Stackification, local morphisms and ordinary inverse images

Written by GPT-6.1 Sol (OpenAI), October 2026. Original text: CC0.

Local objects can fit together even when the category we started with has no object representing their union. There is a second issue: compatible local arrows might have no global arrow. Stackification supplies both kinds of missing data. Ordinary inverse image of a stack uses this construction, and differs from pulling back the algebra that presents a module stack.

We use the set-sheafification and local equality proof in Sieves, local isomorphisms and presheaf localization, the common-stage rules in Filtered stages and finite limits, and the descent convention in Representing objects from local data. The final example explains the ordinary inverse image used in Relative pullback of locally module stacks, Exercise 8.1.

1. Categories over opens and their descent data

Fix a topological space \(X\) and ambient universes containing its opens, the categories under consideration, their Hom sets and the required choices. Covers and all families below are sets in that setting. A category-valued presheaf \(\mathsf F\) assigns a category \(\mathsf F(U)\) to every open \(U\), restriction functors, and specified invertible comparisons between successive and direct restrictions. They satisfy the unit and associativity coherence laws. We allow all arrows in each category, including noninvertible ones.

Write \(a|_V\) for restriction. Repeated restrictions in an equation are identified using the specified comparisons. For three nested inclusions, the two comparisons to direct restriction are equal by the coherence axiom. For a longer chain, compare first with the fully left-associated restriction; induction on the chain length, applying that three-inclusion identity to each adjacent reassociation, gives the same comparison. Thus the notation does not impose strict restriction functors.

For \(a,b\in\mathsf F(U)\), the presheaf of arrows on \(U\) is

A descent datum on a cover \(U=\bigcup_i U_i\) consists of objects \(a_i\in\mathsf F(U_i)\) and isomorphisms \(\theta_{ji}:a_i|_{U_i\cap U_j}\to a_j|_{U_i\cap U_j}\), with

In particular \(\theta_{ij}=\theta_{ji}^{-1}\). An arrow between two data on the same cover is a family of arrows commuting with their transitions; these arrows need not be invertible. Call \(\mathsf F\) a stack if every \(H_{a,b}\) is a sheaf and every such object datum is effective: it is isomorphic to the restrictions of an object over \(U\). The empty cover is included, so a stack over the empty open is equivalent to the terminal category.

For a stack, restriction to any cover is an equivalence onto its descent category. Indeed the sheaf condition on (1.1) identifies global arrows with compatible local arrows, giving full faithfulness. Effectiveness gives essential surjectivity. Conversely these two properties for every cover give the two stack axioms. This is an equivalence of categories, with all arrows, rather than an equivalence only between the groupoids of isomorphisms.

A morphism of category-valued presheaves is a family of functors with invertible restriction comparisons satisfying their coherence law. An arrow between two such morphisms is a compatible family of natural transformations. Its components may be noninvertible. Denote this category of morphisms and transformations by \(\operatorname{Fun}_X(\mathsf F,\mathsf G)\).

2. First supply the missing morphisms

Let \(aH\) denote the associated sheaf of a presheaf of sets. We use two concrete properties: its sections have local representatives in \(H\), and two such representatives have equal images precisely when they agree locally. Both follow from the matching-family construction in the linked sheafification lesson. Sheafification restricts compatibly to an open subset, by restricting those representatives and their local equality tests.

Lemma 2.1. There is a category-valued presheaf \(\mathsf F^h\), with the same objects as \(\mathsf F\), and

Its morphism presheaves are sheaves. For every category-valued presheaf \(\mathsf B\) whose morphism presheaves are sheaves, restriction induces an equivalence

Proof. Given arrows in (2.1) from \(a\) to \(b\) and from \(b\) to \(c\), cover \(U\) so that both have representatives in \(\mathsf F\). Intersections of the two representing covers give a common cover. Compose the representatives there. On overlaps, equality of each pair of representatives holds on a further cover, so their composites agree locally. Their images in the sheaf \(aH_{a,c}\) agree and glue. Changing representatives produces the same glued section by that same local equality test. This defines composition.

