Noncommutative stacks.

Maxim Kontsevich, Alexander Rosenberg

Introduction

One of the purposes of this work is to introduce a noncommutative analogue of Artin’s and Deligne-Mumford algebraic stacks in the most natural and sufficiently general way. We start with quasi-coherent modules on fibered categories, then define stacks and prestacks. We define formally smooth, formally unramified, and formally étale cartesian functors. This provides us with enough tools to extend to stacks the glueing formalism we developed in [KR3] for presheaves and sheaves of sets.

Quasi-coherent presheaves and sheaves on a fibered category.

Quasi-coherent sheaves on geometric (i.e. locally ringed topological) spaces were introduced in fifties. The notion of quasi-coherent modules was extended in an obvious way to ringed sites and toposes at the moment the latter appeared (in SGA), but it was not used much in this generality. Recently, the subject was revisited by D. Orlov in his work on quasi-coherent sheaves in commutative an noncommutative geometry [Or] and by G. Laumon an L. Moret-Bailly in their book on algebraic stacks [LM-B].

Slightly generalizing [R4], we associate with any functor F (regarded as a category over a category) the category of ’quasi-coherent presheaves’ on F (otherwise called ’quasicoherent presheaves of modules’ or simply ’quasi-coherent modules’) and study some basic properties of this correspondence in the case when the functor defines a fibered category. Imitating [Gir], we define the quasi-topology of 1-descent (or simply ’descent’) and the quasi-topology of 2-descent (or ’effective descent’) on the base of a fibered category (i.e. on the target of the functor F).

If the base is endowed with a quasi-topology, τ, we introduce the notion of a ’sheaf of modules’ on (F, τ). We define the category Qcoh(F, τ) of quasi-coherent sheaves on (F, τ) as the intersection of the category Qcoh(F) of quasi-coherent presheaves on F and the category of sheaves of modules on (F, τ).

If the quasi-topology τ is coarser than the quasi-topology of 1-descent, than every quasi-coherent module on F is a sheaf of modules on (F, τ), i.e. Qcoh(F, τ) = Qcoh(F). In this case, we show, under certain natural conditions on a presheaf of sets on the base X, the existence of a ’coherator’ which is, by definition, a right adjoint to the embedding of the category Qcoh(F/X) of quasi-coherent modules on X into the category of sheaves of modules on X (that is on F/X). This fact is important, because the existence of the coherator on X garantees the existence of the direct image functor (between quasi-coherent modules) for any morphism from a presheaf of sets to X.

The relation of this formalism with the classical notions and those used in [Or] is as follows. With any ringed category (A, O), one can naturally associate a fibered category F: its fiber over an object T of A is the category opposite to the category of O(T)-modules. The category Qcoh(F) of quasi-coherent modules on the fibered category F is equivalent to the category of quasi-coherent O-modules in the sense of [Or]. If τ is a topology on A, then the category Qcoh(F, τ) is equivalent to the category of quasi-coherent sheaves of O-modules in the classical (i.e. [SGA]) sense. In particular, if F is the fibered category of modules over (commutative) affine schemes and the presheaf X is represented by a scheme (or an algebraic space) X, then the category Qcoh(F/X) is naturally equivalent to the category of quasi-coherent sheaves on the scheme (resp. on the algebraic space) X.

A standard noncommutative example is the ringed category (Aff, O), where Aff is the category opposite to the category of associative rings and the presheaf of rings O assigns to any object SpecR of Aff (corresponding to the ring R) the ring R. Thus, to any presheaf of sets X on Aff, we assign the category Qcoh(X) of quasi-coherent modules on X. If the presheaf X is representable by SpecR, then Qcoh(X) is equivalent to the category R − mod of left R-modules. If X is a locally affine space, the category Qcoh(X) is described via affine covers and relations.

We show, among other facts, that the canonical topology on Aff (i.e. the strongest topology such that every representable presheaf of sets on Aff is a sheaf) is precisely the topology of 1-descent. In the commutative case, this fact was established by D. Orlov [Or].

Noncommutative stacks.

A stack is a sheaf of categories on a (pre)topology. One of the classical examples is the category of (commutative) affine schemes with Zariski topology with the category of quasi-coherent sheaves assigned to each affine scheme. The sheaf property means an effective descent property for quasi-coherent sheaves.

In noncommutative algebraic geometry, (pre)topologies are replaced by Q-categories (where Q stands for ’quotient’), and stacks are sheaves of categories (in an appropriate sense) on a Q-category. We introduce monopresheaves of categories, otherwise called ’prestacks’, and epipresheaves of categories the meaning of which is explained below.

A standard example of a noncommutative stack is the Q-category of associative unital rings with faithfully flat morphisms (the category of rings is a quotient of the category of flat covers) with the category of left R-modules assigned to each ring R. The Q-category in this example gives rise to a quasi-topology which is coarser than the quasi-topology of effective descent. The latter is equivalent to the fact that this is, indeed, an example of a noncommutative stack.

In general, every quasi-topology on a category A induces a structure of a Q-category on the opposite category. A fibered category over the category A is a stack (resp. prestack) if the quasi-topology is coarser that the quasi-topology of 2-descent (resp. of 1-descent).

There is another interpretaion of a Q-category illustrated by the following example: the Q-category CRings of commutative unital rings with ring epimorphisms having a nilpotent kernel. This example suggests that Q-categories might be regarded also as ”categories of thickennings”.

In this case, sheaves (resp. monopresheaves) of categories over the category CAff of commutative affine schemes (i.e. the category opposite to the category CRings) can be interpreted as ’formally etale’ (resp. formally unramified) fibered categories; and epipresheaves of categories are formally smooth fibered categories over CAff. An appropriate (not quite obvious) choice of a Q-category produces a noncommutative version of formally étale, formally unramified, and formally smooth fibered categories over the category Aff of noncommutative affine schemes.

We consider the relative situation (i.e. morphisms of fibered categories) and define the notions of formally étale, formally unramified, and formally smooth cartesian functors. Note that even in the case of stacks over the site of commutative schemes, our notions are applicable to a much larger class of cartesian functors (morphisms of stacks) than the conventional ones, since they do not require any representability assumptions.

We define smooth, unramified, and étale morphisms as formally smooth (resp. formally unramified, resp. formally étale) locally finitely presentable morphisms.

Given a quasi-pretopology (i.e a class of coverings) τ on the category of stacks (or fibered categories) over a Q-category, we define ’τ-locally affine stacks’ as stacks which are locally representable. Using the notions of smooth and étale cartesian functors, we introduce smooth, étale, and Zariski quasi-topologies on the category of stacks over a given Q-category. In particular, we define smooth, étale and Zariski quasi-topologies on the category of stacks over the category Aff of noncommutative affine schemes endowed with the flat quasi-topology.

Locally affine stacks on Aff with respect to smooth (resp. étale) quasi-topology are noncommutative versions of algebraic Artin’s (resp. Deligne-Mumford’s) stacks.

We define ’local constructions’ on fibered categories (in particular, on stacks) which is a device to transfer certain functorial constructions of (noncommutative) ’varieties’ defined over an affine base to constructions of ’varieties’ over stacks, in particular, over arbitrary locally affine spaces. Among them are affine and projective vector-fibers corresponding to a quasi-coherent module on a locally ringed category, and Grassmannians corresponding to a pair of locally projective quasi-coherent modules on a locally ringed category.

The paper is organized as follows.

In Section 1, we introduce and study quasi-coherent presheaves and sheaves of modules on an arbitrary fibered category. This includes quasi-topologies of 1- and 2-descent, and the existence (under certain conditions) of a coherator.

In Section 2, we apply the facts and constructions of Section 1 to the fibered categories associated with ringed categories (in particular, to ringed sites and toposes).

The second part of the work is dedicated to noncommutative stacks.

Section 3 contains preliminaries on representability of fibered categories and cartesian functors.

Noncommutative stacks and prestacks are defined in Section 4.

In Section 5, we give some standard examples of stacks.

In Section 6, we obtain (as a natural extension of the results and definitions of the first part of the work) the notions of (formally) smooth and (formally) étale morphisms of fibered categories.

In Section 7, we introduce locally affine stacks. In particular, we define noncommutative versions of resp. Artin’s stacks and Deligne-Mumford’s stacks.

In Section 8, we define ’local constructions’ on fibered categories (in particular, on stacks) and illustrate them with several applications.

Our introduction to stacks is restricted to several basic notions illustrated by a couple of examples which we have chosen to sketch here because they are a natural extension of the sheaf theory and the facts on formally smooth and formally étale morphisms presented in [KR3], and also the next step after introducing quasi-coherent presheaves on fibered categories and descent quasi-topologies discussed in Section 1.

A more detailed exposition of stacks will appear in a subsequent paper.

The paper has two appendices. The first appendix is a short summary on Q-categories. Appendix 2 contains preliminaries on fibered categories.

A considerable part of this manuscript was written while the second author was visiting Max-Planck Institute für Mathematik in Bonn during the summer of 2001. He would like to thank the Institute for excellent working conditions.

The work of the second author was partially supported by the NSF grant DMS-0070921.

I. Quasi-coherent sheaves on fibered categories.

1. Quasi-coherent modules on a fibered category.

1.1. Modules and quasi-coherent modules on a category over a category. Let E be a category (which belongs to some universum 𝔘) and 𝔉 = (F, F →^{π} E) a category over E. Denote by Mod(𝔉) the category opposite to the category of all sections of 𝔉. We shall call objects of Mod(𝔉) modules on 𝔉.

We denote by Qcoh(𝔉) the category opposite to the category Cart_{E}(E, 𝔉) of cartesian sections of 𝔉. In other words, Qcoh(𝔉) = (Lim𝔉)^{op} (cf. A2.5.5). Objects of Qcoh(𝔉) will be called quasi-coherent modules on 𝔉.

Any morphism 𝔉 ⟶ 𝔊 of E-categories induces a functor Mod(𝔉) ⟶ Mod(𝔊). Thus we have a functor Mod : Cat/E ⟶ Cat from the category of E-categories to the category of categories.

Similarly, the map 𝔉 ⟼ Qcoh(𝔉) extends to a functor Qcoh : Cart_{E} ⟶ Cat from the category of cartesian functors over E to Cat.

1.1.1. Proposition. The functor Qcoh : Cart_{E} ⟶ Cat preserves small products.

Proof. In fact, by A2.6.6 and A2.6.6.2, the functor Lim : Cart_{E} ⟶ Cat preserves small products. The functor Qcoh is, by definition, the composition of Lim and the canonical automorphism Cat ⟶ Cat, C ⟼ C^{op}.

1.1.2. Modules and quasi-coherent modules on a fibered category. Let 𝔉 be a fibered category corresponding to a pseudo-functor E^{op} ⟶ Cat,

ObE ∋ X ⟼ F_{X}, HomE ∋ f ⟼ f^{∗}, HomE ×_{ObE} HomE ∋ (f, g) ⟼ c_{f,g} (1)

(cf. A2.7, A2.7.1). Then the category Mod(𝔉) can be described as follows. An object of Mod(𝔉) is a function which assigns to each T ∈ ObE an object M(T) of the fiber F_{T}

and to each morphism T ⟶^{f} T′ a morphism f^{∗}(M(T′)) ⟶^{ξ_{f}} M(T) such that ξ_{gf} ∘ c_{f,g} = ξ_{f} ∘ f^{∗}(ξ_{g}). Morphisms are defined in a natural way.

An object (M, ξ) of Mod(𝔉) belongs to the subcategory Qcoh(𝔉) iff ξ_{f} is an isomorphism for all f ∈ HomE.

1.1.3. Proposition. Let 𝔉 be a fibered category over E. Suppose the category E has a final object, T_{•}. Then

(a) The category Qcoh(𝔉) is equivalent to the category F_{T_{•}}^{op} dual to the fiber of 𝔉 at the final object T_{•}.

(b) The inclusion functor Qcoh(𝔉) ⟶ Mod(𝔉) has a right adjoint.

Proof. (a) The equivalence is given by the functor Qcoh(𝔉) ⟶ F_{T_{•}} which assigns to every quasi-coherent module M on 𝔉 the object M(T_{•}) of F_{T_{•}}. The quasi-inverse functor maps any object L of F_{T_{•}} to the quasi-coherent module L^{∼} which assigns to each object S of E the object f^{∗}(L). Here f is the unique morphism S ⟶ T_{•}.

(b) The composition of the functor Mod(𝔉) ⟶ F_{T_{•}}^{op}, M ⟼ M(T_{•}), with the equivalence F_{T_{•}}^{op} ⟶ Qcoh(𝔉) constructed in (a) is a right adjoint to the inclusion functor Qcoh(𝔉) ⟶ Mod(𝔉).

1.1.4. Base change and quasi-coherent modules.

1.1.4.1. Proposition. Let 𝔉 = (F, F →^{π} E) be a category over E and E′ ⟶ E a functor. Let 𝔉 ×_{E} E′ denote the induced category (F ×_{E} E′, F ×_{E} E′ ⟶ E′).

Then Qcoh(𝔉 ×_{E} E′) is isomorphic to the full subcategory of Hom_{E}(E′, 𝔉)^{op} objects of which are E-functors that transform any morphism into a cartesian morphism.

If 𝔉 is a fibered category over E and F_{c} is the subcategory of F formed by all cartesian morphisms of F, then ObQcoh(𝔉 ×_{E} E′) ≃ ObHom_{E}(E′, F_{c}).

Proof. The assertion follows from A2.6.7.2.

Let 𝔘, 𝔙 be two universums such that 𝔘 ∈ 𝔙. Denote by Cart_{𝔘,𝔙} the 2-category objects of which are categories over categories 𝔉 = (F →^{π} E) such that the base E belongs to 𝔙 and each fiber belongs to 𝔘. Denote by 𝔐Cart_{𝔘,𝔙} all cartesian functors (1-morphisms of Cart_{𝔘,𝔙})

\begin{matrix}F′ & ⟶^{u} & F \\ π′↓ & & ↓ π \\ E′ & ⟶^{v} & E \end{matrix}  (1)

such that the functors induced on fibers are category equivalences. It follows that 𝔐Cart_{𝔘,𝔙} is a 2-subcategory of the 2-category Cart_{𝔘,𝔙}.

1.1.4.2. Proposition. The map 𝔉 ⟼ Qcoh(𝔉) extends to a pseudo-functor

Qcoh : 𝔐Cart_{𝔘,𝔙}^{op} ⟶ Cat_{𝔙}.

Proof. Let (1) be an arbitrary cartesian morphism. It can be decomposed in two cartesian morphisms

\begin{matrix}F′ & ⟶^{u′} & F ×_{E} E′ & ⟶^{v^{∼}} & F \\ π′↓ & & ↓ & & ↓ π \\ E′ & ⟶^{Id_{E′}} & E′ & ⟶^{v} & E \end{matrix} ,  (2)

where the right square is the canonical pull-back. By 1.1.4.1, the right square of (2) induces a functor

Qcoh(𝔉) ⟶ Qcoh(𝔉 ×_{E} E′)  (3)

and by 1.1.1, the left square of (2) induces a functor

Qcoh(𝔉′) ⟶ Qcoh(𝔉 ×_{E} E′).  (4)

The morphism (1) belongs to 𝔐Cart iff u′ in (2) is a category equivalence, which implies that (4) is a category equivalence. Taking the composition of (3) with a quasi-inverse to (4), we assign to the morphism (1) a functor Qcoh(𝔉) ⟶ Qcoh(𝔉′). This correspondence defines a pseudo-functor 𝔐Cart^{op} ⟶ Cat.

1.1.5. Quasi-coherent modules on presheaves of sets. Let X be a presheaf of sets on the base E, i.e. a functor E^{op} ⟶ Sets. Then we have a functor E/X ⟶ E and the category 𝔉/X := 𝔉 ×_{E} E/X over E/X obtained via a base change (as usual, we identify E with a full subcategory of the category E^{∧} of presheaves of sets on E formed by representable presheaves). Notice that any morphism of the category E/X over E is cartesian. Therefore, by 1.1.4.1, the category Qcoh(𝔉/X) is equivalent to the category Cart_{E}(E/X, 𝔉)^{op} opposite to the category of cartesian functors E/X ⟶ 𝔉.

1.1.5.1. The canonical extension of a fibered category. Following [Gir], we denote the category Cart_{E}(E/X, 𝔉) = Qcoh(𝔉/X)^{op} by 𝔉^{+}(X). The correspondence X ⟼ 𝔉^{+}(X) extends to a pseudo-functor, hence defines a fibered category over E^{∧} which is called (in [Gir]) the canonical extension of 𝔉 onto E^{∧}.

1.1.5.2. Proposition. Let 𝔉 be a fibered category over E and X an object of the category E^{∧} of presheaves of sets on E.

(a) If the functor X is a representable by an object, x, of the category E, then the category Qcoh(𝔉/X) is equivalent to the category 𝔉_{x}^{op} opposite to the fiber 𝔉_{x} over x.

(b) Suppose X = colim(X_{i}) for some diagram I ⟶ E^{∧}, i ⟼ X_{i}. Then the natural functor Qcoh(𝔉/X) ⟶ limQcoh(𝔉/X_{i}) is an isomorphism.

(c) X ⟼ Qcoh(𝔉/X) is a sheaf of categories on E^{∧} for the canonical topology.

Proof. (a) This fact is a consequence of 1.1.3.

(b) The assertion follows from the isomorphism Cart(E/X, 𝔉) ⟶^{∼} lim(Cart(E/X_{i}, 𝔉) proven in [Gir] 3.2.4.

The assertion (c) follows from (b).

1.2. The quasi-topology and topology of 𝔉 − i-descent. Recall that a functor is called 0-faithful (resp. 1-faithful, resp. 2-faithful) if it is faithful (resp. fully faithful, resp. an equivalence ([Gir], 0.5.1.1).

1.2.1. Definition. Let 𝔉 be a fibered category over E, h_{X} the presheaf represented by an object X of E. A subpresheaf T of h_{X} is called a subpresheaf of 𝔉 − i-descent, i = 0, 1, 2, if the corresponding functor Qcoh(𝔉/X) ⟶ Qcoh(𝔉/T) (or, equivalently, 𝔉_{X}^{op} ⟶ Qcoh(𝔉/T)) is i-faithful.

A family of morphisms 𝔛 = {X_{i} → X| i ∈ I} is said to be of 𝔉 − i-descent if the subpresheaf of h_{X} associated with 𝔛 is of 𝔉 − i-descent.

The difinition 1.2.1 is equivalent to the usual definition of a sieve of 𝔉 − i-descent (cf. [Gir], II.1.1.1 and II.1.1.1.1). Another terminology: 𝔉 − 1-descent is called simply 𝔉-descent and 𝔉 − 2-descent is called also effective descent.

1.2.2. The quasi-topology of 𝔉−i-descent. For any X ∈ ObE, denote by 𝔗_{𝔉,i}(X) the set of all subpresheaves of h_{X} which are of 𝔉−i-descent. This defines a quasi-topology, 𝔗_{𝔉,1}, which we call the quasi-topology of 𝔉 − i-descent.

1.2.2.1. Proposition. The quasi-topology of 𝔉−1-descent is the finest quasi-topology such that for any X ∈ ObE and any x, y ∈ Ob𝔉_{X}, the presheaf

Hom_{X}(x, y) : E/X ⟶ Sets, (Y →^{f} X) ⟼ Hom_{Y} (f^{∗}(x), f^{∗}(y)),  (1)

is a sheaf on E/X for the induced quasi-topology.

Proof. The presheaf Hom_{X}(x, y) being a sheaf for all x, y ∈ Ob𝔉_{X} is equivalent to the full faithfulness of the functor 𝔉_{X}^{op} ⟶ Qcoh(𝔉/T) for any T ∈ 𝔗(X). The assertion follows now from the definition of the quasi-topology of 𝔉 − 1-descent (see also 1.2.1).

1.2.3. Definition. Let 𝔉 be a fibered category over E, h_{X} the presheaf represented by an object X of E. A subpresheaf T in h_{X} is called a subpresheaf of universal 𝔉 − idescent, i = 0, 1, 2, if for any morphism Y ⟶ X in E, the subpresheaf T ×_{h_{X}} h_{Y} of h_{Y} is of 𝔉 − i-descent.

1.2.4. The topology of 𝔉 − i-descent. For any X ∈ ObE, denote by 𝔗_{𝔉,i}^{u}(X) the set of all subpresheaves of h_{X} which are of universal 𝔉 − i-descent. This defines a topology which is called the topology of 𝔉 − i-descent.

