Stackification by cartesian sections on covering sieves

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

A covering sieve records every arrow through which local data can be compared. Its category of arrows supplies the overlap information even when the base category has no fibre products. We construct the associated stack by first sheafifying morphisms and then allowing objects presented by cartesian sections over covering sieves. Morphisms in the resulting categories, and transformations between functors, may be noninvertible.

Use the topology and set-sheafification proofs in Sieves, local isomorphisms and presheaf localization, Sections 1–2. The topological stackification lesson treats actual intersections of opens and ordinary inverse images. Here the base is an arbitrary small Grothendieck sieve site; its objects need not be spaces, its topology need not be subcanonical, and it need not have points or fibre products.

1. Sieve descent includes all base arrows

Fix a small category \(\mathcal C\), a Grothendieck topology \(J\), and ambient universes containing all categories, Hom sets and choices used below. A sieve \(R\) on \(U\) is a collection of arrows into \(U\) closed under precomposition. For \(f:V\to U\), its pullback sieve is

This is a set of arrows; it requires no pullback object in \(\mathcal C\). We use the maximal sieve, stability of covering sieves under (1.1), and the local character axiom. In particular, an intersection of two covering sieves covers. If \(Q\) covers \(U\) and \(R_q\) covers the domain of every \(q\in Q\), the sieve of composites \(qr\), with \(r\in R_q\), covers \(U\): its pullback along \(q\) contains \(R_q\), so local character applies. An empty covering sieve is allowed.

Let \(\mathsf F\) be a category-valued presheaf: restriction functors \(f^*\) have specified invertible unit and composition comparisons, with the usual coherence laws. Fibre categories have arbitrary morphisms. We insert these comparisons in repeated restrictions; coherence identifies all the reassociations, as explained in Section 1 of the topological lesson.

Regard a sieve \(R\) as its full category of elements in \(\mathcal C/U\). Its objects are arrows \(f:V\to U\) in \(R\). An arrow from \(fg\) to \(f\) is a specified factorization arrow \(g\). Distinct factorization arrows remain distinct. A cartesian section over \(R\) consists of objects \(x_f\in\mathsf F(V)\) and isomorphisms

They respect units and satisfy, after the restriction comparison,

An arrow of sections \(\alpha:x\to y\) is a family \(\alpha_f:x_f\to y_f\) with

Its components need not be invertible. Write \(\operatorname{Cart}(R,\mathsf F)\) for this category. For \(a,b\in\mathsf F(U)\), the arrow presheaf on \(\mathcal C/U\) is \(H_{a,b}(f)=\operatorname{Hom}(f^*a,f^*b)\), with restrictions transported by the given comparisons.

A stack has sheaves \(H_{a,b}\), and every cartesian section over a covering sieve is isomorphic to the restriction section of some object over \(U\). Equivalently,

Indeed a compatible arrow between two restriction sections is exactly a matching family in \(H_{a,b}\) on \(R\); the sheaf axiom glues it uniquely. This gives full faithfulness, and effective descent gives essential surjectivity. Conversely full faithfulness for every cover gives precisely that sheaf axiom, and essential surjectivity gives effectiveness. These statements apply on every slice, whose covers over \(f:V\to U\) are the covering sieves on \(V\).

2. Sheafify arrows without changing objects

Lemma 2.1. There is a category-valued presheaf \(\mathsf F^h\), with the same objects over every \(U\), such that

where \(a\) denotes set-sheafification on the slice site. Its Hom presheaves are sheaves. For every \(\mathsf B\) with sheaves of morphisms, precomposition gives an equivalence between the categories of morphisms \(\mathsf F^h\to\mathsf B\) and \(\mathsf F\to\mathsf B\), including all compatible natural transformations.

Proof. A section of \(aH_{a,b}\) has representatives on a covering sieve, and two original sections have the same image if and only if their equality sieve covers. These are the proved local rules of the linked set-sheafification construction. To compose new arrows from \(a\) to \(b\) and from \(b\) to \(c\), intersect their representing covering sieves and compose the arrows in \(\mathsf F\) on each member. On a further covering refinement, different representatives agree, so their composites agree. The resulting sections of \(aH_{a,c}\) glue uniquely. The local equality rule proves independence of representatives.