The images of identity arrows are identities: this equality holds on each representing open. Three arrows can be represented simultaneously after taking finite intersections of their covers; associativity there is associativity in \(\mathsf F\). Equality of the resulting global sections follows from the sheaf axiom. This proves the category laws without choosing a single cover uniformly for every arrow.

Open restriction uses the restriction of the sheaves in (2.1). The original restriction comparisons on objects remain invertible in the new category. Their naturality and coherence can be checked on local representatives, where they are the given laws in \(\mathsf F\). The Hom presheaf of the resulting objects is \(aH_{a,b}\), including on each smaller open, so it is a sheaf.

To extend \(T:\mathsf F\to\mathsf B\), keep its values on objects. Its arrow map extends uniquely from \(H_{a,b}\) to the sheaf of arrows between \(T(a)\) and \(T(b)\), using the restriction comparison of \(T\) to identify their local objects. The extension preserves composition and identities because these equations hold on representatives and the target arrow presheaves are sheaves. Its restriction comparisons are the original ones; their naturality is checked locally.

A transformation between two functors on \(\mathsf F\) has a component at every object of \(\mathsf F^h\), since the objects have not changed. Naturality for a new arrow follows by representing that arrow locally and using the original naturality there. Thus restriction is fully faithful on transformations and every functor extends. This proves (2.2). \(\square\)

3. Then make object descent effective

Put \(\mathsf P=\mathsf F^h\). Define \(\mathsf P^+(U)\) as follows. An object \(A\) is a cover \((U_i)\) of \(U\), objects \(a_i\in\mathsf P(U_i)\), and transitions \(\theta_{i'i}:a_i\to a_{i'}\) on intersections satisfying (1.2). Different covers remain different presentations; they may be isomorphic objects. The empty cover gives an object over the empty open.

For a second object \(B=((V_j),b_j,\psi_{j'j})\), an arrow \(A\to B\) is a family

satisfying, on every fourfold intersection,

Every arrow in (3.1) is in \(\mathsf P\), and is not required to be invertible. This definition uses all chart intersections, so no arbitrary choice of refinement is part of a morphism.

Lemma 3.1. These data form a category-valued presheaf. The functor \(j:\mathsf P\to\mathsf P^+\) using the one-open presentation is fully faithful on every open. Every object of \(\mathsf P^+\) is locally isomorphic to an object in the image of \(j\).

Proof. For \(A\xrightarrow\alpha B\xrightarrow\beta C\), with charts \(W_k,c_k\) for \(C\), define its \((k,i)\) component by gluing

These are arrows between the same two objects restricted from \(U_i\cap W_k\). On an overlap with indices \(j,j'\), (3.2) gives \(\alpha_{j'i}=\psi_{j'j}\alpha_{ji}\) and \(\beta_{kj'}\psi_{j'j}=\beta_{kj}\). Their composites therefore agree. The sheaf of arrows in \(\mathsf P\) glues them uniquely. Equation (3.2) for the glued composite holds on the covering opens with index \(j\), where it follows from the two original compatibility equations. It thus holds everywhere.

The identity of \(A\) has components \(\theta_{i'i}\). Equations (1.2) and (3.2) show that composing with it returns every component of an arrow. For three arrows, restrict to intersections of a chart from each of the four presentations. Both iterated composites are the same three-arrow composite in \(\mathsf P\). Such intersections cover each required first-to-last intersection; uniqueness of gluing proves associativity.

Restriction intersects every chart with the smaller open and restricts all arrows. Composition commutes with this operation because the glued section is characterized by its restrictions. The restriction comparisons and their coherence are those of \(\mathsf P\); their equalities hold on each chart. This constructs the category-valued presheaf.