1.2.4.1. Proposition. The topology of 𝔉 − 1-descent is the finest topology such that for any X ∈ ObE and any x, y ∈ Ob𝔉_{X}, the presheaf

Hom_{X}(x, y) : E/X ⟶ Sets, (Y →^{f} X) ⟼ Hom_{Y} (f^{∗}(x), f^{∗}(y)),  (1)

is a sheaf on E/X for the induced topology.

Proof. The assertion follows from 1.2.2.1.

1.2.5. Coinduced topologies and 𝔉 − i-descent. Let 𝔗 be a topology on the category A, and let 𝔗^{∧} denote the coinduced topology on the category A^{∧} of prespaces. Recall that the topology 𝔗^{∧} is defined as follows: for any prespace X, a sieve V ↪ h_{X} = Hom(−, X) is a refinement of X for 𝔗^{∧} iff for any S ∈ ObA and any morphism S ⟶ X, the sieve V ×_{h_{X}} h_{S} ↪ h_{S} is a refinement of S for 𝔗.

1.2.5.1. Example. If 𝔗 is the discrete topology on E, then the coincduced topology on E^{∧} coincides with the canonical topology on E^{∧}.

A morphism f : X ⟶ Y is called covering (resp. bicovering) if the induced morphism of associated sheaves, f^{a} : X^{a} ⟶ Y^{a} is an epimorphism (resp. an isomorphism).

1.2.6. Proposition. Let 𝔉 be a fibered category over a category E and i an integer 0 ≤ i ≤ 2. Let 𝔉^{+} denote the canonical extension of 𝔉 onto E^{∧} (cf. 1.1.5.1).

(a) The topology of 𝔉^{+} − i-descent is the topology coinduced by the topology of 𝔉 − idescent.

(b) A morphism X ⟶^{f} Y in E^{∧} is bicovering for the topology of 𝔉^{+} − i-descent iff for any morphism Y′ ⟶ Y, the corresponding functor

Qcoh(𝔉/Y′) ⟶ Qcoh(𝔉/X ×_{Y} Y′)  (1)

is i-faithful. The converse is true if i = 2 (i.e. (1) is a category equivalence), or if f is a monomorphism.

Proof. The assertions (a) and (b) are equivalent to the assertions resp. (iii) and (iv) of II.11.3 in [Gir].

1.2.7. Proposition. Let 𝔛 = {X_{i} → X| i ∈ I} be a family of arrows in E^{∧}, and let X′ →^{τ} X be the image of 𝔛. Then 𝔛 is of 𝔉 − i-descent iff the corresponding inverse image functor τ^{∗} = Qcoh(τ) : Qcoh(𝔉/X) ⟶ Qcoh(𝔉/X′) is i-faithful.

Proof. The assertion is equivalent to the assertion II.1.1.3.1 in [Gir] (which is a part of the argument of II.1.1.3).

1.2.8. Canonical topology on presheaves of sets and the effective descent. If 𝔛 = {X_{i} → X| i ∈ I} is a cover for the canonical topology on E^{∧}, then the image X′ of 𝔛 coincides with X. By 1.2.7, the family 𝔛 is a cover for the effective 𝔉-descent (or 𝔉 − 2-descent) topology. In particular, the topology of the effective 𝔉^{+}-descent is finer than the canonical topology (hence any subcanonical topology) on E^{∧}.

1.2.8.1. Remark. One can deduce the latter fact directly from the part (a) of 1.2.6 as follows. Let 𝔉 be a fibered category over E. If the topology of 𝔉 − i-descent is finer than a topology 𝔗 than, evidently, the coinduced topology 𝔗^{∧} on E^{∧} is coarser than the topology of 𝔉^{+} − i-descent, because by 1.2.6 the topology of 𝔉^{+} − i-descent is coinduced by the topology of 𝔉 − i-descent). In particular, the topology 𝔗_{d}^{∧} coinduced by the discrete topology is coarser than the topology of the effective 𝔉^{+}-descent. But, as it has been already observed (in 1.2.5.1), 𝔗_{d}^{∧} coincides with the canonical topology on E^{∧}.

1.3. Sheaves of modules.

1.3.1. Sheaves of modules on a cofibered category. Let 𝔉 be a cofibered category over E corresponding to a pseudo-functor E ⟶ Cat,

ObE ∋ X ⟼ F_{X}, HomE ∋ f ⟼ f_{•}, HomE ×_{ObE} HomE ∋ (f, g) ⟼ ((fg)_{•} ⟶^{c_{f,g}} f_{•}g_{•})

(1)

(cf. A2.7, A2.7.1). Then the category Mod(𝔉) can be described as follows. An object of Mod(𝔉) is a function which assigns to each T ∈ ObE an object M(T) of the fiber F_{T} and to

each morphism T ⟶^{f} T′ a morphism M(T′) ⟶^{ξ_{f}} f_{•}(M(T)) such that c_{f,g}∘ξ_{gf} = f_{•}(ξ_{g})∘ξ_{f}. Morphisms are defined in a natural way.

Let M be an object of Mod(𝔉). For any object X of the category E and any subpresheaf R ↪ h_{X}, we have a cone

{M(X) ⟶^{ξ_{f}} f_{•}(M(Y))| Y →^{f} X factors by R ↪ h_{X}}.  (2)

Denote by 𝔗_{M}(X) the set of all subpresheaves R ↪ h_{X} such that the cone (2) is terminal. The correspondence 𝔗_{M} : X ⟼ 𝔗_{M}(X) is a quasi-topology on E.

Let 𝔗 be a quasi-topology on E. We say that M ∈ ObMod(𝔉) is a sheaf, or a sheaf of modules on (𝔉, 𝔗), if the quasi-topology 𝔗_{M} is finer than 𝔗; i.e. for any X ∈ ObE and any R ∈ 𝔗(X), the cone (2) is terminal. We denote by Mod(𝔉, 𝔗) the full subcategory of Mod(𝔉) formed by sheaves of modules.

1.3.2. Sheaves of modules on a fibered category. Notice that the cone 1.3.1(2) above is terminal iff for any z ∈ Ob𝔉_{X}, the cone

{Hom_{𝔉_{X}} (z, M(X)) ⟶^{𝔉_{X}(z,ξ_{f})} Hom_{𝔉_{X}} (z, f_{•}(M(Y)))| Y →^{f} X factors by R ↪ h_{X}} (3)

is terminal. But the cone (3) is naturally isomorphic to the cone

{Hom_{𝔉_{X}} (z, M(X)) ⟶ Hom_{𝔉_{Y}} (f^{∗}(z), M(Y))| Y →^{f} X factors by R ↪ h_{X}} (4)

Here the map Hom_{𝔉_{X}} (z, M(X)) ⟶ Hom_{𝔉_{X}} (f^{∗}(z), M(Y)) sends a z →^{α} M(X) to the composition of f^{∗}(α) : f^{∗}(z) ⟶ f^{∗}(M(X)) and the morphism f^{∗}(M(X)) ⟶ M(Y).

This observation gives rise to the following

1.3.2.1. Definition. Let 𝔉 be a fibered category over E, and let 𝔗 be a quasitopology on E. We call a presheaf of modules M on 𝔉 a sheaf if for any X ∈ ObE and any R ∈ 𝔗(X), the cone (4) is terminal for all z ∈ Ob𝔉_{X}.

If 𝔉 is a bifibered category, then this definition is equivalent to that of 1.3.1. We denote by Mod(𝔉, 𝔗) the category of sheaves of modules on (𝔉, 𝔗)

1.3.2.2. Quasi-coherent sheaves. We denote by Qcoh(𝔉, 𝔗) the intersection of the categories Qcoh(𝔉) and Mod(𝔉, 𝔗) and call objects of this category quasi-coherent sheaves on (𝔉, 𝔗).

1.3.3. Proposition. Let 𝔉 be a fibered category over E, and let 𝔗 be a quasi-topology on E. The following conditions are equivalent:

(a) The quasi-topology of 𝔉 − 1-descent is finer than 𝔗.

(b) For any X ∈ ObE, the category Qcoh(𝔉/X) is a subcategory of the category Mod(𝔉/X, 𝔗/X) of sheaves of modules on E/X (with the induced quasi-topology 𝔗/X).

Proof. By definition, a subpresheaf R of h_{X} is of 𝔉-descent if the inverse image functor Qcoh(𝔉/h_{X}) ⟶ Qcoh(𝔉/R) of the embedding R ↪ h_{X} is a fully faithful functor. By 1.1.5.2(a), the category Qcoh(𝔉/h_{X}) is equivalent to 𝔉_{X}^{op}. Let M be a quasi-coherent module on 𝔉. And let z be any object of 𝔉_{X}. Since M is quasi-coherent, for any morphism f : Y ⟶ X in E, the object M(Y) is isomorphic to f^{∗}(M(X). Thus the cone 1.3.2(4) is isomorphic to the cone

{Hom_{𝔉_{X}} (z, M(X)) ⟶ Hom_{𝔉_{X}} (f^{∗}(z), f^{∗}(M(X)))| Y →^{f} X factors by R ↪ h_{X}}. (5)

The cone (5) is terminal, because if 𝔗 is of 𝔉-descent, then for any X ∈ ObE and any z, x ∈ Ob𝔉_{X}, the presheaf of sets

Hom_{𝔉_{X}} (z, y) : (Y, Y →^{f} X) ⟼ Hom_{𝔉_{Y}} (f^{∗}(z), f^{∗}(x))

is a sheaf on E/X for the induced quasi-topology (see 1.2.2.1). This imply also the assertion (b).

1.3.3.1. Thus the quasi-topology (resp. topology) of 𝔉 − 1-descent is the finest among quasi-topologies (resp. topologies) 𝔗 on E such that for any X ∈ ObE, quasicoherent modules on E/X are sheaves on (E/X, 𝔗/X). In particular, if 𝔗 is coarser than the quasi-topology of 𝔉 − 1-descent, then all quasi-coherent modules on 𝔉 are sheaves on the quasi-site (E, 𝔗).

1.3.4. Fibered category of sheaves of modules over presheaves of sets. Let 𝔉 be a fibered category over E and 𝔗 a quasi-topolgy on E. To any X ∈ ObE^{∧}, we assign the category Mod(𝔉/X, 𝔗/X) of sheaves of modules on E/X (with the induced quasi-topology 𝔗/X). This correspondence extends to a functor (E^{∧})^{op} ⟶ Cat, hence defines a fibered category, 𝔐od^{+}(𝔉, 𝔗).

1.3.4.1. Note. Let 𝔐od(𝔉, 𝔗) be the restriction of the fibered category 𝔐od^{+}(𝔉, 𝔗) to E. Recall that a canonical extension, 𝔐od(𝔉, 𝔗)^{+}, of the fibered category 𝔐od(𝔉, 𝔗) onto E^{∧} is defined by 𝔐od(𝔉, 𝔗)_{X}^{+} = Qcoh(𝔐od(𝔉, 𝔗)/X). It follows from definitions that this extension coincides with the fibered category 𝔐od^{+}(𝔉, 𝔗).

1.3.4.2. Lemma. The quasi-topology 𝔗 is of effective 𝔐od(𝔉, 𝔗)-descent.

Proof. The argument is left to the reader.

1.3.5. Proposition. Let 𝔉 be a fibered category over E and 𝔗 a quasi-topolgy on E which is coarser than the 𝔉 − 1 descent quasi-topology. Let

ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 ⟶^{f} X  (1)

be a diagram in E^{∧} such that ℜ and 𝔘 are representable, f ∘ p_{1} = f ∘ p_{2}, and the morphism Cok(p_{1}, p_{2}) ⟶ X corresponding to f is bicovering. Then the inclusion functor

Qcoh(𝔉/X) ⟶^{𝔮_{X}^{∗}} Mod(𝔉/X, 𝔗/X)

has a right adjoint.

Proof. The condition that the canonical morphism Cok(p_{1}, p_{2}) ⟶ X is bicovering means that the inverse image functor Qcoh(𝔉/X) ⟶ Qcoh(𝔉/Cok(p_{1}, p_{2})) is a category equivalence. Thus, we can (and will) assume that the diagram of presheaves of sets (1) is exact.

Consider the quasi-commutative diagram

\begin{matrix}Qcoh(𝔉/X) & ⟶^{f^{∗}} & Qcoh(𝔉/𝔘) & \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} & Qcoh(𝔉/ℜ) \\ 𝔮_{X}^{∗} ↓ & & ↓ 𝔮_{𝔘}^{∗} & & ↓ 𝔮_{ℜ}^{∗} \\ Mod(𝔉/X, 𝔗/X) & ⟶^{f^{•}} & Mod(𝔉/𝔘, 𝔗/𝔘) & \overrightarrow{⟶}^{p^{•}_{1}}_{p^{•}_{2}} & Mod(𝔉/ℜ, 𝔗/ℜ) \end{matrix}  (2)

corresponding to the diagram (1). Since the functors 𝔘 and ℜ are representable, the functors 𝔮_{𝔘}^{∗} and 𝔮_{ℜ}^{∗} have right adjoints, resp. 𝔮_{𝔘∗} and 𝔮_{ℜ∗}. Recall that the functor 𝔮_{𝔘∗} (resp. 𝔮_{ℜ∗} assigns to every sheaf M its value at 𝔘 (resp. at ℜ) (cf. 1.1.3). This implies that

𝔮_{ℜ∗} ∘ p_{i}^{•} = p_{i}^{∗} ∘ 𝔮_{𝔘∗}.  (3)

By 1.1.5.2, the subdiagram

Qcoh(𝔉/X) ⟶^{f^{∗}} Qcoh(𝔉/𝔘) \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} Qcoh(𝔉/ℜ)

of (2) is exact. This means that the category Qcoh(𝔉/X) can be identified with a category whose objects are pairs (M, ϕ), where M ∈ Ob𝔉_{𝔘} and ϕ an isomorphism p_{1}^{∗}(M) ⟶^{∼} p_{2}^{∗}(M). Morphisms (M, ϕ) ⟶ (M′, ϕ′) are given by morphisms M ⟶^{g} M′ such that p_{2}^{∗}(g) ∘ ϕ = ϕ′ ∘ p_{1}^{∗}(g). The functor f^{∗} maps (M, ϕ) to M and a morphism (M, ϕ) ⟶^{g} (M′, ϕ′) to

M ⟶^{g} M′.

Similarly, it follows from 1.3.4.2 and 1.1.5.2 that the subdiagram

Mod(𝔉/X, 𝔗/X) ⟶^{f^{•}} Mod(𝔉/𝔘, 𝔗/𝔘) \overrightarrow{⟶}^{p^{•}_{1}}_{p^{•}_{2}} Mod(𝔉/ℜ, 𝔗/ℜ)

is exact, hence the category Mod(𝔉/X, 𝔗/X) of sheaves on X admits an analogous description: its objects are pairs (L, ψ), where L ∈ ObMod(𝔉/𝔘, 𝔗/𝔘) and ψ an isomorphism p_{1}^{•}(L) ⟶^{∼} p_{2}^{•}(L), etc.. The functor 𝔮_{X}^{∗} maps an object (M, ϕ) of the category Qcoh(𝔉/X) to the object (𝔮_{𝔘}^{∗}(M), 𝔮_{ℜ}^{∗}(ϕ)). A right adjoint to 𝔮_{X}^{∗} is induced by the right adjoint 𝔮_{𝔘∗} to the inclusion functor 𝔮_{𝔘}^{∗}.

In fact, let (L, p_{1}^{•}(L) →^{ψ} p_{2}^{•}) be an object of Mod(𝔉/𝔘, 𝔗/𝔘). Thanks to (3), we have isomorphisms:

p_{1}^{∗}𝔮_{𝔘∗}(L) ⟶^{∼} 𝔮_{ℜ∗}p_{1}^{•}(L) ⟶^{𝔮_{ℜ∗}(ψ)} 𝔮_{ℜ∗}p_{2}^{•}(L) ⟶^{∼} p_{2}^{∗}𝔮_{𝔘∗}(L)

the composition of which, ψ′, defines an object (𝔮_{𝔘∗}(L), ψ′) of the category Qcoh(𝔉/X). The map (L, ψ) ⟼ (𝔮_{𝔘∗}(L), ψ′) extends to functor

Mod(𝔉/X, 𝔗/X) ⟶^{𝔮_{X∗}} Qcoh(𝔉/X).

It is left to the reader to check that the functor 𝔮_{X∗} is a right adjoint to 𝔮_{X}^{∗}.

1.3.5.1. Remarks. (i) If the quasi-topology in 1.3.5 is of effective descent, it suffices

to require that the canonical morphism Cok(p_{1}, p_{2}) ⟶^{f̂} X is a cover.

(ii) The argument of 1.3.5 is valid in a more general setting. Namely, one can replace representability of 𝔘 and ℜ by the existence of right adjoints to the inclusion functors

Qcoh(𝔉/𝔘) ⟶^{𝔮_{𝔘}^{∗}} Mod(𝔉/𝔘, 𝔗/𝔘) and Qcoh(𝔉/ℜ) ⟶^{𝔮_{ℜ}^{∗}} Mod(𝔉/ℜ, 𝔗/ℜ) which satisfy

the condition (3). Notice that in this case a right adjoint, 𝔮_{X∗}, to 𝔮_{X}^{∗} satisfies this condition too: f^{∗} ∘ 𝔮_{X∗} ≃ 𝔮_{𝔘∗} ∘ f^{•} (cf. the argument of 1.3.5).

1.3.5.2. Corollary. Let 𝔉 be a fibered category over E, and let

ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 ⟶^{f} X  (1)

be an exact diagram in E^{∧} such that ℜ and 𝔘 are representable. Then the inclusion functor 𝔮_{X}^{∗} : Qcoh(𝔉/X) ↪ Mod(𝔉/X) has a right adjoint.

Proof. Let C be a category with the discrete topology, 𝔗_{d}. The corresponding coinduced topology 𝔗_{d}^{∧} on the category C^{∧} of presheaves of sets on C can be described in terms of covers as follows. A set of morphisms {U_{i} ⟶ X| i ∈ J} is a cover iff the corresponding presheaf morphism ∐_{i∈J} U_{i} ⟶ X is surjective; in particular, any surjective presheaf morphism U ⟶ X is a cover in the topology 𝔗_{d}^{∧}. This shows that the topology 𝔗_{d}^{∧} is the canonical topology on C^{∧}.

Taking C = E endowed with the discrete topology, notice that for any S ∈ ObE every quasi-coherent module on S is a sheaf; i.e. the condition of 1.3.5 holds for the discrete topology on E. Moreover, by 1.2.8, the canonical topology on E^{∧} is of effective descent. Thus the assertion follows from 1.3.5 and 1.3.5.1(i).

2. Modules and quasi-coherent modules over a ringed category.

A ringed category is a pair (A, O), where A is a category and O is a presheaf of rings

on A. For any arrow T ⟶^{f} T′, let O(T)−mod ⟶^{f_{∗}} O(T′)−mod be the pull-back functor

corresponding to the ring morphism O(T′) ⟶^{O(f)} O(T). The map which assigs to each object T of A the category O(T) − mod^{op} opposite to the category of left O(T)-modules

and to each morphism T ⟶^{f} T′ the functor (O(T) − mod)^{op} ⟶^{f_{∗}^{op}} (O(T′) − mod)^{op} is a functor A ⟶ Cat which defines a cofibered category, 𝔐(A, O), over A with the fiber (O(T)−mod)^{op} over T ∈ ObA. The category 𝔐(A, O) is fibered (hence bifibered), because for any T ⟶^{f} T′, the functor f_{∗} has a left adjoint,

O(T′) − mod ⟶^{f^{∗}} O(T) − mod, M ⟼ O(T) ⊗_{O(T′)} M,

or, equivalently, f_{∗}^{op} has a right adjoint, (f^{∗})^{op}.

2.1. Modules over noncommutative affine schemes. Our standard example of a ringed category is the category Aff_{k} = Alg_{k}^{op} of affine k-schemes endowed with the presheaf O which assigns to SpecR the k-algebra R. The corresponding bifibered category 𝔐(A, O) will be called the bifibered category of modules over noncommutative affine kschemes.

2.2. Presheaves of modules. Let O − mod denote the category Mod(𝔐(A, O)) of modules on the fibered category 𝔐(A, O). An object of O−mod is a function which assigns to each T ∈ ObA an O-module M(T) and to each morphism T ⟶^{f} T′ an O(T′)-module morphism M(T′) ⟶^{γ_{f}} f_{∗}(M(T)) such that γ_{gf} = g_{∗}(γ_{f}) ∘ γ_{g}. Objects of the category O − mod are called presheaves of O-modules on A.