Identity and associativity follow on a common intersection of the finitely many representing sieves, where they are the category laws in \(\mathsf F\). The sheaf axiom makes the resulting global equations true. No family of base pullback objects is used.

Restriction along \(f:V\to U\) commutes with this sheafification. To verify it, compare the sheaf on \(\mathcal C/V\) obtained from \(aH_{a,b}\) with the sheafification of \(H_{f^*a,f^*b}\). Their original arrow presheaves identify by the restriction comparisons. A representative on a covering sieve of any object over \(V\) gives the same representative in both constructions; its equality tests use exactly the same covering sieves on that object. Thus these rules give inverse section maps, commuting with restrictions. They identify the two sheaves. The original comparison isomorphisms on objects consequently define the comparisons for \(\mathsf F^h\); their naturality and coherence hold locally on original arrows and hence globally.

A morphism \(T:\mathsf F\to\mathsf B\) extends on each arrow presheaf by its sheafification universal property, with the same object values. It preserves identities, composition and restriction comparisons because these equations hold on original local representatives. For a transformation, use its original components at the unchanged objects. Naturality for a new arrow holds on a representing cover and therefore globally in the target Hom sheaf. Restriction is thus fully faithful on transformations, and every functor extends, proving the asserted equivalence. \(\square\)

Put \(\mathsf P=\mathsf F^h\) for the rest of the construction.

3. Present objects over covering sieves

An object \(A\) of \(\mathsf P^+(U)\) is a pair \((R,x)\), where \(R\in J(U)\) and \(x\) is a cartesian section of \(\mathsf P\) over \(R\). Different presenting sieves remain different objects and may give isomorphic objects.

For \(A=(R,x)\) and \(B=(S,y)\), a morphism \(A\to B\) is a morphism of their restricted sections on \(R\cap S\). Thus it is a family \(\alpha_f:x_f\to y_f\), for all \(f\in R\cap S\), satisfying (1.4). We do not quotient away the arrows or the specified factorization maps in this intersection.

Lemma 3.1. These definitions give a category-valued presheaf \(\mathsf P^+\). The functor \(j:\mathsf P\to\mathsf P^+\) using the maximal sieve and restriction sections is fully faithful on every object of the site. Every object of \(\mathsf P^+\) is locally isomorphic to an object in its image.

Proof: composition. Let \(A=(R,x)\), \(B=(S,y)\), \(C=(T,z)\), and arrows \(\alpha:A\to B\), \(\beta:B\to C\). For \(f:V\to U\) in \(R\cap T\), the sieve \(f^*S\) covers \(V\). On its member \(g\), form the arrow

Equations (1.3) and (1.4) say exactly that these arrows form a matching family in the sheaf \(H_{x_f,z_f}\). Glue it to a unique arrow \(\gamma_f:x_f\to z_f\). For a further factorization \(fh\), the restrictions of \(\gamma_f\) and \(\gamma_{fh}\), transported by the section comparisons, agree on the covering sieve \((fh)^*S\), by (3.1). Hom separation gives (1.4) for \(\gamma\). This defines the composite.

The identity of \((R,x)\) has components \(1_{x_f}\). The composite with an identity is the original arrow on a covering sieve, hence globally. For three arrows between presentations on \(R,S,T,L\), fix \(f\in R\cap L\) and restrict to the covering sieve \(f^*S\cap f^*T\). Both iterated composites become the same three-arrow composite, with the same outside transport comparisons, by (1.3). The Hom sheaf proves their equality. This proves the category laws.

Proof: restrictions and local objects. Pullback of a presentation along \(h:V\to U\) uses \(h^*R\) and the section with value \(x_{hg}\) at \(g\). Restrict an arrow by the same rule. Formula (3.1) and uniqueness of its gluing prove preservation of composition. The identity and composite restriction comparisons are induced by the original ones in \(\mathsf P\); their coherence follows at each section value. Thus these are coherent restriction functors.