For two one-open presentations over \(U\), (3.1) is just one arrow in \(\mathsf P(U)\) and (3.2) imposes no new condition. Hence \(j\) is fully faithful. On \(U_i\), the transitions \(\theta_{ki}\) give an isomorphism from \(j(a_i)\) to the restricted presentation of \(A\). Their inverses are the reverse transitions, and (1.2) proves both inverse equations. \(\square\)

Theorem 3.2. The category-valued presheaf \(\mathsf P^+\) is a stack.

Proof: morphisms. Fix presentations \(A,B\) over \(U\), and a cover \((Z_r)\) of an open \(Z\subseteq U\). Compatible arrows over the \(Z_r\) supply, for every \(i,j\), compatible sections of the sheaf of arrows from \(a_i\) to \(b_j\) on the cover \((Z_r\cap U_i\cap V_j)\) of \(Z\cap U_i\cap V_j\). Glue these sections. Equation (3.2) holds on the cover indexed by \(r\), so holds for their gluings. Uniqueness follows for every component from that same sheaf axiom. This is the Hom sheaf condition.

Proof: objects. Suppose that objects \(A_r\in\mathsf P^+(Z_r)\) have isomorphisms \(d_{sr}:A_r\to A_s\) on overlaps satisfying the cocycle law. Choose presentations of \(A_r\) by objects \(a_{ri}\) over opens \(U_{ri}\subseteq Z_r\). The opens \((U_{ri})\) cover \(U=\bigcup_r Z_r\). Define the transition from \(a_{ri}\) to \(a_{sj}\) to be the corresponding component of \(d_{sr}\), restricted to \(U_{ri}\cap U_{sj}\).

These components are isomorphisms. To see this directly, the components of \(d_{rs}\), followed or preceded by those of \(d_{sr}\), are locally the identity by (3.3) and \(d_{rs}d_{sr}=1\). On a chosen pair of charts the same statement follows after restricting to charts covering the intersection, hence globally by the Hom sheaf condition in \(\mathsf P\). The cocycle equation for the \(d\)'s gives the cocycle equation for these chart transitions: restrict to a third presentation's charts in (3.3), then use equality of its glued components. For \(r=s\), they are the original transitions of \(A_r\), since \(d_{rr}=1\).

Thus these objects and transitions define a single presentation \(A\in\mathsf P^+(U)\). Over \(Z_r\), its transitions give an isomorphism to \(A_r\): their components are the corresponding components of the \(d\)'s and satisfy (3.2). The inverse and overlap equations follow on every chart from the original descent laws. They exhibit \(A\) as an effective descent of the given data. For the empty cover the unique empty presentation has exactly one endomorphism, the empty family, so the value on the empty open is equivalent to the terminal category. \(\square\)

4. The universal property, including transformations

Set \(\operatorname{St}(\mathsf F)=(\mathsf F^h)^+\), and let \(\iota:\mathsf F\to\operatorname{St}(\mathsf F)\) be the two constructed functors. Lemma 3.1 gives

Every object on the left-hand stack is locally an \(\iota a\).

Theorem 4.1. For every stack \(\mathsf B\), precomposition is an equivalence of categories

This includes all compatible natural transformations, not only their invertible components.

Proof. By Lemma 2.1 first extend a given functor to \(T:\mathsf P\to\mathsf B\). For a presentation \(A=((U_i),a_i,\theta)\), the objects \(T(a_i)\) and the images of its transitions form descent data in \(\mathsf B\), after using the restriction comparisons of \(T\). Choose an effective descent \(\overline T(A)\) and isomorphisms between its restrictions and these objects. For an arrow \(\alpha:A\to A'\), its chart components under \(T\), conjugated by these chosen isomorphisms, agree on overlaps by (3.2). The Hom sheaves in \(\mathsf B\) glue them uniquely to \(\overline T(\alpha)\).

The identity and composition laws hold on chart intersections, by the construction and the laws of \(T\), and hence hold globally. Restricting a chosen descent object and descending the restricted presentation give two objects with the same specified local data. Full faithfulness of the descent functor in Section 1 supplies a unique comparison respecting those data. These comparisons are invertible, natural in \(A\), and coherent: both sides of every required equation restrict to the same comparison on the presenting charts. This constructs a morphism of category-valued presheaves. The descent isomorphism for a one-open presentation gives \(\overline Tj\simeq T\). Hence (4.2) is essentially surjective on functors.