2.3. Quasi-coherent modules. We define the category Qcoh(A, O) of quasicoherent modules on (A, O) as the category Qcoh(𝔐(A, O)) of quasi-coherent modules on the fibered category 𝔐(O). It follows from definitions that an object of the category Qcoh(A, O) is a presheaf M of O-modules such that for any morphism T ⟶^{f} T′, the dual to γ_{f} morphism

f^{∗}(M(T′)) = O(T) ⊗_{O(T′)} M(T′, ξ′) ⟶^{γ_{f}^{∨}} M(T, ξ)

is an isomorphism.

2.4. Proposition. Suppose the category A has a final object, T_{•}. Then the category Qcoh(A, O) is equivalent to the category O(T_{•}) − mod of left O(T_{•})-modules.

Proof. The assertion is a special case of 1.1.3.

2.5. Example. Let (A, O) be as in 2.1; i.e. A = Aff_{k} and O(SpecR) = R for any k-algebra R. Notice that Speck is a final object of the category Aff_{k}. It follows from 2.4 (or 1.1.3) that the category Qcoh(A, O) is equivalent to the category k − mod of left k-modules.

2.6. Modules on prespaces. Fix a ringed category (A, O). Consider the category A^{∧} of presheaves of sets on A. We shall call presheaves of sets on A prespaces, or Aprespaces, and regard representable presheaves as affine prespaces. For any X ∈ ObA^{∧}, we have the category A/X of objects over X and the canonical functor A/X ⟶^{p_{X}} A. The presheaf of rings O induces a presheaf of rings O_{X} on A/X so that p_{X} becomes a morphism of ringed categories. The bifibered category 𝔐(A/X, O_{X}) is naturally isomorphic to the category obtained from 𝔐(A, O) via the base change by the functor p_{X}.

Thus we have the category O_{X} − mod of presheaves of O_{X}-modules which we denote by Mod_{X} and its full subcategory of quasi-coherent O_{X}-modules which we denote by Qcoh_{X} (instead of Qcoh(A/X, O_{X})).

If X ∈ ObA^{∧} is representable by an object T_{X} of A, then the category A/X has a final object, and by 2.4, the category Qcoh_{X} is equivalent to the category O(T_{X}) − mod of left O(T_{X})-modules.

2.6.1. Fibered category of modules over prespaces. Let X, Y ∈ ObA^{∧}. To any morphism X ⟶^{f} Y, there corresponds a functor

A/X ⟶^{f^{a}} A/Y, (T, ξ) ⟼ (T, f ∘ ξ),  (1)

which lifts to a fibered category morphism

𝔐(O_{X}) ⟶ 𝔐(O_{Y}), (M, (T, ξ)) ⟼ (M, (T, f ∘ ξ)).

The latter induces the ’pull-back’ functor Mod_{Y} ⟶^{f^{•}} Mod_{X} which maps each module M on Y to the module f^{•}(M) on X defined by f^{•}(M)(R.ξ) = M(R, f ∘ξ). The map assigning to any presheaf morphism X ⟶^{f} Y the functor (f^{•})^{op} : Mod_{Y}^{op} ⟶ Mod_{X}^{op} extends to a pseudo-functor, hence defines a fibered category of modules over prespaces. This fibered category is, actually, bifibered because for any prespace morphism f, the functor f^{•} has a right adjoint, f_{•}.

2.6.2. The fibered category of quasi-coherent modules. For any prespace morphism X ⟶^{f} Y, the corresponding functor 𝔐(A/Y, O_{Y}) ⟶ 𝔐(A/X, O_{X}) (cf. 2.6.1)

is cartesian, hence the functor Mod_{Y} ⟶^{f^{•}} Mod_{X} maps quasi-coherent modules to quasicoherent modules, i.e. it induces a right exact functor Qcoh_{Y} ⟶^{f^{∗}} Qcoh_{X} which we call inverse image functor of f.

Suppose X, Y are objects of A. Then a morphism f determines a ring morphism

O(Y) ⟶^{O(f)} O(X), Qcoh_{X} ≃ O(X) − mod, Qcoh_{Y} ≃ O(Y) − mod (cf. 2.4), and the functor

f^{∗} is equivalent to O(X)⊗_{O(Y)} : O(Y) − mod ⟶ O(X) − mod.

In general, one might interpret f^{∗} as the functor M ⟼ O_{X} ⊗_{O_{Y}} M.

The map X ⟼ Qcoh_{X}^{op}, f ⟼ (f^{∗})^{op} extends to a pseudo-functor from the category A^{∧} of A-prespaces to Cat which defines the fibered category 𝔔𝔠𝔬𝔥(A, O) of quasi-coherent modules on A-prespaces.

2.7. Quasi-topology and topology of descent.

2.7.1. Lemma. Let (A, O) be a ringed category, and let 𝔗 be a topology (resp. quasi-topology) on A.

(a) A presheaf M of O-modules is a sheaf on (𝔐(O), 𝔗) (cf. 1.3) iff M is a sheaf of abelian groups.

(b) The topology (resp. quasi-topology) 𝔗 is coarser than the topology (resp. quasitopology) of 𝔐(O) − 1-descent iff for any X ∈ ObA and any O(X)-module L, the presheaf on A/X which assigns to any object (S, S →^{f} X) of A/X the O(X)-module f_{∗}f^{∗}(L) = f_{∗}(O(S) ⊗_{O(X)} L) is a sheaf of O(X)-modules.

In particular, if 𝔗 is coarser than the quasi-topology of 𝔐(O) − 1-descent, than O is a sheaf of rings on (A, 𝔗).

Proof. (a) The assertion follows from definitions.

(b) By 1.3.3, the (quasi-)topology of 𝔐(O) − 1-descent is the finest (quasi-)topology such that for any X ∈ ObA, quasi-coherent modules on 𝔐(O_{X}) are sheaves on (A, 𝔗). Quasi-coherent modules on 𝔐(O_{X}) map each object (S, S →^{f} X) of A/X the O(X)-module f_{∗}f^{∗}(L) = f_{∗}(O(S) ⊗_{O(X)} L) for some O(X)-module L (cf. 2.4), hence the assertion.

2.7.2. Canonical topology and the descent topology on the category of commutative affine schemes. Let A be the category CAff of commutative affine schemes, O a presheaf of rings on A which assigns to any affine scheme (X, O_{X}) the ring ΓO_{X} of global sections of the structure sheaf O_{X}.

2.7.2.1. Lemma. The quasi-topology of 𝔐(O) − 1-descent is a topology.

Proof. Let

{SpecR_{i} ⟶^{ϕ_{i}} SpecR | i ∈ I}  (1)

be a family of scheme morphisms. It follows from 2.7.1(b) (and from the isomorphism Spec(R_{i} ⊗_{R} R_{j}) ≃ X_{i} ×_{X} X_{j}, where X_{i} = SpecR_{i} and X = SpecR) that (1) is a cover for the quasi-topology of 𝔐(O) − 1-descent iff for any R-module M the diagram

M ⟶ ∏_{i∈I} R_{i} ⊗_{R} M \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⊗_{R} R_{j} ⊗_{R} M  (2)

is exact. In particular, for any morphism SpecS ⟶ SpecR, the diagram

S ⊗_{R} M ⟶ ∏_{i∈I} R_{i} ⊗_{R} (S ⊗_{R} M) \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⊗_{R} R_{j} ⊗_{R} (S ⊗_{R} M)  (3)

is exact. The latter means that the family of morphisms {SpecS×_{X} X_{i} ⟶ SpecS | i ∈ I} is a cover for the quasi-topology of 𝔐(O)−1-descent, hence the quasi-topology of 𝔐(O)−1descent is a topology.

2.7.2.2. Proposition. The canonical topology on CAff coincides with the topology of 𝔐(O) − 1-descent.

Proof. The family (1) is a cover for the canonical topology iff for any morphism SpecS ⟶ SpecR the diagram of R-modules

S ⟶ ∏_{i∈I} R_{i} ⊗_{R} S \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⊗_{R} R_{j} ⊗_{R} S  (4)

is exact. If (1) is a cover in the topology of 𝔐(O)−1-descent, then the diagram (4) is exact. This shows that the canonical topology on A is finer than the topology of 𝔐(O)−1-descent.

On the other hand, to any R-module M there corresponds an augmented R-algebra S_{M} = R ⊕ M with zero multiplication on M. One can easily check that the exactness of the diagram (4) for S = S_{M} is equivalent to the exactness of the diagram (2).

2.7.3. Canonical topology and the descent topology on the category of noncommutative affine schemes. Proposition 2.7.2.2 extends to the noncommutative case. Namely, there is the following

2.7.3.1. Proposition. Let (A, O) be the ringed category of (noncommutative) affine k-schemes; i.e. A is the category Aff_{k} = Alg_{k}^{op} of affine k-schemes and the presheaf O is defined by O(SpecR) = R for any associative k-algebra R (cf. 2.1). Then the topology of 𝔐(O) − 1-descent coincides with the canonical topology on A = Aff_{k}.

Proof. A family

{SpecR_{i} ⟶^{ϕ_{i}} SpecR | i ∈ I}  (1)

of morphisms of Aff_{k} is a cover for the descent quasi-topology iff for any R-module M the diagram

M ⟶ ∏_{i∈I} R_{i} ⊗_{R} M \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⋆_{R} R_{j} ⊗_{R} M  (2)

is exact.

The family (1) is a cover for the canonical topology iff for any morphism SpecS ⟶ SpecR, the diagram

S ⟶ ∏_{i∈I} R_{i} ⋆_{R} S \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⋆_{R} R_{j} ⋆_{R} S  (3)

is exact.

(a) Suppose M is an R-bimodule. Let S_{M} be an algebra which is isomorphic to R⊕M as R-bimodule with zero multiplication on M. Then we have a commutative diagram

\begin{matrix}M & ⟶ & ∏_{i∈I} R_{i} ⊗_{R} M & \overrightarrow{⟶} & ∏_{i,j∈I} R_{i} ⋆_{R} R_{j} ⊗_{R} M \\ ↓ & & ↓ & & ↓ \\ S_{M} & ⟶ & ∏_{i∈I} R_{i} ⋆_{R} S_{M} & \overrightarrow{⟶} & ∏_{i,j∈I} R_{i} ⋆_{R} R_{j} ⋆_{R} S_{M} \end{matrix}  (4)

If (1) is a cover for the canonical topology, then lower row in (4) is an exact diagram. Vertical arrows in (4) define a morphism from the diagram (2) to the diagram

S_{M} ⟶ ∏_{i∈I} R_{i} ⋆_{R} S_{M} \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⋆_{R} R_{j} ⋆_{R} S_{M}  (5)

which is a retraction (in particular, vertical arrows are split monomorphisms). This implies that the diagram (2) is exact too.

(b) Let M be an arbitrary left R-module. Denote by S_{M} the algebra S_{M′}, where M′ is the R-bimodule M ⊗_{k} R. The argument (a) applied to S_{M′} proves that the diagram (2) is exact, i.e. (1) is a cover for the 𝔐(O) − 1 descent topology.

(c) Let {SpecR_{i} ⟶^{ϕ_{i}} SpecR | i ∈ I} be a cover for the 𝔐(O) − 1 descent topology. Then for any morphism SpecS ⟶ SpecR, the family {Spec(R_{i} ⋆ S) ⟶ SpecS | i ∈ I} is a cover for the 𝔐(O) − 1 descent topology; i.e. for any S-module M, the diagram

M ⟶ ∏_{i∈I} R_{i} ⋆_{R} S ⊗_{S} M \overrightarrow{⟶} ∏_{i,j∈I} R_{i} ⋆_{R} R_{j} ⋆_{R} S ⊗_{S} M

is exact. Taking M = S, we obtain the exact diagram (3). Thus, {SpecR_{i} ⟶^{ϕ_{i}} SpecR | i ∈ I} is a cover for the canonical topology.

2.7.3.2. Corollary. A subpresheaf T ⟶^{ι} X of a representable presheaf X on Aff_{k}. is a refinement in the canonical topology iff the inverse image functor Qcoh_{X} ⟶^{ι^{∗}} Qcoh_{T} is fully faithful.

Proof. This follows from the fact that the canonical topology on Aff_{k} is the topology of 𝔐(O) − 1-descent.

2.7.3.3. Corollary. Every quasi-coherent module on S ∈ ObAff_{k} is a sheaf for the canonical topology on Aff_{k}/S. In particular, for any subcanonical topology 𝔗_{S} on Aff_{k}/S, all quasi-coherent modules on S are sheaves.

Proof. The fact follows from 2.7.3.1 and 1.3.3.

2.7.3.4. Note. The assertion 2.7.3.2 is proven in [Or] for the commutative case. The corollary 2.7.3.3 is also a result by D. Orlov [Or, Proposition 4.9].

2.8. Sheaves of modules. Let 𝔗 be a topology on the category A. The category of sheaves of modules on the fibered category 𝔐(A, O) coincides with the category Mod(A, 𝔗; O) of sheaves of left O-modules on the site (A, 𝔗) in the conventional sense. The category Mod(A, 𝔗; O) is a Grothendieck category with small products, i.e. an abelian category satisfying the Grothendieck’s conditions AB5 and AB3^{∗} which has a family of generators (cf. [SGA4], II).

For any prespace X, denote by Mod_{X}^{𝔗} the full subcategory of Mod_{X} whose objects are sheaves of modules with respect to the topology 𝔗_{X} induced by 𝔗 on A/X.

Let X ∈ ObA^{∧} and A/X ⟶^{p_{X}} A the canonical functor. A presheaf M of O_{X}-modules is a sheaf on (X, 𝔗_{X}) (i.e. on the site (A/X, 𝔗_{X})) iff for any S ∈ ObA and S ⟶^{ξ} X, the presheaf ξ^{•}(M) is a sheaf on S.

In particular, a quasi-coherent module M on X is a sheaf on (X, 𝔗_{X}) iff for any S ∈ ObA and S ⟶^{ξ} X, the inverse image ξ^{∗}(M) of M is a sheaf on S.

The functor A/X ⟶^{p_{X}} A induces a functor Mod(A, 𝔗) ⟶^{p_{X}^{•}} Mod_{X}^{𝔗}. The functor p_{X}^{•} has a right adjoint, p_{X•}, and a left adjoint, p_{X!}.

Similarly, for any prespace morphism X ⟶^{f} Y, the functor A/X ⟶^{p_{f}} A/Y induces

an ’inverse image’ functor Mod_{Y}^{𝔗} ⟶^{p_{f}^{•}} Mod_{X}^{𝔗} which has a right adjoint, p_{f•}, and a left adjoint, p_{f!}.

2.8.1. Coherator. Suppose every quasi-coherent module on (A, O) is a sheaf on the site (A, 𝔗), i.e. Qcoh(A, O) is a full subcategory of the category Mod(A, 𝔗; O) of the sheaves of O-modules on (A, 𝔗). A right adjoint (if any) to the inclusion functor

Qcoh(A, O) ⟶^{ψ_{X}^{𝔗}} Mod(A, 𝔗; O) is called a coherator on the ringed site (A, 𝔗; O).

Since Mod(A, 𝔗; O) is a Grothendieck category with small products, the existence of the coherator implies that the category Qcoh(A, O) of quasi-coherent modules is a Grothendieck category with small products too (see [BD], 5.39).

If every quasi-coherent module on a prespace X is a sheaf on X, i.e. Qcoh_{X} is a (full) subcategory of Mod_{X}^{𝔗}, we have the notion of a coherator on (X, 𝔗_{X}).

2.8.2. Proposition. Suppose the topology 𝔗 on A is such that for any S ∈ ObA, quasi-coherent modules on S are sheaves on (A/S, 𝔗_{S}). Let X ⟶^{f} Y be a prespace morphism. If there exists a coherator on (Y, 𝔗_{Y}). Then the inverse image functor f^{∗} : Qcoh_{Y} ⟶ Qcoh_{X} has a right adjoint, f_{∗} (a direct image functor of f). In particular, any morphism to an affine space has a direct image.

Proof. In fact, the functor Mod_{Y} ⟶^{f^{•}} Mod_{X} has a right adjoint, f_{•}. The pair of adjoint functors f^{•}, f_{•} induces a pair of adjoint functors

Mod_{Y}^{𝔗} ⟶^{f_{𝔗}^{•}} Mod_{X}^{𝔗}, Mod_{X}^{𝔗} ⟶^{f_{𝔗•}} Mod_{Y}^{𝔗} .

The composition of f_{𝔗}^{•} with the inclusion functor Qcoh_{Y} ⟶^{j_{Y}} Mod_{Y}^{𝔗} equals to the

composition of Qcoh_{Y} ⟶^{f^{∗}} Qcoh_{X} and the inclusion functor Qcoh_{X} ⟶^{j_{X}} Mod_{X}^{𝔗}.

Since the functor j_{Y} has a right adjoint, a coherator Mod_{Y}^{𝔗} ⟶^{ψ_{Y}} Qcoh_{Y}, the functor f_{𝔗}^{•} ∘ j_{Y} = j_{X} ∘ f^{∗} is left adjoint to the functor ψ_{Y} ∘ f_{𝔗•}. Denote by f_{∗} the composition ψ_{Y} ∘ f_{•} ∘ j_{X} : Qcoh_{X} ⟶ Qcoh_{Y}. Thus defined functor f_{∗} is a right adjoint to f^{∗}. In fact, for any L ∈ ObQcoh_{Y} and M ∈ ObQcoh_{X}, we have functorial isomorphisms:

Qcoh_{Y} (L, ψ_{Y} ∘ f_{𝔗•} ∘ j_{X}(M)) ≃ Mod_{X}(f_{𝔗}^{•} ∘ j_{Y} (L), j_{X}(M))
= Mod_{X}(j_{X} ∘ f^{∗}(L), j_{X}(M)) ≃ Qcoh_{X}(f^{∗}(L), M),

hence the assertion.

2.8.3. A formula for the coherator. Assume that the inclusion functor j_{X} has a

right adjoint, Mod_{X}^{𝔗} ⟶^{ψ_{X}} Qcoh_{X}. Since ψ_{X} is left exact, it preserves kernels of pairs of morphisms. In particular, it maps the exact diagram (3) to the exact diagram

ψ_{X}(M) ⟶ ψ_{X}π_{•}(M(U, π)) \overrightarrow{⟶} ψ_{X}ν_{•}(M(R, ν)).  (4)

The equality j_{U} ∘ π^{∗} = π^{•} ∘ j_{X} (reflecting the fact that π^{•} maps quasi-coherent modules to quasi-coherent modules) implies that π_{∗} ∘ ψ_{U} ≃ ψ_{X} ∘ π_{•}. Similarly, ν_{∗} ∘ ψ_{R} ≃ ψ_{X} ∘ ν_{•}. Therefore the diagram (4) is isomorphic to the diagram

ψ_{X}(M) ⟶ π_{∗}ψ_{U}(M(U, π)) \overrightarrow{⟶} ν_{∗}ψ_{R}(M(R, ν)).  (5)

Since the diagram (4) is exact, the diagram (5) is exact too; i.e. ψ_{X}(M) is isomorphic to the kernel of the pair of arrows π_{∗}ψ_{U}π^{•}(M) \overrightarrow{⟶} ν_{∗}ψ_{R}ν^{•}(M).

2.8.4. Proposition. Suppose the topology 𝔗 on A is such that for any S ∈ ObA, quasi-coherent modules on S are sheaves on (A/S, 𝔗_{S}). Let X be a prespace such that there exists a diagram

ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 ⟶^{π} X  (1)

where R and U are representable, π ∘ p_{1} = π ∘ p_{2}, and the morphism Cok(p_{1}, p_{2}) ⟶ X induced by π is bicovering. Then the inclusion functor j_{X} : Qcoh_{X} ⟶ Mod_{X}^{𝔗} has a right adjoint.

Proof. The assertion follows from 1.3.5 and 1.3.5.2.

2.8.5. Quasi-coherent modules on prespaces and quasi-coherent modules on the associated spaces. Fix a ringed site (A, 𝔗; O).

2.8.5.1. Lemma. Let (A, 𝔗) be a site, X a prespace on A and X^{a} the associated space (i.e. the associated sheaf of sets). The canonical morphism X ⟶ X^{a} is a cover in the coinduced topology 𝔗^{∧}.