An arrow between \(j(a)\) and \(j(b)\) is a compatible family on the maximal sieve. Its value at \(1_U\), using the unit comparison, is an arrow \(a\to b\); equation (1.4) makes every other value its restriction. This proves full faithfulness of \(j\). For \(f\in R\), the sieve \(f^*R\) is maximal. The maps \(\xi_{f,g}\), for all \(g\), give an isomorphism

Its naturality is (1.3); the inverse uses the inverse comparisons. Hence every presentation has the claimed local form. \(\square\)

4. Gluing morphisms and objects

Theorem 4.1. The category-valued presheaf \(\mathsf P^+\) is a stack on \((\mathcal C,J)\).

Proof: morphisms. Fix \(A=(R,x)\), \(B=(S,y)\) over \(U\), an arrow \(t:V\to U\), and a covering sieve \(Q\) on \(V\). A matching family of arrows between \(t^*A\) and \(t^*B\) gives, for each \(q\in Q\), an arrow between their restrictions along \(q\), compatible with every factorization in \(Q\).

For \(f:W\to V\) with \(tf\in R\cap S\), use the cover \(f^*Q\) of \(W\). The given arrow over \(fg\in Q\) has a component from \(x_{tfg}\) to \(y_{tfg}\). Transport it by \(\xi^x_{tf,g}\) and \(\xi^y_{tf,g}\) to an arrow from \(g^*x_{tf}\) to \(g^*y_{tf}\). The matching condition and (1.3) make these a matching family in \(H_{x_{tf},y_{tf}}\). Glue to \(\alpha_f:x_{tf}\to y_{tf}\).

For a further arrow \(h\), compatibility (1.4) follows after restriction to \((fh)^*Q\), where it is the original matching condition. It therefore holds globally. The \(\alpha_f\)'s define an arrow \(t^*A\to t^*B\). Its restriction to each \(q\) is the given arrow, by its componentwise construction. Uniqueness holds for each component on \(f^*Q\). This proves the Hom sheaf axiom on every slice.

Proof: objects. Let \(Q\in J(U)\) and let \(A_q\in\mathsf P^+(\operatorname{dom}q)\) be a cartesian section over \(Q\), with comparisons

Choose a presentation \((R_q,a^q)\) of each \(A_q\). Form the sieve

It is closed under precomposition because every \(R_q\) is a sieve. It covers by the composite-cover argument of Section 1, and is contained in \(Q\). For each \(f\in R\), choose one factorization \(f=q_fr_f\) of this form, and put \(a_f=a^{q_f}_{r_f}\). Combining (3.2) for that chart with (4.1) gives a specified isomorphism

For \(g:W\to\operatorname{dom}f\), define the comparison \(g^*a_f\to a_{fg}\) as the unique arrow whose image under \(j\) is

The restriction comparison for \(j\) identifies its source with \(j(g^*a_f)\). Full faithfulness of \(j\) gives the unique arrow, and lifting the inverse of (4.4) proves it invertible. The unit and composition laws (1.3) hold: after applying the faithful functor \(j\), the intervening \(e\)'s cancel, and the equations become the cartesian-section laws for \(d\). Thus the \(a_f\)'s form a cartesian section of \(\mathsf P\) over \(R\). Let \(A=(R,a)\in\mathsf P^+(U)\).

We verify that it descends the original datum, not merely its chosen charts. For \(q\in Q\), restrict to the covering sieve \(q^*R\). At \(r\in q^*R\), the local identification of \(A\) with \(j(a_{qr})\), followed by

is a local isomorphism from \(r^*q^*A\) to \(r^*A_q\). The definition (4.4) makes these maps compatible with every further arrow. The Hom sheaf axiom just proved glues them to \(q^*A\to A_q\). Their inverses glue too, so it is an isomorphism. Its compatibility with (4.1) can be tested on the same covering sieves, where it is the coherence law for \(d\). Hence the original cartesian section is effective.

If an empty sieve covers \(U\), its empty cartesian section gives an object of \(\mathsf P^+(U)\). The Hom sheaf axiom on that empty cover makes the Hom set between any two objects a singleton. All those arrows are inverse in pairs. Thus the value is equivalent to the terminal category, including when \(\mathsf F(U)\) had no objects. This also verifies the empty-cover case of the construction. \(\square\)

5. The full universal property

Define \(\operatorname{St}(\mathsf F)=(\mathsf F^h)^+\), with unit \(\iota\). A morphism of category-valued presheaves is a family of functors with coherent invertible restriction comparisons. Its transformations are compatible natural transformations, with arbitrary, possibly noninvertible components. Write \(\operatorname{Fun}_{\mathcal C}\) for their category.

