonically isomorphic with their stated universal identifications.

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