Proof. Let H_{𝔗} denote the corresponding Heller’s functor A^{∧} ⟶ A^{∧} defined by H_{𝔗}(X)(T) = colim(A^{∧}(S, X)| S ∈ 𝔗(T)) for all T ∈ ObA. It follows from the definition of H_{𝔗} that that the canonical morphism X ⟶^{τ} H_{𝔗}(X) is a cover in the topology 𝔗^{∧}. The associated sheaf, X^{a}, is isomorphic to H_{𝔗}^{2}(X) and the canonical morphism X ⟶ X^{a} corresponds to the composition τH_{𝔗}(X) ∘ τ(X) of two covers, hence it is a cover itself.

The following fact is well known (see [SGA4], II, or [Or], 2.4).

2.8.5.2. Proposition. For any prespace X, the canonical morphism X ⟶^{𝔧_{X}} X^{a}

induces an equivalence of categories Mod_{X^{a}}^{𝔗} ⟶^{𝔧_{X}^{•}} Mod_{X}^{𝔗}.

Proof. The assertion follows from the fact that X ⟶ X^{a} is a cover in the coinduced topology 𝔗^{∧}. Details are left to the reader.

2.8.5.3. Corollary. Suppose the topology 𝔗 on A is of 1-descent, i.e. quasi-coherent modules on S are sheaves on (A/S, 𝔗_{S}). Let X be a prespace and X ⟶^{𝔧_{X}} X^{a} the canonical

morphism. Then the inverse image functor Qcoh_{X^{a}} ⟶^{𝔧_{X}^{∗}} Qcoh_{X} is fully faithful.

Proof. In the commutative diagram

Qcoh_{X^{a}}  ⟶^{𝔧_{X}^{∗}} Qcoh_{X}
↓  ↓
Mod_{X^{a}}^{𝔗} ⟶^{𝔧_{X}^{•}} Mod_{X}^{𝔗}

the vertical arrows are full embeddings and the lower horizontal arrow, 𝔧_{X}^{•}, is a category equivalence, hence 𝔧_{X}^{∗} is fully faithful.

2.9. A description of the category of quasi-coherent modules. Let (A, O) be a ringed category. Let X be a presheaf of sets on A such that there exists a diagram

ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 ⟶^{π} X  (1)

where π ∘ p_{1} = π ∘ p_{2}, and the morphism Cok(p_{1}, p_{2}) ⟶ X induced by π is bicovering. Then the category Qcoh_{X} of quasi-coherent modules on X is equivalent to the category Ker(p_{1}^{∗}, p_{2}^{∗}) whose objects are pairs (M, ϕ), where M is an object of Qcoh_{𝔘} and ϕ is an isomorphism p_{1}^{∗}(M) ⟶^{∼} p_{2}^{∗}(M). Morphisms from (M, ϕ) to (M′, ϕ′) are given by arrows M ⟶^{g} M′ which make the diagram

p_{1}^{∗}(M) ⟶^{ϕ} p_{2}^{∗}(M)
p_{1}^{∗}(g) ↓  ↓ p_{2}^{∗}(g)
p_{1}^{∗}(M′) ⟶^{ϕ′} p_{2}^{∗}(M′)

commute. If 𝔘 and ℜ are representable by objects resp. U and R, then Qcoh_{𝔘} = O(U) − mod, Qcoh_{ℜ} = O(R) − mod, and inverse image functor p_{i}^{∗}, i = 1, 2, is the tensoring

L ⟼ O(R) ⊗_{O(U)} L corresponding to the ring morphism O(U) ⟶^{O(p_{i})} O(R). This follows from the argument of 1.3.5. In particular, we have the following

2.9.1. Proposition. Let (A, O) be a ringed category and 𝔗 a quasi-topology on A

which is coarser than the quasi-topology of effective descent. Let ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 ⟶^{π} X be an

exact diagram of sheaves of sets on A. Then the category Qcoh_{X} of quasi-coherent modules on X is equivalent to the category Ker(p_{1}^{∗}, p_{2}^{∗}).

If ℜ, 𝔘 are representable by objects resp. R and U of the category A, then the category Ker(p_{1}^{∗}, p_{2}^{∗}) is described by a linear algebra data: its objects are pairs (L, ϕ), where L is an O(U)-module and ϕ is an O(R)-module isomorphism p_{1}^{∗}(L) →^{∼} p_{2}^{∗}(L).

2.9.2. Remark. The category Ker(p_{1}^{∗}, p_{2}^{∗}) in 2.9.1 is equivalent to the category of

quasi-coherent modules on the cokernel of the pair ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 If Cok(p_{1}, p_{2}) ⟶^{g} X is a

presheaf morphism such that the corresponding map of associated sheaves is an isomorphism (and 𝔗 is coarser than the quasi-topology of 2-descent), than g^{∗} is an equivalence of Qcoh_{X} and Qcoh_{Cok(p_{1},p_{2})}. This allows to find the category of quasi-coherent modules on a space without finding the space itself. We illustrate this observation in the following examples.

2.9.3. Application: quasi-coherent modules on Grassmannians. Let R be a k-algebra, M and V projective left R-modules of finite type. Then the functors G_{M,V} and ℜ_{M,V} in the exact diagram

ℜ_{M,V} \overrightarrow{⟶}^{p_{1}}_{p_{2}} G_{M,V} ⟶^{π} Gr_{M,V}  (1)

defining the functor Gr_{M,V}, are corepresentable by k-algebras over R resp. R ⟶ G_{M,V} and R ⟶ R_{M,V} (cf. [KR3, 10.1.3]), which implies that the category Qcoh_{Gr_{M,V}} of quasicoherent modules on Gr_{M,V} is defined by a linear algebra data: it is equivalent to the category Ker(p_{1}^{∗}, p_{2}^{∗}) whose objects are pairs (L, ϕ), where L is a G_{M,V} -module and ϕ is an R_{M,V} -module isomorphism p_{1}^{∗}(L) ⟶^{∼} p_{2}^{∗}(L). The inverse image functor p_{i}^{∗}, i = 1, 2, is isomorphic to the tensoring L ⟼ R_{M,V} ⊗_{G_{M,V}} L corresponding to an algebra morphism

G_{M,V} ⟶^{p̃_{i}} R_{M,V} representing p_{i}.

Let 𝔗 be a quasi-topology on the category Aff_{k}/SpecR of affine k-schemes over SpecR. Let Gr𝔗_{M,V} be the 𝔗-Grassmannian corresponding to Gr_{M,V}, i.e. a sheaf of sets (a ’space’) associated to Gr_{M,V}. If 𝔗 is coarser than the quasi-topology of effective descent, then the category Qcoh_{Gr_{M,V}} of quasi-coherent modules on Gr_{M,V} is naturally equivalent

to the category Qcoh_{Gr{𝔗 \atop M,V}} of quasi-coherent modules on Gr𝔗_{M,V}.

2.9.4. Noncommutative projective space. Let M be the free R-module of the rank n+1, V the free R-module of the rank 1. In this case, we denote the functor Gr_{M,V} by Pn_{R}. If a quasi-topology 𝔗 on the category Aff_{k}/SpecR of affine k-schemes over SpecR is coarser than the quasi-topology of 2-descent, than the category Qcoh_{P_{R}^{n}} of quasi-coherent modules on Pn_{R} is equivalent to the category of quasi-coherent modules on the associated projective space Pn^{𝔗}_{R}.

2.9.5. The commutative case. Let R = k. Denote by Grc_{M,V} the restriction of the presheaf Gr_{M,V} to the subcategory CAff_{k} of commutative affine k-schemes (i.e. the opposite category to the category CAlg_{k} of commutative k-algebras). We assume that the rank of the k-module M at each point of Spec(k) is greater than, or equal to the rank of the k-module V at this point; otherwise the functor Grc_{M,V} maps every commutative k-algebra to the empty set. The exact diagram (1) induces an exact diagram

ℜc_{M,V} \overrightarrow{⟶}^{𝔭_{1}}_{𝔭_{2}} Gc_{M,V} ⟶^{π} Grc_{M,V} ,  (2)

where ℜc_{M,V} and Gc_{M,V} denote the restrictions of presheaves resp. ℜ_{M,V} and G_{M,V} to the subcategory CAff_{k}. If ℜ_{M,V} and G_{M,V} are representable by the algebras resp. R_{M,V} and G_{M,V}, then the presheaves ℜc_{M,V} and Gc_{M,V} are representable by abelianizations (quotients by the commutant) of these algebras, Rc_{M,V} and Gc_{M,V}. By 2.9.2, the category of quasicoherent modules on Grc_{M,V} is isomorphic to the kernel Ker(𝔭_{1}^{∗}, 𝔭_{2}^{∗}) of the pair of the inverse image functors

Gc_{M,V} − mod \overrightarrow{⟶}^{𝔭^{∗}_{1}}_{𝔭^{∗}_{2}} Rc_{M,V} − mod.  (3)

Now we regard (2) as an exact sequence of presheaves of sets on the ringed site (CAff_{k}, O) of commutative k-schemes for a flat (fpqc, or fppf) topology. The presheaves ℜc_{M,V} and Gc_{M,V} are sheaves because the flat topology is subcanonical (all representable presheaves are sheaves). The presheaf Grc_{M,V} is not a sheaf, but the sheaf associated to Grc_{M,V} is isomorphic to the Grassmannian Grassm_{M,V}. Since the fpqc topology (hence the fppf topology) is coarser than the 2-descent topology, the category of quasi-coherent modules on the Grassmannian Grassm_{M,V} is equivalent to the category of quasi-coherent modules on the presheaf Grc_{M,V}, i.e. to the kernel of the pair of functors (3).

II. Noncommutative stacks.

A stack is a sheaf of categories on a (pre)topology. A classical example is the category of (commutative) affine schemes with Zariski topology with the category of quasi-coherent sheaves assigned to each affine scheme. The sheaf property means an effective descent property for quasi-coherent sheaves.

In noncommutative geometry, (pre)topologies are replaced by Q-categories, and stacks are sheaves of categories (in an appropriate sense) on a Q-category. A standard example is the Q-category of rings with faithfully flat covers (see [KR3, 9.3]) with the category of left R-modules assigned to each ring R.

3. Preliminaries: representable fibered categories and representable cartesian functors. Finiteness conditions.

3.1. Categories over E representable by a presheaf of sets. Fix a category E. For any presheaf E^{op} ⟶^{S} Sets, we have a category E/S over E. For any X ∈ ObE, the fiber (E/S)_{X} is a discrete category formed by all objects (X, X → S). In particular, it is empty if S(X) = ∅. Any morphism X ⟶^{f} Y of the category E induces a functor

(E/S)_{Y} ⟶^{f^{∗}} (E/S)_{X}, (Y, Y →^{ξ} S) ⟼ (X, ξ ∘ f).

The map f ⟼ f^{∗} is a functor E^{op} ⟶ Cat, and E/S is a fibered category corresponding to this functor. Note that every morphism of the category E/S is cartesian.

3.1.1. Proposition. The map

S ⟼ E/S, S ∈ ObE^{∧}, (S →^{g} T) ⟼ (E/S ⟶ E/T, (X, ξ) ⟼ (X, g ∘ ξ), )

is a fully faithful functor, 𝔥^{E}, from E^{∧} to the category of categories over E. The functor 𝔥^{E} preserves finite limits.

Proof is left to the reader.

3.2. Definition. Let E′ be a full subcategory of E^{∧}. A category F over E is called E′-representable if it is E-equivalent to the category E/S for some object S of E′.

In particular, any E′-representable category over E is fibered.

3.2.1. Standard choices of E′. For an arbitrary category E, the standard choices of E′ are the category E itself, the category E^{∧} of presheaves on E, and the subcategory of left exact functors E^{op} ⟶ Sets.

3.2.2. Standard choices in the case of a Q-category. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, and let E = A^{op}. Main examples of E′ in this case are:

(a) The subcategory of 𝔸-spaces, i.e. sheaves of sets on 𝔸.

(b) The subcategory of locally affine 𝔸-spaces.

(c) The subcategory of 𝔸-monopresheaves on 𝔸.

(d) The category A^{op}. We shall call A^{op}-representable fibered categories affine.

3.3. Representable cartesian functors. Fix a full subcategory E′ of E^{∧}. A cartesian functor F ⟶^{Φ} G between categories over E is called E′-representable if for any T ∈ ObE′ and any cartesian functor E/T ⟶ G, the fibered product E/T ×_{G} F is E′- representable.

3.3.1. Affine cartesian functors. We call a cartesian functor between categories over E affine if it is E-representable.

3.3.2. Proposition. (a) Any E-equivalence is E′-representable for any full subcategory E′ of E^{∧}. In particular it is affine.

(b) Suppose E′ has finite products taken in E^{∧}. Then for any E′-representable category F over E, the structure morphism of F is E′-representable.

(c) Suppose E′ has a final object. Then a structure morphism of a category F over E is E′-representable iff F is E′-representable.

(c’) Suppose E has a final object. Then a structure morphism of a category F over E is affine iff F is affine.

Proof. (a) The assertion follows from definitions.

(b) Let X ∈ ObE′. The only E-functor E/X ⟶ E is the canonical functor,

(V, ξ) ⟼ V, ((V, ξ) →^{f} (V′, ξ′)) ⟼ (V →^{f} V′).

If F ≃ E/S for some S ∈ ObE′, then F ×_{E} E/X ≃ E/S ×_{E} E/X ≃ E/(X × S), hence the structure morphism F ⟶ E is E′-representable.

(c) Suppose the structure morphism F : F ⟶ E is E′-representable. This means that F ×_{E} E/X is E′-representable for any X ∈ ObE′. Taking as X a final object of the category E′, we obtain that F ×_{E} E ≃ F is representable.

(c’) This assertion is a special case of (c).

3.3.3. Proposition. Let F be a category over E. Suppose E′ is closed under finite limits taken in E^{∧}. Then the following conditions are equivalent:

(i) Any cartesian morphism E/S ⟶ F, S ∈ ObE′, is E′-representable.

(ii) The diagonal morphism F ⟶^{Δ_{F}} F ×_{E} F is E′-representable.

Proof. (i) ⇒ (ii). Let X ∈ ObE′, and let f, g : E/X ⟶ F be arbitrary cartesian morphisms over E. Consider the following canonical commutative diagram

\begin{matrix}F ×_{F×_{E} F} E/X & ⟶ & E/X ×_{F} E/X & ⟶ & F \\ ↓ & & ↓ & & ↓ \\ E/X & ⟶^{Δ_{E/X}} & E/X ×_{E} E/X & ⟶^{(f,g)} & F ×_{E} F \end{matrix}  (1)

formed by two universal squares. By (i), the fiber product E/X ×_{F} E/X is E′-representable. Since E′ has products, X × X exists in E′, and E/X ×_{E} E/X ≃ E/(X × X), i.e. it is representable by X × X. Since E′ has fibered products and the embedding

E ⟶ Cat/E, X ⟼ E/X,

preserves limits (cf. 2.2), the category F ×_{F×_{E} F} E/X over E is E′-representable.

(ii) ⇒ (i). For any two morphisms, E/X ⟶^{f} F ⟵^{g} E/Y, of Cart_{E}, the square

E/X ×_{F} E/Y ⟶ E/X ×_{E} E/Y
↓  ↓ f × g
F  ⟶^{Δ_{F}}  F ×_{E} F

is cartesian, hence the assertion.

3.4. Definition. Let (P) be a property of morphisms of the category E′ stable under base change. We say that an E′-representable morphism F ⟶^{f} G of categories over E satisfies (P) if for any X ∈ ObE′ and any morphism E/X ⟶ G, the (morphism of E′ representing the) projection E/X ×_{G} F ⟶ E/X satisfies (P).

3.4.1. Proposition. Let P be the property of morphisms of the category E′ ⊆ E^{∧} stable under base change.

(a) The class of E′-representable morphisms of Cat/E having the property P is stable under base change.

(b) If the property P of morphisms of E′ is stable under composition, then same holds for the property P of E′-representable morphisms of Cat/E.

Proof is left to the reader.

3.4.2. Examples. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category. Take E = A^{op} and E′ = E^{∧} = A^{∨}. The following classes of morphisms of A^{∨} are stable under composition and base change (cf. 5.4 and 5.6):

3.4.2.1. The class P_{fsm}^{𝔸} of formally 𝔸-smooth morphisms.

3.4.2.2. The class P_{funr}^{𝔸} of formally 𝔸-unramified morphisms.

3.4.2.3. The class Pfét^{𝔸} of formally 𝔸-étale morphisms.

By 3.4.1, the class of formally 𝔸-smooth (resp. formally 𝔸-étale, resp. formally 𝔸unramified) A^{∨}-representable morphisms of Cat/A^{op} is stable under composition and base change.

3.5. Finitely presentable and locally finitely presentable cartesian functors. Let E be a category and E′ a full subcategory of E^{∧}. We assume that E′ contains all representable functors and is closed under limits of filtered projective systems.

Let F and G be fibered categories over E. A cartesian functor F ⟶^{Φ} G over E will be called E′-finitely presentable, if for any filtered projective system D ⟶^{𝔇} E′, the canonical square

colim F_{𝔇_{μ}}^{+} ⟶ F_{lim 𝔇}^{+}
Φ′ ↓  ↓ Φ_{lim 𝔇}
colim G_{𝔇_{μ}}^{+} ⟶ G_{lim 𝔇}^{+}

is 2-cartesian. Here F^{+} and G^{+} are the canonical extensions of the fibered categories resp. F and G onto E^{∧} (cf. 1.1.5.1); and lim 𝔇 is taken in E^{∧}.

We call an E′-finitely presentable cartesian functor locally finitely presentable if E′ = E and finitely presentable if E′ = E^{∧}.

4. Stacks and prestacks over a Q-category.

Fix a Q-category 𝔸 = (Ā ⇄^{u} A). Let F be a category over A^{op}. Any object ȳ of Ā defines two categories over A, the category ȳ\u^{∗} and the category u_{∗}(ȳ)\A, and the A-functor

θȳ : ȳ\u^{∗} ⟶ u_{∗}(ȳ)\A, (z, ȳ →^{g} u^{∗}(z)) ⟼ (z, η_{u}^{−1}(z) ∘ u_{∗}(g)).  (1)

Here η_{u} denotes an adjunction isomorphism Id_{A} ⟶ u_{∗}u^{∗}. Note that all morphisms of the A-categories u_{∗}(ȳ)\A and ȳ\u^{∗} are cartesian and cocartesian. Hence any functor ȳ\u^{∗} ⟶ u_{∗}(ȳ)\A, in particular θȳ, is (co)cartesian. Therefore θȳ induces a functor

Cart_{A^{op}}((u_{∗}(ȳ)\A)^{op}, F) ⟶ Cart_{A^{op}}((ȳ\u^{∗})^{op}, F), Φ ⟼ Φ ∘ θȳ.  (2)

Note that the A-categories u_{∗}(ȳ)\A and ȳ\u^{∗} are cofibered over A, hence their opposite categories are fibered over A^{op}.

4.1. Definition. We call a fibered category F over A^{op} an 𝔸-stack (resp. an 𝔸prestack) if for any ȳ ∈ Ā, the functor (2) is an equivalence (resp. fully faithful).

4.1.1. The 2-categories of 𝔸-stacks and 𝔸-prestacks. We denote by Stacks_{𝔸} (resp. P rest_{𝔸}) the full 2-subcategory of the 2-category F ib_{/A^{op}} (of fibered categories over A^{op}, see A2.6.1) objects of which are 𝔸-stacks (resp. 𝔸-prestacks).

4.1.2. Note. In the case 𝔸 is the Q-category associated with a Grothendieck site, the definition of an 𝔸-stack is equivalent to the usual one (cf. [Giraud], Ch. II, 1.2.1).