For full faithfulness, take morphisms \(H,K:\mathsf P^+\to\mathsf B\) and a transformation \(\gamma:Hj\to Kj\). On a presentation \(A\), use its local isomorphisms \(j(a_i)\simeq A|_{U_i}\) to transport the components \(\gamma_{a_i}\) to arrows \(H(A)|_{U_i}\to K(A)|_{U_i}\). Naturality of \(\gamma\) for the transition arrows makes them agree on overlaps. They glue uniquely to an arrow \(\gamma_A\). For any arrow \(A\to A'\), naturality holds on the intersections of presenting charts, where it is naturality for the arrows (3.1). The Hom sheaves make it hold globally. Restriction compatibility is checked on the same charts. A transformation with the prescribed restriction to \(j\) must have these local components, so is unique. No component was inverted in this argument. Combining this with (2.2) proves (4.2). \(\square\)

The construction is canonical up to coherent equivalence. Indeed, (4.2) extends any \(\mathsf F\to\mathsf G\) to the associated stacks; full faithfulness makes comparisons for identities and compositions unique after their specified restriction to \(\mathsf F\). Their coherence equations follow from that uniqueness. If \(\mathsf F\) already is a stack, every presentation has an effective descent and its Hom presheaves are unchanged. Thus \(\iota\) is an equivalence on every open. Applying this observation to two constructions satisfying (4.2) gives inverse equivalences compatible with their units, and unique compatible transformations between the choices.

Proposition 4.2 (local equivalences). A morphism \(T:\mathsf A\to\mathsf B\) between stacks is an equivalence if it induces isomorphisms of all Hom sheaves and every object of \(\mathsf B\) is locally isomorphic to an image object.

Proof. Taking sections of the Hom sheaves gives full faithfulness on every open. For \(b\in\mathsf B(U)\), choose a cover and isomorphisms \(T(a_i)\simeq b|_{U_i}\). Full faithfulness on intersections lifts their transition isomorphisms uniquely to arrows between the \(a_i\)'s. Their inverses lift too, so these arrows are invertible. Faithfulness lifts the cocycle identities. Descend the \(a_i\) in \(\mathsf A\), then glue the isomorphisms of their images to obtain \(T(a)\simeq b\). Hence every functor on an open is essentially surjective as well. Choosing quasi-inverses, their restriction comparisons are uniquely determined by full faithfulness and their counits; those comparisons satisfy coherence for the same reason. This gives an equivalence of stacks. \(\square\)

5. Ordinary inverse image along a continuous map

Let \(f:Y\to X\) be continuous and \(\mathsf A\) a stack on \(X\). For an open \(V\subseteq Y\), let \(I_V\) be the opens \(U\subseteq X\) with \(V\subseteq f^{-1}U\), ordered toward smaller opens. It is nonempty, since it contains \(X\), and intersections give common later objects. Define a category \(\mathsf A_f^0(V)\) by the following explicit filtered construction.

An object is a pair \((U,a)\), with \(U\in I_V\) and \(a\in\mathsf A(U)\). An arrow \((U,a)\to(U',b)\) is represented by an open \(W\subseteq U\cap U'\) with \(V\subseteq f^{-1}W\), and an arrow \(a|_W\to b|_W\). Two representatives are equal when their restrictions agree on a smaller such open. Equivalently,

Compose representatives after intersecting their representing opens. Changing a representative changes the composite only after a further intersection, so gives the same class. Identities and associativity follow after a common finite intersection, using the restriction comparisons of \(\mathsf A\). This proves all category laws. A smaller \(V\) allows more representing opens and hence defines restriction on objects and arrow classes. The identity and composition laws for these restrictions follow from their representatives, so this is a category-valued presheaf.