Theorem 5.1. For any stack \(\mathsf B\), precomposition induces an equivalence

The unit identifies the Hom sheaf between two original objects with the sheafification of their original Hom presheaf, and every associated-stack object is locally in its essential image.

Proof. Lemma 2.1 first extends a given functor to \(T:\mathsf P\to\mathsf B\). Apply \(T\) to the section of a presentation \((R,x)\), using its restriction comparisons. This gives a cartesian section of \(\mathsf B\) over \(R\). Choose an effective descent object \(\overline T(R,x)\), with the specified local isomorphisms. For a morphism between presentations on \(R,S\), apply \(T\) to its components on \(R\cap S\), and conjugate by those descent isomorphisms. These are a matching family in the Hom sheaf between the chosen descent objects. Glue it uniquely to define \(\overline T\) on the arrow.

Composition and identities are preserved: check them on the covering sieve where all the relevant section components exist, and use the composition rule (3.1). Hom separation then gives the global equations. Restriction of a chosen descent object and descent of the pulled-back presentation have the same specified local data. The full faithfulness of (1.5) supplies a unique invertible comparison between them. Its naturality and coherence follow by checking its prescribed values on the presenting covering sieves. This constructs a morphism \(\overline T:\mathsf P^+\to\mathsf B\), with \(\overline Tj\simeq T\).

Now take functors \(H,K:\mathsf P^+\to\mathsf B\) and a transformation between their restrictions to \(j\). For \(A=(R,x)\), use the local isomorphisms (3.2) to transport its components at \(x_f\) to arrows from \(f^*H(A)\) to \(f^*K(A)\), for every \(f\in R\). Naturality for the section comparisons makes these a matching family. Glue it uniquely. Naturality for any arrow between two presentations can be checked on the intersection of their presenting sieves; there it is the original naturality for the section components. Restriction compatibility is checked on pulled-back covering sieves. A transformation with the given restriction has exactly these local components, so is unique. None of its components has been inverted. This gives full faithfulness on transformations; Lemma 2.1 completes (5.1).

The Hom assertion follows from (2.1) and full faithfulness of \(j\), on every slice. The assertion about objects is (3.2), because \(\mathsf F\) and \(\mathsf F^h\) have the same objects. \(\square\)

The same equivalence gives coherent functoriality and uniqueness. A morphism \(\mathsf F\to\mathsf G\) extends to their associated stacks by (5.1). Comparisons for identities and compositions lift their prescribed comparisons on \(\mathsf F\) uniquely; full faithfulness makes the coherence equations hold. If \(\mathsf F\) already is a stack, every presentation is effective and the Hom sheaves are unchanged, so the unit is an equivalence. Two constructions satisfying (5.1) therefore give inverse equivalences compatible with their units; compatible transformations between such choices are unique after their unit data are specified.

For an open-set site, this construction agrees with the topological lesson. A cover generates the sieve of opens contained in its members. Given chart descent data, choose a containing chart for each such open and use its restricted object. The transition isomorphisms compare the choices and their cocycle law gives (1.3). Conversely, restricting a cartesian section to the covering charts gives the chart objects; on an intersection, the two comparison maps to its section object give the transition between them. Equation (1.3) gives the chart cocycle law. Arrows correspond by their restrictions and the Hom sheaf axiom. Alternatively (5.1) and the full topological universal property give the same compatible equivalence. Thus the sieve construction extends that foundation.

6. Equivalence can be checked locally

Proposition 6.1. Let \(T:\mathsf A\to\mathsf B\) be a morphism of stacks. Suppose it identifies every Hom sheaf, and for every \(b\in\mathsf B(U)\) there is a covering sieve \(R\) on which each \(f^*b\) is isomorphic to \(T(a_f)\). Then \(T\) is an equivalence of stacks.