4.2. A reformulation. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category and 𝔉 = (F, F →^{π} A^{op}) a category over A^{op}. For any ȳ ∈ ObĀ, the category (u_{∗}(ȳ)\A)^{op} is isomorphic (by Yoneda’s lemma) to the category A^{op}/A(u_{∗}(ȳ), −). By a similar reason, the category (ȳ\u^{∗})^{op} is isomorphic to A^{op}/Ā(ȳ, u^{∗}(−)). Therefore,

Cart_{A^{op}}((u_{∗}(ȳ)\A)^{op}, F) ≃ Cart_{A^{op}}(A^{op}/A(u_{∗}(ȳ), −), F) = F_{A(u_{∗}(ȳ),−)}^{+} ≃ F_{u_{∗}(ȳ)}

and

Cart_{A^{op}}((ȳ\u^{∗})^{op}, F) ≃ Cart_{A^{op}}(A^{op}/A(ȳ, u^{∗}(−)), F) = F_{Ā(ȳ,u^{∗}(−))}^{+},

where F^{+} is the canonical extension of F onto A^{∨} defined by F_{X}^{+} = Cart_{A^{op}}(A^{op}/X, F) = Qcoh(F/X)^{op} (cf. 1.1.5.2). Here we identify the category A^{∨} of functors A ⟶ Sets with the category of presheaves of sets on A^{op}. The morphism (2) above is a part of the commutative diagram

Cart_{A^{op}}((u_{∗}(ȳ)\A)^{op}, F) ⟶ Cart_{A^{op}}((ȳ\u^{∗})^{op}, F)
≀↓  ↓≀
F_{A(u_{∗}(ȳ),−)}^{+}  ⟶^{αF ; ȳ}  F_{Ā(ȳ,u^{∗}(−))}^{+}

where the lower horizontal arrow is induced by the canonical morphism

Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −).

Thus, a fibered category F over A^{op} is an 𝔸-prestack (resp. an 𝔸-stack) iff the canonical functor

F_{A(u_{∗}(ȳ),−)}^{+} ⟶^{αF ; ȳ} F_{Ā(ȳ,u^{∗}(−))}^{+}

is fully faithful (resp. an equivalence of categories) for all ȳ ∈ ObĀ.

4.3. 𝔸-stacks and 𝔸-spaces. Representable 𝔸-stacks and 𝔸-prestacks. Let X be a functor A ⟶ Sets. Consider the category A^{op}/X corresponding to the Yoneda’s embedding

A^{op} ⟶ A^{∨} = Sets^{A}, x ⟼ A(x, −).

It is a fibered category over A^{op}.

4.3.1. Proposition. There are canonical isomorphisms

Cart_{A^{op}}((u_{∗}(ȳ)\A)^{op}, A^{op}/X) ⟶^{∼} X ∘ u_{∗}(ȳ) = û^{∗}(X)(ȳ) ≃ Ā^{∨}(ȳ, X ∘ u_{∗}) (1)

and

Cart_{A^{op}}((ȳ\u^{∗})^{op}, A^{op}/X) ⟶^{∼} û^{!}(X)(ȳ) ≃ A^{∨}(Ā(ȳ, u^{∗}(−)), X)  (2)

(cf. 3.6) such that the diagram

\begin{matrix}Cart_{A^{op}}((u_{∗}(ȳ)\A)^{op}, A^{op}/X) & ⟶ & Cart_{A^{op}}((ȳ\u^{∗})^{op}, A^{op}/X) \\ ≀↓ & & ↓≀ \\ A^{∨}(u_{∗}(ȳ), X) ≃ û^{∗}(X)(ȳ) & ⟶ & û^{!}(X)(ȳ) = A^{∨}(Ā(ȳ, u ∗ (−)), X) \end{matrix}  (3)

is commutative.

Proof. The category (z\A)^{op} is naturally identified with A^{op}/z; in particular, we can replace (u_{∗}(ȳ)\A)^{op} by A^{op}/u_{∗}(ȳ). Any A^{op}-functor A^{op}/z ⟶ A^{op}/X, z ∈ ObA, maps an object (V, V → z) to an object (V, V → X) and is uniquely determined by the image, (z, z →^{ξ} X), of the initial object (z, id_{z}). Since by Yoneda’s lemma, A^{∨}(z, X) ≃ X(z), this defines a canonical isomorphism Cart_{A^{op}}(A^{op}/z, A^{op}/X) ⟶^{∼} X(z). Taking z = u_{∗}(ȳ), we obtain the isomorphism Cart_{A^{op}}((u_{∗}(ȳ)\A)^{op}, A^{op}/X) ⟶^{∼} X ∘ u_{∗}(ȳ).

The isomorphism

Cart_{A^{op}}((ȳ\u^{∗})^{op}, A^{op}/X) ⟶^{∼} A^{∨}(Ā(ȳ, u^{∗}(−)), X)

is defined in an obvious way.

4.3.2. Corollary. The fibered category A^{op}/X is an 𝔸-stack (resp. an 𝔸-prestack) if and only if X is a sheaf (resp. a monopresheaf) on 𝔸.

4.3.3. Definition. An 𝔸-stack (resp. an 𝔸-prestack) F is called representable if it is A^{op}-equivalent to the 𝔸-stack (resp. 𝔸-prestack) A^{op}/X for some 𝔸-sheaf (resp. 𝔸monopresheaf) of sets X on 𝔸.

4.4. 𝔸-prestacks and presheaves Hom_{S}(x, y). Let F be a fibered category over E defined by a pseudo-functor E^{op} ⟶ Cat, given by

ObE ∋ X ⟼ F_{X}, HomE ∋ f ⟼ f^{∗}, c_{g,f} : f^{∗}g^{∗} →^{∼} (gf)^{∗}.

For any S ∈ ObE and any pair of objects, x, y of the fiber F_{S}, we denote by Hom_{S}(x, y) a presheaf of sets on E/S defined as follows. Hom_{S}(x, y)(T, f) = Hom_{T} (f^{∗}(x), f^{∗}(y)) for any object (T, T →^{f} S) of E/S. To any morphism h : (T, f) ⟶ (U, g), the presheaf Hom_{S}(x, y) assigns the map

Hom_{S}(x, y)(U, g) := Hom_{U}(g^{∗}(x), g^{∗}(y)) ⟶ Hom_{S}(x, y)(T, f) := Hom_{T} (f^{∗}(x), f^{∗}(y))

given by

(g^{∗}(x) →^{ξ} g^{∗}(y)) ⟼ c_{g,h}(y) ∘ h^{∗}(ξ) ∘ c_{g,h}^{−1}(x) ∈ Hom_{T} (f^{∗}(x), f^{∗}(y)).

4.4.1. Proposition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category. A fibered category F over A^{op} is an 𝔸-prestack iff for any S ∈ ObA and any pair of objects x, y of F_{S}, the presheaf Hom_{S}(x, y) : S\A = (A^{op}/S)^{op} ⟶ Sets is a sheaf on the Q-category S\𝔸 (cf. 2.2.1).

Proof is left to the reader.

4.5. Sub(pre)stacks. Let F be an 𝔸-stack, and let G be a full subcategory of F such that G is a fibered category over A^{op} and the inclusion functor G ⟶ F is cartesian. Then G is an 𝔸-prestack.

4.5.1. Definition. In the setting above, let S be an object of A and x an object of F_{S}. We say that x belongs locally to ObG if there exists S̄ ∈ ObĀ such that u_{∗}(S̄) ≃ S and for any (T, ξ) ∈ ObS̄\u^{∗} the object u_{_{S}∗}(ξ)^{∗}(x) of F_{T} is isomorphic to an object of G_{T}. Here u_{_{S}∗}(ξ) = η_{u}^{−1}(T) ∘ u_{∗}(ξ).

4.5.2. Lemma. A subprestack G of an 𝔸-stack F is an 𝔸-stack iff for any S ∈ ObA, any x ∈ ObF_{S} which locally belongs to ObG is isomorphic to an x′ ∈ ObG.

Proof is left to the reader.

4.6. Local epimorphisms. A canonical decomposition of a cartesian functor.

Fix a Q-category 𝔸 = (Ā ⇄^{u} A). Let X, Y be fibered categories over A^{op}. We call a cartesian functor X ⟶^{F} Y a local epimorphism if for every v ∈ ObA^{op} and any z ∈ ObY_{v}, there exist ȳ ∈ ObĀ and z′ ∈ ObX_{Ā(ȳ,u^{∗}(−))}^{+} such that u_{∗}(ȳ) ≃ v and F^{+}(z′) is isomorphic to the image of z by the (inverse image of the) composition of the canonical morphism Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −) and the isomorphism A(u_{∗}(ȳ), −) ⟶^{∼} A(v, −).

4.6.1. Lemma. The composition of local epimorphisms is a local epimorphism. If X ⟶^{F} Y ⟶^{G} Z are cartesian functors such that G ∘ F is a local epimorphism, then G is a local epimorphism.

Proof is left to the reader.

Let X ⟶^{F} Y be an arbitrary cartesian functor. We denote by Y^{F} a full subcategory of Y whose objects are z ∈ ObY such that there exists ȳ ∈ ObĀ and z′ ∈ ObX_{Ā(ȳ,u^{∗}(−))}^{+} such that u_{∗}(ȳ) ≃ π(z) and F^{+}(z′) is isomorphic to the image of z by the (inverse image of the) composition Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −) ⟶^{∼} A(π(z), −). Here π denotes the projection Y ⟶ A^{op}. The cartesian functor F factorizes into

X ⟶^{F_{𝔢}} Y^{F} ⟶^{F_{𝔪}} Y,  (1)

where F_{𝔢} is a local epimorphism and F_{𝔪} is the natural embedding. This decomposition is

universal in the sense that if X ⟶^{F′} Y′ ⟶^{F′′} Y is an arbitrary factorization of F into a local epimorphism, F′, and a fully faithful functor, F′′, then there is a commutative diagram

\begin{matrix}X ⟶^{F′} Y′ ⟶^{F′′} Y \\ _{F_{𝔢}} ↘ ↓I ↗_{F_{𝔪}} \\ Y^{F} \end{matrix}  (2)

where Y′ ⟶^{I} Y^{F} is a fully faithful cartesian functor.

4.6.2. Proposition. In the above setting, suppose Y is an 𝔸-stack. Then Y^{F} is a substack of Y. If X is a stack too, then the decomposition (1) is unique up to isomorphism; i.e. the functor I in the diagram (2) is an equivalence of categories.

Proof. (a) If Y is a stack, then Y^{F} is a substack of Y by 4.5.2.

(b) Suppose now that both X and Y are 𝔸-stacks. We can and will assume that F is a local epimorphism, i.e. Y^{F} = Y. We need to show that for every v ∈ ObA, the functor

Y′_{v} ⟶^{I_{v}} Y_{v} in (2) is a category equivalence. Since the functor I_{v} is fully faithful, it suffices to show that for every object z of the category Y_{v}, there exists an object z′′ of X_{v} such that F_{v}(z′′) ≃ z.

Let z ∈ ObY_{v}. Since F is a local epimorphism, there exist ȳ ∈ ObĀ and z′ ∈ ObX_{Ā(ȳ,u^{∗}(−))}^{+} such that u_{∗}(ȳ) ≃ v and F^{+}(z′) is isomorphic to the image of z by the (inverse image of the) composition Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −) ⟶^{∼} A(v, −). Since X is a stack, the canonical functor

X_{A(u_{∗}(ȳ),−)}^{+} ⟶^{αX ; ȳ} X_{Ā(ȳ,u^{∗}(−))}^{+}

is an equivalence of categories for all ȳ ∈ ObĀ (see 4.2). Since X_{A(u_{∗}(ȳ),−)}^{+} ≃ X_{A(v,−)}^{+} ≃ X_{v}, this implies that there exists an object z′′ of X_{v} (defined uniquely up to isomorphism) whose image in X_{A(u_{∗}(ȳ),−)}^{+} is isomorphic to z′. Since Y is a stack, we have category equivalence

Y_{v} ≃ Y_{A(u_{∗}(ȳ),−)}^{+} ⟶^{αY ; ȳ} Y_{Ā(ȳ,u^{∗}(−))}^{+}

Thus, we have a 2-commutative diagram

X_{Ā(ȳ,u^{∗}(−))}^{+} ⟵ X_{v}
F^{+} ↓  ↓ F_{v}
Y_{Ā(ȳ,u^{∗}(−))}^{+} ⟵ Y_{v}

whose horizontal arrows are category equivalences. In particular, F_{v}(z′′) ≃ z, hence the assertion.

4.6.3. Corollary. Let X and Y be 𝔸-stacks. A cartesian functor X ⟶^{F} Y is an equivalence iff it is a fully faithful local epimorphism.

4.6.4. Proposition. Let

\begin{matrix}X′ & ⟶^{G′} & X \\ F′ ↓ & & ↓ F \\ Y′ & ⟶^{G} & Y \end{matrix}  (3)

be a 2-cartesian square of morphisms of 𝔸-stacks. If F is a local epimorphism (resp. fully faithful, resp. an equivalence), then F′ is the same.

If F′, G are local epimorphisms, then F is a local epimorphism.

Proof. (a) We can assume that X′ is the canonical pull-back of the pair of functors Y′ ⟶^{G} Y ⟵^{F} X; i.e. objects of X′ are triples (x, y; ϕ), where x ∈ ObX, y ∈ ObY′, and ϕ is an isomorphism F (x) ⟶^{∼} G(y). For any v ∈ ObA, the fiber X′_{v} is the (standard) pull-back of the pair of functors Y′_{v} ⟶^{G_{v}} Y ⟵^{F_{v}} X_{v}; i.e. spanned by objects (x, y; ϕ) such

that π(ϕ) = id_{v}. Here π is the projection Y ⟶ A^{op}. The functor X′_{v} ⟶^{F′_{v}} Y′_{v} maps an

object (x, y; ϕ) to x and a morphism (x, y; ϕ) ⟶^{(f,g)} (x′, y′; ϕ) to x ⟶^{f} x′. It follows from the construction that if F_{v} is fully faithful, then F′_{v} is fully faithful, and if F is a local epimorphism, then F′ is a local epimorphism.

(b) If F′ and G are local epimorphisms, then, by 4.6.1, G ∘ F′ is a local epimorphism, hence F ∘ G′ is a local epimorphism. By 4.6.1, the latter implies that F is a local epimorphism.

4.6.5. Cartesian squares and the canonical decomposition of cartesian functors. Let

\begin{matrix}X′ & ⟶^{G′} & X \\ F′ ↓ & & ↓ F \\ Y′ & ⟶^{G} & Y \end{matrix}  (3)

be a 2-cartesian square of morphisms of 𝔸-stacks. Consider the canonical decomposition X ⟶^{F_{𝔢}} Y^{F} ⟶^{F_{𝔪}} Y of the cartesian functor X ⟶^{F} Y. Then we have a diagram of cartesian functors

X′′ \underrightarrow{_{F′_{𝔢}}}  Ỹ^{F}  ⟶^{F′_{𝔪}} Y′
G′ ↓  ↓ G′′  ↓ G
X ⟶^{F_{𝔢}} Y^{F}  ⟶^{F_{𝔪}} Y

with both squares 2-cartesian: we construct first the right square, and then take pull-back of X ⟶^{F_{𝔢}} Y^{F} ⟵^{G′′} Ỹ^{F}. By 4.6.4, F′_{𝔪} is fully faithful and F′_{𝔢} is a local epimorphism. Since the square

X′′ ⟶^{F′_{𝔪}∘_{F′_{𝔢}}} Y′
G′ ↓  ↓ G
X ⟶^{F} Y

is 2-cartesian, as well as the square (3), there is an equivalence X′ ⟶^{Φ} X′′. Taking F′_{𝔢} = F̃′_{𝔢} ∘ Φ, we obtain a diagram

X′ ⟶^{F′_{𝔢}}  Ỹ^{F}  ⟶^{F′_{𝔪}} Y′
G′ ↓  ↓ G′′  ↓ G
X ⟶^{F_{𝔢}} Y^{F}  ⟶^{F_{𝔪}} Y

with both squares 2-cartesian, with F′_{𝔢} a local epimorphism and F′_{𝔪} fully faithful, and such that F′ = F′_{𝔪} ∘ F′_{𝔢}.

5. Standard examples.

5.1. Stacks and prestacks of categories of quasi-coherent modules. Let 𝔉 = (F → E) be a fibered category and 𝔗 a quasi-topology on E. If 𝔗 is coarser than the quasi-topology of effective descent (resp. the 1-descent quasi-topology), then the fibered category 𝔉 is a stack (resp. a prestack) over (E, 𝔗). This follows from definitions and 1.2.

Let 𝔗^{∧} be the coinduced topology on E^{∧}, and let 𝔉^{+} be a canonical extension of 𝔉 to E^{∧}. Recall that for any presheaf of sets X, the fiber 𝔉_{X}^{+} is the category opposite to the category Qcoh(𝔉/X) of quasi-coherent modules on X (cf. 1.1.5.1). If 𝔗 is coarser than the topology of effective descent (resp. the 1-descent topology), then the fibered category 𝔉^{+} over E^{∧} is a stack (resp. a prestack) over (E^{∧}, 𝔗^{∧}).

5.2. The fibered category of noncommutative affine schemes. Let 𝔉 be the fibered category 𝔐(Aff_{k}, O) of modules over affine noncommutative k-schemes (associated with the ringed category (Aff_{k}, O) of affines k-schemes); i.e. E = Aff_{k} = Alg_{k}^{op} and the fiber over SpecR is the category R − mod^{op} opposite to the category of left R-modules (see 2.1).

5.2.1. Subcanonical topologies. Let 𝔗 be a topology on Aff_{k}. By 2.7.3.1, 𝔉 is a prestack over (Aff_{k}, 𝔗) iff the topology 𝔗 is subcanonical (i.e. if every representable functor on Aff_{k} is a sheaf).

5.2.2. The fpqc quasi-topology. Let 𝔗_{fpqc} be fpqc quasi-topolgy on Aff_{k} defined via covers consisting of finite conservative families of flat morphisms. The fibered category 𝔉 = 𝔐(Aff_{k}, O) is a stack over (Aff_{k}, 𝔗_{fpqc}).

6. Formally 𝔸-smooth, formally 𝔸-unramified, and formally 𝔸-étale cartesian functors.

Given a Q-category 𝔸, we define the notions of formally 𝔸-smooth, formally 𝔸unramified, and formally 𝔸-étale cartesian functor without using the representability hypothesis.

6.1. Definition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, and let X , Y be categories over A^{op}. We call a cartesian functor X ⟶^{f} Y formally 𝔸-smooth if for any ȳ ∈ ObĀ and for

any pair of morphisms A^{op}/u_{∗}(ȳ) ⟶^{g} Y, (ȳ\u^{∗})^{op} ⟶^{g′} X such that the diagram

\begin{matrix}(ȳ\u^{∗})^{op} & ⟶^{g′} & X \\ θȳ ↓ & & ↓ f \\ A^{op}/u_{∗}(ȳ) & ⟶^{g} & Y \end{matrix}  (1)

2-commutes, there exists a morphism A(u_{∗}(ȳ), −) ⟶^{γ} X such that γ∘θȳ ≃ g′ and f ∘γ ≃ g. In other words, the diagram

\begin{matrix}(ȳ\u^{∗})^{op} ⟶^{g′} X \\ θȳ ↓  ↗_{γ} ↓ f \\ A^{op}/u_{∗}(ȳ) ⟶^{g} Y \end{matrix}  (2)

2-commutes.

6.2. Definition. We call a cartesian functor X ⟶^{f} Y formally 𝔸-unramified if for

any ȳ ∈ ObĀ and any pair of cartesian functors, A(u_{∗}(ȳ), −) ⟶^{g} Y, Ā(ȳ, u^{∗}(−)) ⟶^{g′} X, for which the diagram 6.1(1) 2-commutes, there exists at most one (up to isomorphism) cartesian functor, A^{op}/u_{∗}(ȳ) ⟶^{γ} X, such that the diagram 6.1(2) 2-commutes.

6.2.1. Note. Let a cartesian functor X ⟶^{f} Y is a monofunctor in the following

sense: if Z \overrightarrow{⟶}^{g_{1}}_{g_{2}}X is a pair of cartesian functors such that f ∘ g_{1} ≃ f ∘ g_{2}, then g_{1} ≃ g_{2}.

Then f is formally 𝔸-unramified.

6.3. Definition. We call a cartesian functor X ⟶^{f} Y formally 𝔸-étale if it is both formally 𝔸-smooth and formally 𝔸-unramified.

6.4. Definition. We call a fibered category X over A^{op} formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) iff the functor, X ⟶^{p} A^{op}, is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale).

6.4.1. Note. It follows from definitions that a fibered category over A^{op} is 𝔸-étale if and only if it is an 𝔸-stack. Thus, a formally 𝔸-étale morphism can be regarded as a generalization of a notion of an 𝔸-stack.

6.5. Proposition. (a) A composition of formally 𝔸-smooth (resp. formally 𝔸unramified, resp. formally 𝔸-étale) cartesian functors is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale).

(b) Let X , Y, Z be categories over A^{op}, and let X ⟶^{f} Y and Y ⟶^{h} Z be cartesian functors.