For \(W\subseteq U\) in \(I_V\), the objects \((U,a)\) and \((W,a|_W)\) are canonically isomorphic, by the identity of \(a|_W\); the two inverse equations hold on \(W\). Define

The second assignment is a stack: a cover pulls back to a cover and all its intersections pull back to the same intersections, so both descent axioms follow from those of \(\mathsf B\).

Theorem 5.1. Ordinary inverse image is left adjoint to direct image in the sense of the natural equivalence of categories

Proof. By (4.2), its left side is \(\operatorname{Fun}_Y(\mathsf A_f^0,\mathsf B)\). Given \(T:\mathsf A\to f_*\mathsf B\), send \((U,a)\) over \(V\) to \(T_U(a)|_V\). A representative over \(W\) of an arrow in (5.1) maps, using the restriction comparisons of \(T\), to an arrow over \(f^{-1}W\) between the corresponding restrictions. Restrict it to \(V\). A smaller representative gives the same arrow, and composition is respected on common representing opens. Restriction in \(V\), with the comparisons of \(\mathsf B\), supplies the required coherent morphism. A transformation of the \(T\)'s restricts by exactly this rule.

Conversely, given \(H:\mathsf A_f^0\to\mathsf B\), define \(T_U(a)=H_{f^{-1}U}(U,a)\). For \(W\subseteq U\), restrict to \(f^{-1}W\) and use the canonical isomorphism \((U,a)\simeq(W,a|_W)\). Applying \(H\) supplies the comparison with \(T_W(a|_W)\). Its naturality and coherence follow from the identity representatives just described. Transformations are evaluated on these same objects.

Starting with \(T\), these rules return \(T\) up to its specified unit and restriction comparisons. Starting with \(H\), the object \((U,a)\) over \(V\) is the restriction of that same pair over \(f^{-1}U\). The restriction comparison of \(H\) therefore identifies the reconstructed functor with \(H\), on objects and every representative in (5.1). These identifications respect every transformation. They prove the equivalence, and their formulas show its naturality in both stacks. \(\square\)

The unit sends \(a\in\mathsf A(U)\) to the one-chart image of \((U,a)\) over \(f^{-1}U\). The counit on a generator \((U,b)\), with \(b\in\mathsf B(f^{-1}U)\), is restriction of \(b\) to \(V\); it extends to the associated stack by (4.2). On these generators the triangle comparisons are the identity restrictions. Full faithfulness in (4.2) extends them uniquely to invertible transformations on the stacks, with the triangle coherence. Thus (5.3) supplies the unit and counit as well as the Hom equivalence.

6. Germs of categories and the module boundary

The stalk category \(\mathsf A_x\) has objects represented by \(a\in\mathsf A(U)\) near \(x\); arrows are represented near \(x\) with equality after shrinking, as in (5.1).

Proposition 6.1. For \(y\in Y\), there is a natural equivalence

Proof. First, \(\mathsf F\to\operatorname{St}(\mathsf F)\) induces an equivalence on stalk categories for every category-valued presheaf on a space. Every associated-stack object has a presenting chart containing the point, and is isomorphic there to an original object. This proves essential surjectivity on germs. By (4.1), a germ of an arrow between original objects is the germ of a section of \(aH_{a,b}\). Local representatives and local equality identify its germs exactly with the germs of \(H_{a,b}\). This gives full faithfulness, including noninvertible arrows.

Now take \(\mathsf F=\mathsf A_f^0\). A pair \((U,a)\) on a neighborhood \(V\) of \(y\) has \(f(y)\in U\), hence gives an object germ of \(\mathsf A\) at \(f(y)\). Every such object germ occurs, using \(V=f^{-1}U\). An arrow representative near \(y\) in (5.1) is an arrow near \(f(y)\), and every arrow near \(f(y)\) gives one by taking its inverse-image neighborhood. If two such germs are equal after shrinking in \(X\), shrink in \(Y\) to the inverse image of that neighborhood. Conversely equality in the germ construction on \(Y\) means equality on an allowed representing open \(W\); since its inverse image contains a neighborhood of \(y\), it contains \(f(y)\). Thus equality is equality of germs on \(X\). These rules preserve identities and composition. Combining the two equivalences proves (6.1). \(\square\)

For module stacks this ordinary inverse image keeps local objects coming from the original stack, with morphisms enlarged by local representation. In contrast, the relative pullback of a locally module stack uses all modules over the transported local algebra. The inclusion of ordinary pullback objects can fail to be essentially surjective. Exercise 3 computes that failure on a two-point space; it does not require a separation, compactness or finite-presentation hypothesis.