Proof. Taking sections gives full faithfulness of \(T\) on each object of the site. Choose isomorphisms \(v_f:T(a_f)\to f^*b\). For a factorization \(fg\), compare \(T(g^*a_f)\) and \(T(a_{fg})\) through \(g^*v_f\), the restriction comparison of \(b\), and \(v_{fg}^{-1}\). Full faithfulness lifts this comparison uniquely. Its inverse lifts too; faithfulness then lifts the cartesian-section unit and composition laws. Effective descent in \(\mathsf A\) gives \(a\in\mathsf A(U)\) with these local objects. The \(v_f\)'s glue to \(T(a)\simeq b\), by the Hom sheaf condition in \(\mathsf B\). Thus each fibre functor is an equivalence. Its chosen quasi-inverse receives unique restriction comparisons compatible with the counit; full faithfulness makes their coherence hold, giving an equivalence of stacks. \(\square\)

More generally, a morphism of category-valued presheaves induces an equivalence of associated stacks if it induces isomorphisms after sheafifying the Hom presheaves and is locally essentially surjective on original target objects. Indeed, after the first stage its Hom maps are isomorphisms and its local lifting property persists. For two associated source objects, intersect their presenting sieves. There both objects come from originals, so the map of their Hom sheaves is an isomorphism. A map of sheaves that is an isomorphism on a cover is globally an isomorphism: local inverse images of a section glue by injectivity, and equality of two lifts is tested on the cover. This proves the Hom hypothesis of Proposition 6.1. For an arbitrary associated target object, use its presenting sieve and then the local lifting covers for its original local objects; local character combines those covers. The resulting map of stacks satisfies Proposition 6.1.

Conversely, an equivalence of associated stacks identifies the sheafified Hom presheaves by Theorem 5.1. For an original target object, represent an associated-stack preimage locally by original source objects. The resulting comparison isomorphism in the associated target stack is a section of a sheafified original Hom presheaf. Refine the covering sieve so that this section and its inverse both have original representatives. Refine once more so that the two inverse equations hold for those representatives; the local equality rule permits this finite simultaneous refinement. Local character combines the refinements into a covering sieve of the original object. On it the representatives are actual inverse arrows in the original target categories, and give the required local lifts. This proves both directions.

7. Four exercises with complete solutions

Exercise 1 — a cover with no intersection object

Introductory. Let \(\mathcal C\) have objects \(a,b,u\), two nonidentity arrows \(a\to u\), \(b\to u\), and no others. Give \(a,b\) only their maximal covering sieves, and give \(u\) both its maximal sieve and the sieve \(R\) consisting of the two displayed arrows. Verify the topology and the absence of \(a\times_u b\). For the constant one-object category with automorphism group \(G\), compute the endomorphism group of its original object over \(u\) after stackification.

Solution. Pulling \(R\) back along either of its arrows gives the maximal sieve on that domain. Pulling back a maximal sieve gives a maximal sieve. For local character on the cover \(R\), a sieve locally covering must contain both displayed arrows, so contains \(R\). For the maximal cover, local character includes the identity arrow and returns the same covering condition. These verify the topology axioms. No object admits arrows to both \(a\) and \(b\), so there cannot be a fibre product with those two projection maps.

The original Hom presheaf has values \(G\) at all three objects, with identity restrictions. A matching family on \(R\) is an arbitrary pair in \(G\times G\), since the two objects of its element category have no arrows between them. Sheafification has value \(G\times G\) at \(u\), with projection restrictions, and the original map is \(g\mapsto(g,g)\). The first stage therefore supplies that endomorphism group. Full faithfulness of \(j\) preserves it in the final stack. No intersection object or point argument is needed for this computation.

Exercise 2 — the arrows of a maximal sieve matter

Intermediate. Regard a group \(G\) as a one-object category, with only the maximal sieve covering. Show that every category-valued presheaf on this site is a stack. For a constant category, explain why descent over the maximal sieve does not amount to choosing an arbitrary representation of \(G\).