(i) Suppose h ∘ f is formally 𝔸-unramified. Then f is formally 𝔸-unramified.

(ii) Suppose h is formally 𝔸-unramified. If X ⟶^{h∘f} Z is formally 𝔸-smooth (resp. formally 𝔸-étale, then f is formally 𝔸-smooth (resp. formally 𝔸-étale).

(c) Let S ∈ ObCart_{A^{op}}, and let (X , ξ) ⟶^{f} (Y, μ), (X′, ξ′) ⟶^{f′} (Y′, μ′) be morphisms

of objects over S. The morphisms f, f′ are formally 𝔸-unramified (resp. formally 𝔸smooth, resp. formally 𝔸-étale) iff the morphism f ×_{S} f′ : X ×_{S} X′ ⟶ Y ×_{S} Y′ has the corresponding property.

Proof. The argument is similar to the one of 5.4.

6.6. Smooth, unramified, and étale cartesian functors. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, and let X , Y be fibered categories over A^{op}. We call a cartesian functor X ⟶^{f} Y 𝔸-smooth (resp. 𝔸-unramified, resp. 𝔸-étale) if it is locally finitely presentable (cf. 3.5) and formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale).

7. Locally affine stacks. 𝔸-stacks are viewed as a natural generalization of 𝔸-spaces. Morphisms of spaces are replaced by 1-morphisms of stacks, i.e. cartesian functors. Below we extend to stacks the content of Sections 8 and 9 of [KR3].

7.1. Affine τ-covers, locally affine τ-stacks. Zariski τ-stacks. Let Stacks_{𝔸}^{1} denote the 1-category of 𝔸-stacks: its objects are 𝔸-stacks and morphisms cartesian functors. Let τ be a quasi-topology on the category Stacks_{𝔸}^{1}.

We call a τ-cover {X_{u} →^{u} X | u ∈ U} affine, if all X_{u} are affine (i.e. corepresentable by objects of A).

We call an 𝔸-stack X τ-locally affine if it has an affine τ-cover.

We call a τ-cover {X_{u} →^{u} X | u ∈ U} 2-affine if it is affine and for any u, v ∈ U, the stack X_{u} ×_{X} X_{v} has an affine τ-cover.

An affine τ-cover {X_{u} →^{u} X | u ∈ U} will be called a Zariski affine τ-cover if each u separates non-isomorphic cartesian functors A^{op}/T ⟶ X_{u} for all T ∈ ObA.

We call an 𝔸-stack X a Zariski τ-stack if it has a Zariski affine τ-cover.

A τ-cover {X_{u} →^{u} X | u ∈ U} will be called a Zariski 2-affine τ-cover if it is Zariski affine and for any u, v ∈ U, the space X_{u} ×_{X} X_{v} has a Zariski affine τ-cover. We call an 𝔸-stack X a Zariski τ-stack if it has a Zariski affine τ-cover.

7.2. Semiseparated and weakly separated covers and stacks. We call a τ-cover

{X_{u} →^{u} X | u ∈ U}

– weakly separated if X_{u} and the pull-back X_{u} ×_{X} X_{v} are affine (i.e. corepresentable) for all u, v ∈ U,

– semiseparated if the stack X_{u} and the morphism X_{u} →^{u} X are affine for all u.

Clearly semiseparated covers are weakly separated, and weakly separated covers are 2-covers. In particular, a stack which has a weakly separated cover is locally affine. We call a stack which has a semiseparated (resp. weakly separated) affine cover semiseparated (resp. weakly separated).

7.2.1. Proposition. Suppose that 𝔸 = (Ā ⇄^{u} A) is subcanonical (i.e. all corepresentable functors A ⟶ Sets are 𝔸-stacks), and A has products. Then every separated locally affine 𝔸-stack is τ-semiseparated.

Proof. The argument imitates that of [KR3, 8.4]. Details are left to the reader.

7.3. Natural quasi-topologies on Stacks_{𝔸}. For a cartesian functor Y ⟶^{φ} X, let

Λ_{φ} denote the family of all pairs of cartesian functors Z \overrightarrow{⟶}^{g_{1}}_{g_{2}}Y such that φ∘g_{1} ≃ φ∘g_{2}. We

call a cartesian functor Y ⟶^{φ} X strictly epimorphic if the following condition holds: for any cartesian functor Y ⟶^{ψ} Z such that Λ_{φ} ⊆ Λ_{ψ}, there exists a unique up to isomorphism cartesian functor X ⟶^{γ} Z such that ψ ≃ γ ∘ φ.

We call a set of morphisms {X_{u} →^{u} X | u ∈ U} strictly epimorphic if the corresponding cartesian functor ∐_{u∈U} X_{u} ⟶^{u} X is strictly epimorphic.

Let P be a class of morphisms in Stacks_{𝔸} (cartesian functors) which contains all identical morphisms and is closed under the composition (that is P is a subcategory of Stacks_{𝔸} having same objects as Stacks_{𝔸}). We call a set of morphisms {X_{u} →^{u} X | u ∈ U} from a P-cover of X if

(i) it is strictly epimorphic;

(ii) all morphisms of U belong to P.

This defines a quasi-topology, τ^{P}, on Stacks_{𝔸} and its quasi-compact version, τ_{f}^{P}. It remains to choose the class P.

7.4. Formally étale, formally smooth, and formally Zariski quasi-topologies. Let 𝔸_{1} = (Ā_{1} ⇄ A) be another Q-category with the same underlying category A (thought as the Q-category of thickennings). Then we have the following choices:

– the class Pfét of formally 𝔸_{1}-étale cartesian functors,

– the class P_{fsm} of formally 𝔸_{1}-smooth cartesian functors,

– the class P_{fzar} of formally 𝔸_{1}-open immersions which we define as formally 𝔸_{1}-étale cartesian functors X ⟶^{φ} Y such that Λ_{φ} consists of isomorphic pairs of functors.

We denote the corresponding quasi-topologies resp. by τfét, τ_{fsm}, and τ_{fzar} and call them resp.formally étale, formally smooth and formally Zariski quasi-topology.

Note that the category of Zariski τ-stacks (cf. 7.1) is the same for all three quasitopologies; i.e. it coincides with the category of τ_{fzar}-locally affine 𝔸-stacks. We call them formally Zariski 𝔸_{1}-stacks.

7.4.1. Smooth, étale, and Zariski quasi-topologies. Our main choices are

– the class Pét of 𝔸_{1}-étale cartesian functors,

– the class P_{sm} of 𝔸_{1}-smooth cartesian functors,

– the class P_{zar} of 𝔸_{1}-open immersions.

We denote the corresponding quasi-topologies resp. by τét, τ_{sm}, and τ_{zar} and call them resp.étale, smooth and Zariski quasi-topology.

7.5. Noncommutative DM-stacks and Artin’s algebraic stacks.

7.5.1. Commutative Zariski and algebraic stacks. Let A be the category CAlg_{k} of commutative k-algebras, Ā_{1} the full subcategory of A^{2} formed by k-algebra epimorphisms with nilpotent kernels, Ā the full subcategory of A^{2} formed by faithfully flat morphisms. If τ is the étale topology, τ = τét, then locally affine separated stacks are DM-stacks. If τ is the smooth topology, then the locally affine stacks are Artin’s stacks.

7.5.2. Noncommutative DM-stacks. Let A be the category 𝔄ss_{k} of associative kalgebras with morphisms defined up to conjugation (cf. 4.6). Let Ā be the full subcategory of A^{2} formed by equivalence classes of faithfully flat algebra morphisms, and let Ā_{1} be the full subcategory of A^{2} formed by equivalence classes (with respect to the conjugation, see 4.6) of k-algebra epimorphisms with nilpotent kernels. If τ is the étale quasi-topology (cf. 7.4.1), then locally affine 𝔸-stacks in this setting seem to be an adequate noncommutative version of DM-algebraic stacks.

7.5.3. Noncommutative Artin’s stacks. Let A be the category Alg_{k} of associative k-algebras, Ā_{1} the full subcategory of A^{2} formed by k-algebra epimorphisms with nilpotent kernels, Ā the full subcategory of A^{2} formed by faithfully flat morphisms. If τ is a smooth quasi-pretopology τ_{sm} (cf. 7.3.1), locally affine 𝔸-stacks can be regarded as a noncommutative version of Artin’s algebraic stacks.

8. Representable cartesian functors and local constructions.

Fix a category E and a full subcategory E′ of the category E^{∧} of presheaves of sets on E which contains (the image of) E and is closed under fibered products. We denote by Ẽ′ the fibered category over E defined as follows: for any V ∈ ObE, the fiber Ẽ′_{V} is E′/V. For

any morphism U ⟶^{ϕ} V, its inverse image functor, ϕ^{∗}, assigns to any object (X, X → V) of the category E′/V its pull-back (X ×_{ϕ,V} U, X ×_{ϕ,V} U → U).

8.1. Proposition. Let 𝔛 be a fibered category over E. There is a natural equivalence between the category of E′-representable cartesian functors Y ⟶ 𝔛 and the category of cartesian functors 𝔛 ⟶ Ẽ′.

Proof. An argument is left to the reader.

8.1.1. Local constructions. We call any cartesian functor 𝔛 ⟶ Ẽ′ a local E′- construction on 𝔛. Local E-constructions will be called affine.

8.2. Relative local constructions. Fix a functor A ⟶^{Φ} E. Let Ẽ′_{Φ} denote the fibered category Ẽ′ ×_{E} A = Ẽ′ ×_{Φ,E} A over A.

Let 𝔛 be a fibered category over A. We call any cartesian functor 𝔛 ⟶ Ẽ′ ×_{E} A a local (E′, Φ)-construction on 𝔛, or simply a local construction on 𝔛, if it is clear what are the subcategory E′ and the functor Φ.

Local (E, Φ)-constructions will be called affine.

8.3. Local constructions over a ringed category. Let (A, O) be a ringed category. Let E be the category Aff of noncommutative affine schemes. Let τ be a pretopology on E and E′ the category of τ-locally affine spaces. The presheaf of rings O induces a functor

Φ = Φ_{O} : A ⟶ E, U ⟼ Spec(O(U)).

A local (E, Φ)-construction on a fibered category 𝔛 over A is a family of functors F_{U} : 𝔛_{U} ⟶ (Ẽ′ ×_{E} A)_{U}, U ∈ ObA, such that for any U ∈ ObA and any object x of the fiber 𝔛_{U} over U, F_{U}(x) is a τ-locally affine space over Spec(O(U)); and for any morphism U ⟶^{ϕ} V and any x ∈ Ob𝔛_{V}, we are given an isomorphism of spaces

F_{U}(ϕ^{∗}(x)) ⟶^{∼} F_{V} (x) ∏_{Spec(O(V))} Spec(O(V)).

8.3.1. Affine local constructions. A local construction 𝔛 ⟶^{F} E′ ×_{E} A is affine iff for any U ∈ ObA and any x ∈ Ob𝔛_{U}, the object F_{U}(x) is an affine space over Spec(O(U)), i.e. F_{U}(x) is isomorphic to a pair (Spec(R(U, x)), Spec(R(U, x)) ⟶ Spec(O(U))) corresponding to a ring morphism O(U) ⟶ R(U, x) defined uniquely up to isomorphism.

Thus, an affine local construction on 𝔛 can be described as a function which assigns to every pair (U, x), where U ∈ ObA and x ∈ Ob𝔛_{U}, a ring morphism O(V) ⟶ R(U, x) and

to any morphism U ⟶^{ϕ} V of A and any x ∈ Ob𝔛_{V}, a morphism R(V, x) ⟶^{ξ_{ϕ}} R(U, ϕ^{∗}(x)) such that the square

R(V, x) ⟶^{ξ_{ϕ}} R(U, ϕ^{∗}(x))
↑  ↑
O(V) ⟶^{O(ϕ)}  O(U)

is cocartesian. In other words, the induced ring morphism

R(V, x) ⋆_{O(V )} O(U) ⟶^{ξ′_{ϕ}} R(U, ϕ^{∗}(x))

is an isomorphism. The morphisms ξ_{ϕ} should satisfy standard compatibility conditions with respect to the composition of morphisms of A.

8.3.2. Vector fibers over a ringed category. Let 𝔅𝔪𝔬𝔡(A, O) denote the fibered category of (A, O)-bimodules determined by the pseudo-functor A^{op} ⟶ Cat which assigns to each object V of A the category opposite to the category of O(V)-bimodules and to any morphism U ⟶^{ϕ} V the functor opposite to the functor

ϕ̃^{∗} : M ⟼ O(U) ⊗_{O(V)} M ⊗_{O(V)} O(U).

Let 𝔛 be a fibered category over A. We define O-bimodules on 𝔛 as cartesian functors M : 𝔛 ⟶ 𝔅𝔪𝔬𝔡(A, O). This means that M is a function which assigns to each pair (V, x), where V ∈ ObA and x ∈ Ob𝔛_{V}, an O(V)-bimodule M(V, x) and to each morphism U ⟶^{ϕ} V a bimodule isomorphism

O(U) ⊗_{O(V)} M(V, x) ⊗_{O(V)} O(U) ⟶^{ζ_{ϕ}} M(U, ϕ^{∗}(x))  (1)

which satisfies the usual compatibility conditions.

For any V ∈ ObA and x ∈ 𝔛_{V}, let R(V, x) be the tensor algebra, T_{O(V )}(M(V, x)), of

the O(V)-bimodule M(V, x). Fix a morphism U ⟶^{ϕ} V of A. The O(V)-bimodule mor-

phism M(V, x) ⟶ M(U, ϕ^{∗}(x)) and the ring morphism O(V) ⟶^{O(ϕ)} O(U) induce ring morphisms T_{O(V )}(M(V, x)) ⟶ T_{O(U)}(M(U, ϕ^{∗}(x))) ⟵ O(U) which, in turn, determines a morphism

O(U) ⋆_{O(V )} T_{O(V )}(M(V, x)) ⟶ T_{O(U)}(M(U, ϕ^{∗}(x))).  (2)

Since the O-bimodule M is quasi-coherent, (1) is a bimodule isomorphism, which implies that the ring morphism (2) is an isomorphism.

We denote the affine scheme Spec(T_{O(V )}(M(V, x))) by 𝕍_{O(V)}(M(V, x)) and the local construction on 𝔛 given by (V, x) ⟼ 𝕍_{O(V)}(M(V, x)) by 𝕍_{O}(M).

8.3.3. Vector fibers associated with pairs of quasi-coherent modules. Let 𝔐(A, O) be the fibered category of (A, O)-modules (cf. 2). Let 𝔛 be a fibered category over A. We define O-modules on 𝔛 as cartesian functors M : 𝔛 ⟶ 𝔐(A, O). This means that M is a function which assigns to each pair (V, x), where V ∈ ObA and x ∈ Ob𝔛_{V}, an O(V)-module M(V, x) and to each morphism U ⟶^{ϕ} V an O(U)-module isomorphism

O(U) ⊗_{O(V)} M(V, x) ⟶^{ζ_{ϕ}} M(U, ϕ^{∗}(x))  (3)

which satisfies the usual compatibility conditions.

Let M and P be two O-modules on 𝔛. The pair (M, P) defines a functor

Hom_{O,𝔛}(M, P) : 𝔛 ⟶ Ẽ^{∧} ×_{E} A  (4)

which assigns to each pair (U, x), U ∈ ObA, x ∈ Ob𝔛_{U}, the functor

O(U)\Rings ⟶ Sets, (O(U) ⟶^{ϕ} R) ⟼ Hom_{R}(ϕ^{∗}(M(U, x)), ϕ^{∗}(P(U, x))) (5)

naturally defined on morphisms.

8.3.3.1. Proposition. Suppose that for every U ∈ ObA and x ∈ Ob𝔛_{U}, the O(U)- module P(U, x) is projective of finite type. Then the functor (4) is an affine local construction.

Proof. Set for convenience M = M(U, x) and P = P(U, x). Then

Hom_{R}(ϕ^{∗}(M), ϕ^{∗}(P)) ≃ Hom_{O(U)}(M, ϕ_{∗}ϕ^{∗}(P)) = Hom_{O(U)}(M, R ⊗_{O(U)} P) ≃

Hom_{O(U)}(M, Hom^{O(U)}(P^{∨}, R)) ≃ Hom_{O(U)}^{e}(M ⊗ P^{∨}, R) ≃
O(U)\Rings(T_{O(U)}(M ⊗ P^{∨}), R)

Here Hom^{O(U)}(P^{∨}, S) is the (left) O(U)-module of right O(U)-module morphisms from P^{∨} to S, O(U)^{e} := O(U) ⊗ O(U)^{o}, and T_{O(U)}(M ⊗ P^{∨}) is the tensor algebra of the O(U)- bimodule M ⊗ P^{∨}. This shows that (4) is isomorphic to the vector fiber 𝕍_{O}(M ⊗ P^{∨}) of the quasi-coherent O-bimodule M ⊗ P^{∨}.

It is useful to have an analogue of 8.3.3.1 for a family of modules. Let {M_{i}, P_{i} | i ∈ J} be a family of O-modules on 𝔛. These family defines a functor

∏_{i∈J} Hom_{O,𝔛}(M_{i}, P_{i}) : 𝔛 ⟶ Ẽ^{∧} ×_{E} A  (6)

which assigns to each pair (U, x), U ∈ ObA, x ∈ Ob𝔛_{U}, the functor

O(U)\Rings ⟶ Sets, (O(U) ⟶^{ϕ} R) ⟼ ∏_{i∈J} Hom_{R}(ϕ^{∗}(M_{i}(U, x)), ϕ^{∗}(P_{i}(U, x)))

naturally defined on morphisms.

8.3.3.2. Proposition. Suppose that for every U ∈ ObA and x ∈ Ob𝔛_{U} and for all i ∈ J, the O(U)-module P(U, x) is projective of finite type. Then the functor (6) is an affine local construction on 𝔛 isomorphic to 𝕍(⊕_{i∈J}(M_{i} ⊗ P_{i}^{∨})).

Proof. The argument is similar to that of 8.3.3.1. Details are left to the reader.

8.3.4. The construction of isomorphisms. Fix a ringed category (A, O) and a fibered category 𝔛 over A. Let M and P be O-modules on 𝔛. We denote by GO,𝔛_{M,V} the functor 𝔛 ⟶ Ẽ^{∧} ×_{E} A which assigns to every pair (U, x), U ∈ ObA, x ∈ Ob𝔛_{U}, the functor

GO(U)_{M(U,x),V(U,x)} : O(U)\Rings ⟶ Sets  (7)

defined as follows: to every ring morphism O(U) ⟶^{ϕ} R, the functor (7) assigns the set of all pairs of R-module morphisms ϕ^{∗}(P(U, x)) ⟶ ϕ^{∗}(M(U, x)) ⟶ ϕ^{∗}(P(U, x)) the composition of which is the identical morphism.

8.3.4.1. Proposition. Let M(U, x) and P(U, x) be projective O(U)-modules of finite type for all U ∈ ObA and x ∈ Ob𝔛. Then the functor GO,𝔛_{M,V} is an affine local construction on 𝔛.

Proof. (a) For convenience, we set M = M(U, x) and P = P(U, x). For any ring morphism O(U) ⟶^{ϕ} R, the set GO(U)_{M,P}(R, ϕ) is the kernel of the pair of morphisms

Hom_{R}(ϕ^{∗}(M), ϕ^{∗}(P)) × Hom_{R}(ϕ^{∗}(P), ϕ^{∗}(M)) \overrightarrow{⟶} Hom_{R}(ϕ^{∗}(P), ϕ^{∗}(P)) (8)

where one arrow assigns to each pair (u, v) the composition, u ∘ v, of morphisms u and v, and the other one maps each pair (u, v) to the identity morphism, id_{ϕ^{∗}(P)}. Since (8) depends functorially on everything, the functor GO,𝔛_{M,V} is the kernel of a pair of functor morphisms

Hom_{O,𝔛}(M, P) × Hom_{O,𝔛}(P, M) \overrightarrow{⟶} Hom_{O,𝔛}(P, P)  (9)

Since M(U, x) and P(U, x) are projective O(U)-modules of finite type for all U ∈ ObA and x ∈ Ob𝔛, the functors Hom_{O,𝔛}(M, P), Hom_{O,𝔛}(P, M), and Hom_{O,𝔛}(P, P) are affine local constructions on 𝔛 resp. 𝕍_{O,𝔛}(M ⊗ P^{∨}), 𝕍_{O,𝔛}(P ⊗ M^{∨}), and 𝕍_{O,𝔛}(P ⊗ P^{∨}). Thus the diagram (9) is equivalent to the diagram

𝕍_{O,𝔛}(M ⊗ P^{∨} ⊕ P ⊗ M^{∨}) \overrightarrow{⟶} 𝕍_{O,𝔛}(P ⊗ P^{∨})  (10)

(see 8.3.3.2). The kernel of a pair of morphisms between two affine local constructions is an affine local construction.