7. Four exercises with complete solutions

Exercise 1 — new morphisms before new objects

Introductory. Let \(X\) be two discrete points and \(G\) a group. On every nonempty open put the category with one object and endomorphism group \(G\), with identity restrictions; on the empty open put the terminal category. Compute the endomorphisms of the original object over \(X\) after each of the two stages. Determine the isomorphism classes of objects of its associated stack.

Solution. Initially the endomorphism group over \(X\) is \(G\). The two-point cover has no nonempty overlap, so sheafifying its Hom presheaf gives \(G\times G\), with the original map \(g\mapsto(g,g)\). On a singleton the group remains \(G\). Lemma 3.1 preserves these endomorphisms in the second stage. Any presentation is isomorphic, on each singleton, to its unique original object. The two singleton objects have a unique compatibility condition on the empty intersection, so together give a presentation over \(X\). Every other presentation is isomorphic to this one: choose its local isomorphism on each singleton and use the Hom sheaf to glue the resulting components. Thus there is exactly one isomorphism class over \(X\), with automorphism group \(G\times G\). There is likewise one class over each singleton and the empty open. The first stage can change morphisms even when the final stage adds no new isomorphism class in this example.

Exercise 2 — a line bundle is a descent presentation

Intermediate. Let \(\mathcal R\) be a sheaf of commutative rings on a space. Let \(\mathsf P(U)\) have one object, the free module \(\mathcal R|_U\), and all its module endomorphisms. Show that \(\mathsf P\) has sheaves of morphisms. Identify its associated stack, with all morphisms, and explain which data are invertible.

Solution. Evaluation at \(1\) identifies the endomorphism presheaf with \(\mathcal R|_U\); restriction is restriction of sections. It is a sheaf. A presentation in \(\mathsf P^+\) has one free rank-one module on each chart. Its transition maps are multiplication by units \(u_{ji}\in\mathcal R(U_i\cap U_j)^\times\), with \(u_{kj}u_{ji}=u_{ki}\).

Construct a module sheaf by taking, on an open \(W\), the families \(m_i\in\mathcal R(W\cap U_i)\) satisfying \(m_j=u_{ji}m_i\). This is a sheaf because compatible families of such tuples glue componentwise in \(\mathcal R\). On \(U_i\), projection to the \(i\)-th component is an isomorphism: its inverse sends \(m_i\) to the family \(u_{ji}m_i\). Thus the module is locally free of rank one. Conversely local frames of any such module give these units and the cocycle law, and reconstruct the original sheaf by its own gluing axiom.

An arrow between two presentations is locally multiplication by a section \(v_{ji}\) satisfying (3.2). Those sections give a module map between the constructed sheaves by multiplication in each frame; compatibility makes its local values agree. Every module map has those local coefficients, uniquely. These constructions therefore give an equivalence with the stack of locally free rank-one module sheaves and all module maps. Only the transition coefficients, and coefficients of an invertible map, must be units. A zero map is a valid morphism. Keeping only unit-valued morphisms would instead give its groupoid of isomorphisms, a different category.

Exercise 3 — ordinary pullback does not give every module

Advanced calculation. Let \(f:Y\to\{*\}\), where \(Y=\{0,1\}\) has opens \(\varnothing,\{1\},Y\), and let \(k\ne0\) be a commutative ring. On the point take the stack of \(k\)-modules. Compute the ordinary inverse-image stack on the two nonempty opens. Compare its global objects with all sheaves of modules over the inverse-image ring.