Solution. The category of elements of the maximal sieve is the slice category \(\mathcal C/*\). Its objects are the elements \(g\in G\); an arrow from \(gh\) to \(g\) is the uniquely specified element \(h\). It has terminal object \(1\), and there is a unique arrow from \(g\) to \(1\), namely \(g\). A cartesian section is therefore determined, up to its specified comparisons, by its object at \(1\): the comparison along that arrow identifies its value at \(g\) with the restriction of its value at \(1\). Equation (1.3) identifies every remaining comparison with the resulting one. A morphism of sections is likewise determined by its value at \(1\), by (1.4). Evaluation at \(1\) is an equivalence inverse to the restriction-section functor. This proves (1.5), for the only cover, for every category-valued presheaf.

For a constant category with identity restrictions, this says that a section is isomorphic to the constant section on its value at \(1\). The element category has a separate object for each \(g\) and a unique arrow between any two such objects. Its automorphism group at each object is trivial. It is not the one-object category \(BG\) itself. Consequently its cartesian sections carry no extra monodromy representation. For example, if both the base and the fibre automorphism group have order two, choosing the comparison between the two section values gives two possible presentations, but they are isomorphic; the endomorphisms of a fixed presentation remain the original fibre group, not a new representation category.

Exercise 3 — an empty covering sieve adds an object

Advanced boundary case. On the one-object identity-only category, declare the empty sieve covering. Starting with any category \(C\), including the empty category, compute its associated stack. Check the universal property when \(C\) is empty.

Solution. The topology contains the empty and maximal sieves. A sheaf of sets must have exactly one section on the unique object, since a matching family on the empty sieve is unique. Sheafifying any original Hom presheaf therefore makes its value a singleton. If \(C\) has objects, its first stage has exactly one arrow between each ordered pair; these arrows are mutually inverse. If \(C\) is empty, that stage still has no objects.

The empty cartesian section is nevertheless a presentation in the second stage. Between it and any other presentation the intersection of the presenting sieves is empty, so there is exactly one morphism, the empty family. Its two composites are the identities, also uniquely determined. Thus the final category is equivalent to the terminal category in both cases.

Every target stack \(\mathsf B\) also has its value equivalent to the terminal category by descent on the empty sieve. If \(C\) is empty, there is exactly one functor from \(C\) to that category, with its unique transformation. The functor category from the associated terminal category to \(\mathsf B\) is equivalent to \(\mathsf B\), hence to the terminal category. This verifies the universal property even though the unit began with no objects at all.

Exercise 4 — why the source must have effective descent

Advanced hypotheses. On the fork site of Exercise 1, let \(\mathsf A(u)\) be empty and let \(\mathsf A(a),\mathsf A(b)\) be terminal, with the unique restriction functors. Let \(\mathsf B\) be the terminal stack. Show that \(\mathsf A\to\mathsf B\) identifies all Hom sheaves attached to existing source objects and is locally essentially surjective, but is not an equivalence. Compute its associated-stack map.

Solution. Over \(u\) there are no source objects, so the assertion about their Hom sheaves is vacuous. Over \(a\) and \(b\), the only Hom presheaves are terminal and their maps to the corresponding sheaves in \(\mathsf B\) are isomorphisms. There are no other objects or base arrows to check. The unique object of \(\mathsf B(u)\) lifts on the covering sieve \(R\), using the unique objects over \(a,b\). On those two domains it lifts already on the maximal cover. Thus local essential surjectivity holds everywhere.

The fibre functor over \(u\) is empty-to-terminal, so is not an equivalence. Its missing descent object is exactly the section on \(R\) consisting of the two unique local objects: there is no object in \(\mathsf A(u)\) from which it can descend. In \(\operatorname{St}(\mathsf A)(u)\), the presentation on \(R\) supplies it. Every presentation over \(u\) has the same local objects, and its Hom sets are singletons, so every such presentation is uniquely isomorphic to this one. The two other fibres remain terminal. Hence the associated-stack map is an equivalence to \(\mathsf B\). This is the distinction between Proposition 6.1, whose source is a stack, and its following stackification criterion.

8. Further reading

The Stacks Project gives Grothendieck sieve topologies, the definition of stacks, and stackification of fibred categories with its universal property. All are freely accessible. The construction above uses cartesian sections on sieves throughout, including the arrows of their element categories.