Let M and P be O-modules on 𝔛. We denote by IsoO,𝔛_{M,P} the functor 𝔛 ⟶ Ẽ^{∧} ×_{E} A which assigns to every pair (U, x), U ∈ ObA, x ∈ Ob𝔛_{U}, the functor

IsoO(U)_{M(U,x),P(U,x)} : O(U)\Rings ⟶ Sets

that assigns every ring morphism O(U) ⟶^{ϕ} R the set of isomorphisms ϕ^{∗}(M(U, x)) ⟶^{∼} ϕ^{∗}(P(U, x)).

8.3.4.2. Proposition. Let M(U, x) and P(U, x) be projective O(U)-modules of finite type for all U ∈ ObA and x ∈ Ob𝔛. Then the functor IsoO,𝔛_{M,V} is an affine local construction on 𝔛.

Proof. The functor IsoO,𝔛_{M,V} is naturally identified with the fiber product of the pair of morphisms

GO,𝔛_{M,P} ⟶^{φ} Hom_{O,𝔛}(M, P) × Hom_{O,𝔛}(P, M) ⟵^{ψ} GO,𝔛_{P,M},  (11)

where φ is the natural embedding, ψ is the composition of the natural imbedding

GO,𝔛_{P,M} ⟶^{φ} Hom_{O,𝔛}(P, M) × Hom_{O,𝔛}(M, P)

and the isomorphism

Hom_{O,𝔛}(P, M) × Hom_{O,𝔛}(M, P) ⟶^{∼} Hom_{O,𝔛}(M, P) × Hom_{O,𝔛}(P, M)

defined by (u, v) ⟼ (v, u). By 8.3.3.2, the diagram (11) is isomorphic to the diagram

GO,𝔛_{M,P} ⟶ 𝕍_{O,𝔛}(M ⊗ P^{∨} ⊕ P ⊗ M^{∨}) ⟵ GO,𝔛_{P,M}.  (12)

By 8.3.4.1, the functors GO,𝔛_{M,P} and GO,𝔛_{P,M} are affine local constructions on 𝔛. Therefore the pull-back of (5) is an affine local construction on 𝔛.

8.4. Grassmannians. For any associative unital ring S and any pair M, P of left S-modules, we define the ’Grassmannian’ GrS_{M,P} as the functor S\Rings ⟶ Sets which

assigns to every ring morphism S ⟶^{ϕ} T the isomorphism class of coretractions (– splittable epimorphisms) ϕ^{∗}(M) ⟶ ϕ^{∗}(P). We have a canonical exact sequence of functors

ℜS_{M,P} \overrightarrow{⟶}^{p_{1}}_{p_{2}} GS_{M,P} ⟶^{π} GrS_{M,P} .  (1)

Here π is the natural epimorphism (sending a split pair of arrows ϕ^{∗}(P) →^{v} ϕ^{∗}(M) →^{u} ϕ^{∗}(P) to the class of the epimorphism u) and ℜS_{M,P} is the ’functor of relations’, i.e. ℜS_{M,P} =

GS_{M,P} ∏_{Gr_{M,P}^{S}} GS_{M,P} .

Fix a ringed category (A, O) and a fibered category 𝔛 over A. Let M and P be O-modules on 𝔛. Since the functors GrS_{M,P} and ℜS_{M,P} (as well as GS_{M,P}) in the diagram (1) depend functorially on S, M, and P, they determine functors 𝔛 ⟶ Ẽ^{∧} ×_{E} A which we denote resp. by GrO,𝔛_{M,V} and ℜO,𝔛_{M,V}. Thus, the diagram (1) induces an exact diagram

ℜO,𝔛_{M,V} \overrightarrow{⟶}^{p_{1}}_{p_{2}} GO,𝔛_{M,V} ⟶^{π} GrO,𝔛_{M,V}  (2)

of functors 𝔛 ⟶ Ẽ^{∧} ×_{E} A.

8.4.1. Proposition. Let M(U, x) and P(U, x) be projective O(U)-modules of finite type for all U ∈ ObA and x ∈ Ob𝔛. Then (2) is the exact diagram of local constructions on 𝔛 (in particular, GrO,𝔛_{M,P} is a local construction), and the local constructions ℜO,𝔛_{M,V} and GO,𝔛_{M,V} are affine.

Proof. If the S-modules M and V are projective of finite type, then, by [KR3, 10.1.3], the functor ℜS_{M,P} (and GS_{M,P}) is representable. By [KR3, 10.1.4], all three functors in (1)

are compatible with the base change. The latter means that if S ⟶^{ϕ} T is a ring morphism, and 𝔊S_{M,P} denotes any of the functors in the diagram (1), then

𝔊T_{ϕ^{∗}(M),ϕ^{∗}(P)} ≃ 𝔊S_{M,P} ∏_{SpecS} SpecT.

This implies the assertion.

Appendix 1: Q-categories and quasi-topologies.

The following is a glossary extracted from the first 3 sections of [KR3].

A1.1. Q-categories and Q^{o}categories. A Q-category is a pair, Ā\overrightarrow{⟵}^{u∗}_{u^{∗}}A, of functors

such that the functor u^{∗} is fully faithful and left adjoint to u_{∗}, which implies that A is a quotient category of Ā and u_{∗} is a localization functor.

A morphism from a Q-category Ā ⇄^{u} A to a Q-category Ā′ ⇄^{u′} A′ is a triple (Φ, Φ̄, ϕ), where Φ : A ⟶ A′ and Φ̄ : Ā ⟶ Ā′ are functors and ϕ is a functor isomorphism Φu_{∗} ⟶ u′_{∗}Φ̄. The composition of two morphisms, is defined by

(Φ′, Φ̄′, ϕ′) ∘ (Φ, Φ̄, ϕ) = (Φ′Φ, Φ̄′Φ̄, Φ̄′ϕ ∘ ϕ′Φ)

A Q^{o}-category is a pair of functors Ā\overrightarrow{⟵}^{u∗}_{u^{∗}}A such that the functor u^{∗} is fully faithful

and a right adjoint to u_{∗}. In other words, the pair of functors Ā\overrightarrow{⟵}^{u∗}_{u^{∗}}A is a Q^{o}-category iff

the pair of dual functors, Ā^{op} ⟶^{u_{∗}^{op}} A^{op} ⟶^{u^{∗op}} Ā^{op}, is a Q-category.

A1.2. Examples of Q-categories and Q^{o}-categories.

A1.2.1. The Q-category of cosieves. Fix a category A. Denote by SA the category of cosieves on A defined as follows. Objects of SA are pairs (x, R), where x ∈ ObA and R is a cosieve in x\A. Morphisms from (x, R) to (x′, R′) are given by morphisms x ⟶^{f} x′ such that R′_{f} ⊆ R. Here R′_{f} is a cosieve in x\A objects of which are all pairs (v, ξ ∘ f) such that (v, ξ) ∈ ObR′. There is a functor u^{∗} : A ⟶ SA which assigns to each object x of A the pair (x, x\A). The functor u^{∗} is fully faithful and has a canonical right adjoint,

u_{∗} : SA ⟶ A, (x, R) ⟼ x.

This defines a Q-category of cosieves, SA ⇄ A.

Cosieves in x\A are in a natural one-to-one correspondence with subfunctors of the functor A(x, −). Thus the Q-category of cosieves is isomorphic to a Q-category, Ā ⇄ A, defined as follows. Objects of Ā are pairs (x, R), where x ∈ ObA, R is a subfunctor of A(x, −). Morphisms from (x, R) to (y, S) are morphisms x ⟶^{f} y such that the morphism A(f, −) : A(y, −) ⟶ A(x, −) induces a morphism S ⟶ R of the subfunctors. The functor u_{∗} maps a pair (x, R) to x. The functor u^{∗} assigns to any object x of A the pair (x, A(x, −)).

A1.2.2. Quasi-(co)sites and (co)sites. Let A be a category, and let 𝔗 be a map which assigns to every object x of A a set 𝔗(x) of subfunctors of A(x, −) which contains A(x, −) itself. We identify the pair (A, 𝔗) with the full Q-subcategory Ā_{𝔗} ⇄ A of the Q-category SA ⇄ A of cosieves objects of which are all pairs (x, R), where x ∈ ObA and R ∈ 𝔗(x).

We call 𝔗 a quasi-cotopology, and the pair (A, 𝔗) a quasi-cosite, if the following two conditions hold:

(a) for any pair R, R′ ∈ 𝔗(x), R ∩ R′ ∈ 𝔗(x),

(b) if R ∈ 𝔗(x) and R′ is a subfunctor of A(x, −) containing R, then R′ ∈ 𝔗(x).

Quasi-topologies and quasi-sites correspond to quasi-cotopologies and quasi-cosites on the dual category A^{op}.

Grothendieck sites are quasi-sites. Recall that a site is a pair (A, 𝔗), where 𝔗 is a topology, i.e. a map which assigns to each x ∈ ObA a set 𝔗(x) of subfunctors of A(−, x) (called refinements of x) satisfying the conditions:

(i) for any R ∈ 𝔗(x) and any arrow f : y ⟶ x, the subfunctor R_{f} = R×_{A(−,x)}A(−, y) of A(−y) is a refinement of y (i.e. it belongs to 𝔗(y).

(ii) If R ∈ 𝔗(x) and R̃ is a subfunctor of A(−, x) such that R̃_{f} ∈ 𝔗(y) for any f ∈ R(y), y ∈ ObA,, then R̃ ∈ 𝔗(x).

Q-categories dual to Grothendieck sites are called cosites. The Q-category of cosieves, (SA ⇄ A) and its Q-subcategory 𝔸_{dis} = (Ā_{dis} ⇄ A), where Ā_{dis} is formed by all pairs (x, x\A), x ∈ ObA, are two extreme examples of cosites.

Cosites might be regarded as a topological data in terms of “closed sets”. In particular,

if (Ā ⇄^{u} A) is a cosite, then A might be viewed as the category of closed sets of a would-be space.

A1.2.3. A quasi-cosite associated with a Q-category. Fix a Q-category (Ā ⇄^{u} A). To any ȳ ∈ ObĀ, we assign the comma category ȳ\u^{∗} of pairs (f, x), where f is a morphism ȳ ⟶ u^{∗}(x). The functor u_{∗} induces a morphism, Φ^{∼} = (Φ, id_{u_{∗}}, Id_{A}), from

𝔸 = (Ā ⇄^{u} A) to the cosite SA ⇄ A. Here Φ is a functor Ā ⟶ SA which assigns to any object ȳ of Ā the pair (u_{∗}(ȳ), Rȳ), where Rȳ denotes the cosieve in u_{∗}(ȳ)\A formed

by all (v, u_{∗}(ȳ) →^{ξ} v) such that ξ = η_{u}^{−1}(v) ∘ ξ̄ for some ξ̄: ȳ ⟶ u^{∗}(v). The quasi-cosite, 𝔗𝔸 = (𝔗_{𝔸} ⇄ A), associated with 𝔸 is the smallest quasi-cosite containing the image of the functor Φ. The triple (Id_{A}, Φ, id) is a canonical morphism from 𝔸 to the Q-category (SA ⇄ A) of cosieves on A.

A1.2.3.1. Note. If 𝔸 is a quasi-cosite, then 𝔗𝔸 is naturally isomorphic to 𝔸.

Dually, with every Q^{o}-category 𝔸, one can associate a quasi-site which is naturally isomorphic to 𝔸 if 𝔸 is a quasi-site.

Suppose 𝔸 has the property:

(*) for any ȳ ∈ ObĀ and any morphism x ⟶^{f} u_{∗}(ȳ), there exists a morphism f̄ : x̄ ⟶ ȳ and an isomorphism α : u_{∗}(x̄) ⟶ x such that u_{∗}(f̄) = f ∘ α.

Then the quasi-cosite associated with 𝔸 is a cosite.

A1.2.4. Quasi-pretopologies. Let A be a category and τ a function which assigns to each object x of A a family, τ_{x}, of sets of arrows to x (’covers’ of x) which contains {x →^{id_{x}} x}. This data defines a category, A_{τ}, whose objects are all pairs (x, U), where x ∈ ObA, U ∈ τ_{x}; we shall call them covers. Morphisms from (x, U) to (y, V) are morphisms x ⟶^{f} y such that any arrow x_{u} →^{u} x in U factors through an arrow of V. The functor A_{τ} ⟶ A which assigns to every pair (x, U) the object x and to every morphism (x, U) ⟶^{f} (y, V) the morphism x ⟶^{f} y is right adjoint to the fully faithful functor A ⟶ A_{τ} which maps every object x of A to (x, {id_{x}}). This defines a Q^{o}-category 𝔸_{τ} = (A_{τ} ⇄ A).

The function τ is called quasi-pretopology if the following conditions hold:

(a) The composition of covers is a cover; i.e. if {x_{u} → x | u ∈ U} is a cover (i.e. it belongs to τ_{x}), and {x_{u}^{ν} → x_{u} | ν ∈ U_{u}} is a cover for any u ∈ U, then the set of compositions of arrows, {x_{u}^{ν} → x | u ∈ U, ν ∈ U_{u}}, is a cover.

(b) For any two covers, U, U′, of x, there exists a third cover, U′′, and morphisms of covers U ← U′′ → U′.

The dual notion carries an awkward name quasi-precotopology.

A1.2.4.1. One of the main fpqc-cocovers. Let A be the category Alg_{k} of associative unital k-algebras. We call a finite set of k-algebra morphisms {R ⟶^{ϕ_{i}} R_{i} | i ∈ J} an fpqc-cocover of R if {R_{i} ⊗_{R} − | i ∈ J} is a conservative family of exact functors. The class of fpqc-cocovers forms a quasi-precotopology on Alg_{k}, or a quasi-pretopology on the dual category Aff_{k} of affine noncommutative k-schemes.

A1.2.4.2. Quasi-topology associated with a quasi-pretopology. Consider the quasi-site 𝔗𝔸_{τ} associated with 𝔸_{τ}. The functor Φ : A_{τ} ⟶ 𝔗A_{τ} assigns to every cover (x, U) the pair (x, R_{U}), where R_{U} is the sieve associated with the set of arrows U: it consists of all arrows to x which factor through some of the arrows of U.

If for any morphism f : y ⟶ x and any U ∈ τ_{x}, there exists V ∈ τ_{y} such that f is a morphism (y, V) ⟶ (x, U), then the quasi-site associated with 𝔸_{τ} is a site. In particular, if τ is a Grothendieck pretopology, we obtain this way the site associated with τ.

A1.2.4.3. Covers. Let 𝔸 = (Ā ⇄^{u} A) be a quasi-site. A set of arrows U = {x_{i} → x| i ∈ J} in A is called a cover (or an 𝔸-cover) of x, if the pair (x, R_{U}), where R_{U} is the sieve associated to U, is an object of Ā.

It follows from the definition of a quasi-site that

(i) every set of arrows to x which contains a cover is a cover;

(ii) if U and U′ are covers of x, then U ×_{x} U′ = {x_{u} ×_{x} x_{v} ⟶ x | u ∈ U, v ∈ U′} is a cover of x, provided the pull-backs x_{u} ×_{x} x_{v} exist for all u ∈ U, v ∈ U′.

A1.2.5. The Q-category and the Q^{o}-category of morphisms of a category. Fix a category A. Consider the category A^{2} objects of which are morphisms of the category

A, and morphisms from x →^{f} y to x′ →^{f′} y′ are commutative squares

\begin{matrix}x & ⟶^{g} & x′ \\ f ↓ & & ↓ f′ \\ y & ⟶^{h} & y′ \end{matrix}  (1)

Denote by u^{∗} the functor A ⟶ A^{2} which assigns to any object x of A the object x ⟶^{id_{x}} x and to any morphism f the corresponding commutative square. The functor u^{∗} is fully faithful and has a right adjoint, u_{∗}, which maps any object x →^{f} y of A^{2} to x and any morphism (1) to x →^{g} x′. In fact, u_{∗}u^{∗} = Id_{A}, and there is a natural morphism ϵ_{u} : u^{∗}u_{∗} → Id_{A^{2}} which assigns to any object x →^{f} y of the category A^{2} the morphism

x ⟶^{id_{x}} x
id_{x} ↓  ↓ f
x ⟶^{f} y

from u^{∗}u_{∗}(x →^{f} y) to (x →^{f} y). One can see that id : Id_{A} → u_{∗}u^{∗} and ϵ_{u} are adjunction morphisms.

Dually, the functor u^{∗} has a natural left adjoint, u_{!}, which assigns to any object x →^{f} y of A^{2} the object y and to any morphism (1) the morphism y →^{h} y′.

A1.2.5.1. Q-subcategories of (A^{2} ⇄ A). Let Ā be a full subcategory of the category A^{2} which contains all objects x ⟶^{id_{x}} x. Then the functor u^{∗} takes values in the

subcategory Ā, hence it induces a structure of a Q-subcategory, 𝔸 = (Ā ⇄^{u} A), of the Q-category A^{2} ⇄ A. The functor u_{!} : A^{2} ⟶ A induces a functor Ā ⟶ A left adjoint to u^{∗}.

A1.2.6. The Q-category of infinitesimal algebra epimorphisms. Let A be the category Alg_{k} of associative unital k-algebras, and let Ā be the full subcategory of the category Alg_{k}^{2} of k-algebra morphisms whose objects are epimorphisms with a nilpotent kernel.

A1.2.6.1. The commutative version of A1.2.6 (i.e. A is the category CAlg_{k} of commutative algebras and Ā is the category of commutative algebra epimorphisms with nilpotent kernel) can be interpreted as the category of infinitesimal extensions of affine schemes over k.

A1.2.6.2. The Q-category of thickennings of a scheme. A non-affine version of the example A1.2.6.1 is the Q-category of thickennings of a scheme. Fix a scheme X. Let A be the category of (Zariski) open subschemes of X, and Ā the category of thickennings: objects of the category Ā are nilpotent scheme closed immersions U ⟶ T, where U is any open subscheme of X. The fully faithful functor u^{∗} : A ⟶ Ā, U ⟼ (U ⟶^{id_{U}} U), is left adjoint to the functor u_{∗} : Ā ⟶ A sending an immersion U ⟶ T to U.

A1.2.8. Q-categories of functors. Fix a category C. To any Q-category 𝔸 =

(Ā ⇄^{u} A), we assign a Q-category C^{𝔸} = (CĀ ⇄^{C^{u}} C^{A}). Here C^{A} denotes the category of functors A ⟶ C and C^{u} is a morphism with the inverse image functor

C^{u_{∗}} : C^{A} ⟶ CĀ, F ⟼ F ∘ u_{∗},

and the direct image functor C^{u^{∗}} : G ⟼ G ∘ u^{∗}. If C is the category Sets, we shall write

𝔸^{∨} = (Ā^{∨} ⇄^{u^{∨}} A^{∨}) instead of Sets^{𝔸} = (SetsĀ ⇄ Sets^{A}).

A1.3. Sheaves, monopresheaves and epipresheaves in a Q-category. Given

a Q-category 𝔸 = (Ā\overrightarrow{⟵}^{u∗}_{u^{∗}}A), an object x of the category A is called an 𝔸-sheaf if the

canonical map

Ā(ȳ, u^{∗}(x)) ⟶ A(u_{∗}(ȳ), x), g ⟼ η_{x}^{−1} ∘ u_{∗}(g),  (1)

is an isomorphism for all ȳ ∈ ObĀ. Here η_{u} is an adjunction isomorphism Id_{A} ⟶^{∼} u_{∗}u^{∗}. We denote by 𝔉𝔸 the full subcategory of the category A generated by 𝔸-sheaves.

A1.3.1.2. 𝔸-monopresheaves. We call an object x of the category A an 𝔸monopresheaf, or an 𝔸-separated presheaf, if the canonical map (1) is injective for any ȳ ∈ ObĀ. We denote by 𝔐𝔸 the full subcategory of A formed by 𝔸-monopresheaves.