Solution. For a nonempty \(V\subseteq Y\), the only open \(U\) of the point with \(V\subseteq f^{-1}U\) is \(\{*\}\). Thus \(\mathsf A_f^0(V)=k\text{-}\mathsf{Mod}\), with identity restriction between the nonempty opens. Its morphism presheaves are already sheaves: every cover of \(Y\) contains \(Y\), and every cover of \(\{1\}\) contains \(\{1\}\). The same fact makes object descent effective. Explicitly a datum on such a cover is isomorphic to the restriction datum of its object on the full open, using its transition isomorphisms; the cocycle law makes these isomorphisms compatible. Its value over the empty open is terminal. Hence stackification changes neither nonempty value up to equivalence.

The inverse-image ring has value \(k\) on both nonempty opens, with identity restriction. A module sheaf on \(Y\) is an arbitrary arrow \(M_0\to M_1\) of \(k\)-modules: the covering conditions above impose no other condition. An ordinary inverse-image object \(M\) corresponds to \(M\xrightarrow{1}M\). Its morphisms are the maps \(M\to N\) acting on both components, as follows also from the commuting square for an arrow between identity diagrams. The sheaf \((k\to0)\) is not isomorphic to any identity diagram, since its two component isomorphisms would imply \(M\simeq k\) and \(M\simeq0\). Transporting the algebra and taking all its modules does include this sheaf. This is the distinction used in the relative-pullback lesson.

Exercise 4 — an open inclusion and its counit

Advanced hypotheses. Let \(j:Y\hookrightarrow X\) be an open inclusion. Prove that ordinary inverse image of stacks is restriction and that \(j^{-1}j_*\mathsf B\simeq\mathsf B\). Must \(\mathsf A\to j_*j^{-1}\mathsf A\) be an equivalence?

Solution. Every open \(V\) of \(Y\) is open in \(X\). In \(I_V\), ordered toward smaller opens, \(V\) itself is a terminal object. More explicitly, restriction sends a pair \((U,a)\) to \(a|_V\); inclusion sends \(b\in\mathsf A(V)\) to \((V,b)\). Their first composite is the identity up to the restriction unit, and the other has the canonical isomorphism from \((U,a)\) to \((V,a|_V)\) constructed after (5.1). This equivalence is natural in \(V\). Restriction of a stack to \(Y\) is already a stack, so its stackification is equivalent to itself by Theorem 4.1. Therefore \(j^{-1}\mathsf A\simeq\mathsf A|_Y\).

For \(V\subseteq Y\), direct image followed by this restriction has value \(\mathsf B(V\cap Y)=\mathsf B(V)\). Its comparisons and counit are those same identity restrictions, so \(j^{-1}j_*\mathsf B\simeq\mathsf B\).

The unit need not be an equivalence. Take \(X\) to be two discrete points and \(Y\) one of them. Choose categories \(C,D\), with \(D\) not equivalent to the terminal category, and let \(\mathsf A\) be the product stack: value \(C\times D\) on \(X\), values \(C,D\) on the respective singletons, and terminal value on the empty open. Descent for the disjoint cover is exactly this product description; every other cover refines it, so the stack axioms follow componentwise. The unit on \(X\) is the projection \(C\times D\to C\), after identifying \(C\times1\) with \(C\). For example take \(C\) terminal and \(D\) a discrete category with two objects; that projection is not essentially injective on isomorphism classes and is not an equivalence. The invertible counit therefore does not force an invertible unit.

8. Further reading

For a construction on arbitrary small Grothendieck sites, using covering sieves and all their factorization arrows, see Stackification by cartesian sections on covering sieves, Sections 1–6. The Stacks Project gives the site-theoretic definitions in Stacks, Section 8.4, stackification and its universal property in Section 8.8, and inverse/direct image comparisons in Section 8.12. These are freely accessible references. The constructions and proofs above use open covers and their actual intersections; no groupoid-only hypothesis is imposed on their fibre categories.