A1.3.1.3. The canonical morphism ρ_{u}. Let x be an object of the category A such that the functor A(u_{∗}(−), x) is representable, i.e. A(u_{∗}(−), x) ≃ Ā(−, u^{!}(x)) for some u^{!}(x) ∈ ObĀ. There is a canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) corresponding to the isomorphism η_{u}^{−1}(x) : u_{∗}u^{∗}(x) ⟶ x. It follows from the definitions that x is an 𝔸monopresheaf iff the morphism ρ_{u}(x) is a monomorphism. Note, however, that x can be a monopresheaf without the functor A(u_{∗}(−), x) being representable.

A1.3.1.4. 𝔸-epipresheaves. We call an object x of A an 𝔸-epipresheaf if the functor A(u_{∗}(−), x) is representable and the canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) (cf. A1.3.1.3) is a strict epimorphism.

We denote by 𝔈𝔸 the full subcategory of the category A formed by 𝔸-epipresheaves. It follows from A1.3.1.3 that an object x of A is an 𝔸-sheaf iff it is an 𝔸-monopresheaf

and an 𝔸-epipresheaf.

A1.4. Examples.

A1.4.1. Sheaves on a quasi-pretopology. Suppose A is a category with fibered products and τ a quasi-pretopology on A identified with the corresponding Q-category 𝔸_{τ} = (Ā_{τ} ⇄ A) (cf. A1.2.4). A presheaf F of sets on A is a sheaf iff for any τ-cover {x_{u} ⟶ x | u ∈ U} the canonical diagram

F (x) ⟶ ∏_{u∈U} F (x_{u}) \overrightarrow{⟶} ∏_{u,v∈U} F (x_{u} ×_{x} x_{v})

is exact. A presheaf F on A with values in a category C is a sheaf in C^{𝔸{op \atop τ}} iff the presheaf of sets C(z, F (−)) is a sheaf for any z ∈ ObC. Thus, if τ is a pretopology, we recover the usual notion of a sheaf.

A1.4.2. Epipresheaves and sheaves on the Q-category of thickennings. Let A be the category of associative unital k-algebras and Ā the category of k-algebra epimor-

phisms with a nilpotent kernel (see A1.2.6). Epipresheaves of sets on 𝔸 = (Ā ⇄^{u} A) are formally smooth functors.

Appendix 2: Fibered categories.

Main references are Exposé VI in [SGA1] and Exposé VI in [SGA4]. The purpose of this appendix is to recall basic notions and fix notations. All categories we consider belong to a fixed universum, 𝔘.

A2.1. Categories over a category. Fix a category E. Let (A, A →^{F} E) and (B, B →^{G} E be objects of the category Cat/E. For any two morphisms, Φ, Ψ from (A, F) to (B, G) (called E-functors), an E-morphism Φ → Ψ is defined as any functor morphism ϕ : Φ ⟶ Ψ such that G(ϕ(x)) = id_{F (x)} for all x ∈ ObA. This defines a subcategory, Hom_{E}((A, F), (B, G)), of the category Hom(A, B) of all functors from A to B. The composition

Hom(A, B) × Hom(B, C)⟶Hom(A, C)

induces a composition

Hom_{E}((A, F), (B, G)) × Hom_{E}((B, G), (C, H))⟶Hom_{E}((A, F), (C, H)).

The map ((A, F), (B, G)) ⟼ Hom_{E}((A, F), (B, G)) defines a functor

(Cat/E)^{op} × Cat/E ⟶ Cat.

A2.2. Inner hom. For any two categories F, G over E and any category H, there is an isomorphism

Hom(H, Hom_{E}(F, G)) ⟶^{∼} Hom_{E}(F × H, G)

functorial in all three arguments).

A2.3. Base change. If F and E′ are two categories over E, F ×_{E} E′ denotes their product in Cat/E. Recall that Ob(F ×_{E} E′) = ObF ×_{ObE} ObE′ and Hom(F ×_{E} E′) = HomF ×_{HomE} HomE′. Fixing γ : E′ ⟶ E, we obtain the base change functor

Cat/E ⟶ Cat/E′, (F ⟼ F ×_{E} E′, π_{E′}),

where π_{E′} is the canonical projection.

For any two categories, F, G, over E, the projection G ×_{E} E′ ⟶ G induces a category isomorphism

Hom_{E′}(F ×_{E} E′, G ×_{E} E′) ⟶ Hom_{E}(F ×_{E} E′, G).

The inverse morphism sends any E-functor Φ to the E′-functor Φ ×_{E} E′.

A2.3.1. Proposition. If an E-functor Φ : F ⟶ G is fully faithful, then for any base change E′ ⟶ E, the corresponding E′-functor Φ ×_{E} E′ : F ×_{E} E′ ⟶ G ×_{E} E′ is fully faithful too.

A2.3.2. Definition. An E-functor Φ : F ⟶ G is called an E-equivalence if there exists E-functor Ψ : G ⟶ F and E-isomorphisms Φ ∘ Ψ →^{∼} Id_{G}, Ψ ∘ Φ →^{∼} Id_{F}.

A2.3.3. Proposition. The following conditions on an E-functor Φ : F ⟶ G are equivalent:

(i) Φ is an E-equivalence.

(ii) For any category E′ over E, the functor Φ ×_{E} E′ : F ×_{E} E′ ⟶ G ×_{E} E′ is an equivalence of categories.

(iii) Φ is an equivalence of categories, and for any X ∈ ObE, the functor Φ_{X} : F_{X} ⟶ G_{X} induced by Φ is an equivalence of categories.

A2.4. Cartesian morphisms. Inverse image functors. Fix a category E and an object A = (A, A →^{F} E) of the category Cat/E. For any X ∈ ObE, we denote by A_{X} the fiber of F in X which is the subcategory F^{−1}(id_{X}) of A. For any f : X → Y of E and x, y ∈ ObA such that F (x) = X, F (y) = Y, we set A_{f}(x, y) := {ξ : x → y| F (ξ) = f}.

A2.4.1. Cartesian morphisms. A morphism ξ ∈ A(x, y) is called cartesian if for any x′ ∈ ObA_{X} and any ξ′ : x′ → y such that F (ξ′) = f := F (ξ), there exists a unique X-morphism u : x′ → x (that is F u = id_{X}) such that ξ′ = ξ ∘ u. In other words, for any y ∈ A_{F (x)}, the map

A_{X}(x′, x)⟶A_{f}(x′, y), v ⟼ ξ ∘ v,

is bijective. This means also that the pair (x, ξ) represents the functor

A_{X}^{op} ⟶ Sets, x′ ⟼ A_{f}(x′, y).

If for a morphism f ∈ E(X, Y), there exists a cartesian morphism ξ : x → y such that F (ξ) = f, then the object x is defined uniquely up to isomorphism and is called inverse image of y by f. The standard notation: x = f^{∗}(y). The morphism ξ : f^{∗}(y) → y is then denoted by ξ_{f}, or by ξ_{f}(y).

A2.4.2. Inverse image functor. Suppose an inverse image exists for all y ∈ A_{Y}. Then the map y ⟼ (f^{∗}(y), ξ_{f}(y)) defines a functor f^{∗} : A_{Y} ⟶ A_{X}.

In fact, fix objects y, y′ of A_{Y} and a cartesian morphisms ξ_{f}(y) : f^{∗}(y) ⟶ y and ξ_{f}(y′) : f^{∗}(y′) ⟶ y′. For any morphism ϕ : y → y′ of A_{Y}, there exists a unique morphism, f^{∗}(ϕ) : f^{∗}(y) ⟶ f^{∗}(y′), such that the diagram

\begin{matrix}f^{∗}(y) & ⟶^{ξ_{f} (y)} & y \\ f^{∗}(ϕ) ↓ & & ↓ ϕ \\ f^{∗}(y′) & ⟶^{ξ_{f} (y′)} & y′ \end{matrix}  (1)

commutes.

A2.4.3. Note. Let

x ⟶^{ξ} y
ψ ↓  ↓ ϕ
x′ ⟶^{ξ′} y′

be a commutative diagram in A such that ψ ∈ IsoA_{X} and ϕ ∈ IsoA_{Y}. Then ξ is cartesian iff ξ′ is cartesian.

A2.5. Cartesian functors. Let A = (A, F), B = (B, G) be E-categories. An Efunctor Φ : A ⟶ B is called a cartesian functor if it transforms cartesian morphisms to cartesian morphisms. The full subcategory of Hom_{E}(A, B) formed by cartesian functors is denoted by Cart_{E}(A, B).

A2.5.1. Proposition. (a) Any E-equivalence is a cartesian functor.

(a’) Given an E-equivalence Φ : A = (A, F) ⟶ B, a morphism ξ of A is cartesian iff Φ(ξ) is cartesian.

(b) Any E-functor which is isomorphic to a cartesian functor is cartesian.

(c) Composition of cartesian functors is a cartesian functor.

A2.5.2. Corollary. Let Φ : A = (A, F) ⟶ B be an E-equivalence. Then for any E-category C, the functors Ψ ⟼ Ψ ∘ Φ and Ψ ⟼ Φ ∘ Ψ induce equivalence of categories:

Cart_{E}(B, C) ⟶^{∼} Cart_{E}(A, C)

Cart_{E}(C, A) ⟶^{∼} Cart_{E}(C, B).

A2.5.3. The category Cart_{E}. We denote by Cart_{E} the category objects of which are same as objects of Cat/E and morphisms are cartesian functors.

A2.5.3.1. The category Cart. We denote by Cart the subcategory of Cat^{2} whose objects are same as objects of Cat^{2} and morphisms from A′ ⟶^{F′} E′ to A ⟶^{F} E are commutative diagrams

A′ ⟶^{G} A
F′ ↓  ↓ F
E′ ⟶^{G′} E

such that G is a cartisian functor from (A′, G′ ∘ F′) to (A, F).

A2.5.4. Colimits. Let B, C be categories and S a family of morphisms of B. Denote by Hom_{S}(B, C) the category of functors B → C which transform morphisms of S into isomorphisms.

Let A = (A, F) be a category over E and S_{A} the family of cartesian morphisms of A. The E-category A defines two functors Cat ⟶ Sets:

C ⟼ Hom_{S_{A}}(A, C),  (1)
C ⟼ Cart_{E}(A, (C × E, P_{E})).  (2)

Here P_{E} is the natural projection C × E ⟶ E.

A2.5.4.1. Lemma. The functors (1) and (2) are canonically isomorphic.

Proof. Cartesian morphisms of C × E are all morphisms of the form (m, f), where m is an isomorphism of A.

A2.5.4.2. Corollary. For any E-category A = (A, F), the functor

Cat ⟶ Sets, C ⟼ Cart_{E}(A, (C × E, P_{E})),  (2)

is representable by the category (S_{A})^{−1}A.

A2.5.4.3. Definition. The functor (2) and the category (S_{A})^{−1}A representing it are denoted by ColimA/E and are called a colimit of A over E.

A2.5.5. Limits.

A2.5.5.1. Proposition. Let A = (A, F) be a category over E. The functor

Cat ⟶ Sets, C ⟼ Cart_{E}((C × E, P_{E}), A),

is representable by the category Cart_{E}(E, A) of E-cartesian functors E → A.

Proof. Let A be a category and G : (C × E) ⟶ A a cartesian functor. For any z ∈ ObC, the functor E ⟶ A, X ⟼ G(z, X), is cartesian. This gives a map Cart_{E}((C × E, P_{E}) ⟶ Cart_{E}(E, A) functorial in A. This map is a bijection.

A2.5.5.2. Definition. The category Cart_{E}(E, F) is called the category of cartesian sections of A over E. It is also called a limit of A over E and is denoted by LimA/E.

A2.6. Fibered and prefibered categories.

A2.6.1. Definitions. (a) A category A = (A, A →^{F} E) over E is called prefibered if for any morphism f : X → Y of E, an inverse image functor exists.

(b) A prefibered category A over E is called fibered if the composition of cartesian morphisms is a cartesian morphism.

A2.6.1.1. The 2-category of fibered categories over E. We denote by F ib_{/E} the 2-category of fibered categories over E. Its 1-morphisms are cartesian functors and 2-morphisms natural transformation of (cartesian) functors.

A2.6.2. Fibered and prefibered subcategories. Let A = (A, F) be a fibered (resp. prefibered) category over E. A subcategory B of A is called a fibered subcategory of A (resp. a prefibered subcategory of A) if B = (B, F |_{B}) is a fibered (resp. prefibered) category and the inclusion functor is a cartesian functor B ⟶ A.

A2.6.2.1. Lemma. Let A = (A, F) be a fibered (resp. prefibered) category over E. If B is a full subcategory of A, then B is a fibered (resp. prefibered) subcategory of A iff for any morphism f : X → Y of E and for any y ∈ ObB_{Y}, the inverse image, f_{A}^{∗}(y) is A_{X}-isomorphic to an object of B_{X}.

A2.6.2.2. Example. Let A = (A, F) be a fibered category over E. And let B be a subcategory of A having same objects; morphisms of B are cartesian morphisms of A. In particular, for any X ∈ ObE, morphisms of B_{X} are all isomorphisms of A_{X}. The subcategory B is a fibered subcategory of A.

A2.6.3. Proposition. Let F : A ⟶ E be a functor. The following conditions are equivalent:

(a) All morphisms of A are cartesian.

(b) A = (A, F) is a fibered category and all fibers are groupoids.

If the equivalent conditions (a), (b) hold, (A, F) is called fibered category of groupoids. If E is a groupoid, then the conditions (a), (b) are equivalent to the condition

(c) The category A is a groupoid, and the functor F : A ⟶ E is transportable. The latter means that for any isomorphism f : X → Y of E and any object x of A_{X}, there exists an isomorphism ξ : x → y such that F (ξ) = f.

A2.6.4. Proposition. Let Φ : A ⟶ B be an E-equivalence. Then A is a fibered (resp. prefibered) category over E iff B is such.

Proof. The assertion follows from the fact that a morphism ξ of A is cartesian iff Φ(ξ) is cartesian.

A2.6.5. Proposition. Let A_{1}, A_{2} be categories over E and let ξ = (ξ_{1}, ξ_{2}) be a morphism of A = A_{1} ×_{E} A_{2}. The morphism ξ is cartesian iff ξ_{1}, ξ_{2} are cartesian.

A2.6.6. Proposition. Let A = A_{1} ×_{E} A_{2}, and let Ψ = (Ψ_{1}, Ψ_{2}) be an E-functor B → A. The functor Ψ is cartesian iff Ψ_{1} and Ψ_{2} are cartesian. Thus one has a category isomorphism

Cart_{E}(B, A_{1} ×_{E} A_{2}) ⟶^{∼} Cart_{E}(B, A_{1}) × Cart_{E}(B, A_{2})

In particular, there is a natural isomorphism of categories

Lim(A_{1} ×_{E} A_{2}/E) ⟶^{∼} Lim(A_{1}/E) × Lim(A_{2}/E)

Proof. The assertion follows from A2.6.5.

A2.6.6.1. Corollary. Let A_{1}, A_{2} be fibered (resp. prefibered) categories over E. Then A_{1} ×_{E} A_{2} is a fibered (resp. prefibered) category over E.

A2.6.6.2. Remark. The results above hold for fibered products of any (small) set of categories over E.

A2.6.7. Proposition. Let A = (A, F) be a category over E and G : E′ ⟶ E a functor. Let A′ = (A′, F′), where A′ = A ×_{E} E′and F′ is the projection A′ → E′. A morphism, ξ′, of A′ is cartesian iff its image, ξ, in A is cartesian.

A2.6.7.1. Corollary. For any cartesian functor F : A ⟶ B of categories over E and any functor G : E′ ⟶ E, the functor F′ = F ×_{E} E′ : A′ = A ×_{E} E′⟶B′ = B ×_{E} E′ is cartesian.

Thus the functor Hom_{E}(A, B)⟶Hom_{E′}(A′, B′) induces a functor

Cart_{E}(A, B)⟶Cart_{E′}(A′, B′).

Taking into consideration the canonical isomorphism

Hom_{E′}(A′, B′) ⟶^{∼} Hom_{E}(A ×_{E} E′, B),

one can see that cartesian E′-functors correspond to E-functors A×_{E}E′⟶B transforming any morphism the first projection of which is cartesian into a cartesian morphism of B.

A2.6.7.2. Corollary. The category Lim(A′/E′) is isomorphic to the full subcategory of Hom_{E}(E′, A) formed by E-functors which transform any morphism into a cartesian morphism.

In particular, if A is a fibered category over E and A_{c} is the subcategory of A morphisms of which are all cartesian morphisms of A, then there is a bijection

ObLim(A′/E′) ⟶^{∼} ObHom_{E}(E′, A_{c}).

A2.6.8. Proposition. Let A be a fibered (resp. prefibered) category over E. Then for any functor E′ ⟶ E, A′ := A ×_{E} E′ is a fibered (resp. prefibered) category over E′.

A2.6.9. Proposition. Let A and B be prefibered categories over E, Φ a cartesian functor A ⟶ B. The functor Φ is faithful (resp. fully faithful, resp. E-equivalence) iff for any X ∈ ObE, the induced functor Φ_{X} : A_{X} ⟶ B_{X} is faithful (resp. fully faithful, resp. an equivalence).

Proof. The fact follows from definitions.

A2.6.10. Proposition. Let A = (A, F) be a prefibered category over E. It is fibered iff the following condition holds:

(Fib) Let ξ : x → y be a cartesian morphism over f : X → Y (i.e. F (ξ) = f). For any morphism g : V → X and any v ∈ ObA_{V}, the map

Hom_{g}(v, x)⟶Hom_{fg}(v, x), u ⟼ ξ ∘ u,

is bijective.

A2.6.10.1. Corollary. Let A = (A, F) be a category over E, ξ a morphism of A.

(a) If ξ an isomorphism, then ξ is cartesian and F (ξ) is an isomorphism.

(b) If A is fibered, then the inverse is true.

A2.6.10.2. Corollary. Let ξ : x → y and α : v → x be morphisms of a fibered category A over E. Suppose ξ is cartesian. Then α is cartesian iff ξ ∘ α is cartesian.

A2.7. Fibered categories and pseudo-functors. A pseudo-functor E^{op} ⟶ Cat is given by the following data:

(i) A map ObE ⟶ ObCat, X ⟼ A_{X}.

(ii) A map HomE ⟶ HomCat which associates to any f : X → Y a functor f^{∗} : A_{Y} ⟶ A_{X}.

(iii) A map which associates to any pair of composable morphisms, S ⟶^{f} T ⟶^{g} U a functor morphism c_{g,f} : f^{∗}g^{∗} ⟶ (gf)^{∗}.

This data should satisfy the following conditions:

(a) c_{f,id_{S}} = id_{f^{∗}} = c_{id_{T} ,f} for any morphism f : S → T of E,

(b) For any composable morphsims, f : S → T, g : T → U, h : U → V of E, the diagram

f^{∗}g^{∗}h^{∗} ⟶^{c_{f,g}h^{∗}} (gf)^{∗}h^{∗}
f^{∗}c_{h,g} ↓  ↓ c_{h,gf}
f^{∗}(hg)^{∗} ⟶^{c_{hg,f}} (hgf)^{∗}

commutes.

Pseudo-functors E^{op} ⟶ Cat form a category defined in a natural way.

A2.7.1. Prefibered categories and pseudo-functors. Let A = (A, F) be a prefibered category over E. Then there is a function which assigns to any morphism f of E its inverse image functor, f^{∗} in such a way that (id_{X})^{∗} = Id_{A_{X}} for all X ∈ ObE.

Let f : X → Y and g : V → X be morphisms of E and y an object of A_{Y}. There exists a unique V -morphism

c_{f,g}(y) : g^{∗}f^{∗}(y) ⟶ (fg)^{∗}(y)

such that the diagram

g^{∗}f^{∗}(y) ⟶^{ξ_{g}(f^{∗}(y))} f^{∗}(y)
c_{f,g}(y) ↓  ↓ ξ_{f}(y)
(fg)^{∗}(y) ⟶^{ξ_{fg}(y)}  y

is commutative. These morphisms are functorial in y, i.e. c_{f,g} = {c_{f,g}(y)|y ∈ ObA_{Y} } is a morphism g^{∗}f^{∗} ⟶ (fg)^{∗} of functors A_{Y} ⟶ A_{V}. One can check that they satisfy the conditions (a), (b) of A2.7.1.

Conversely, let E^{op} ⟶ Cat, X ⟼ A_{X}, f ⟼ f^{∗} be a pseudo-functor. Set ObA = ∐_{X∈ObE} A_{X} = {(X, x)| X ∈ ObE, x ∈ ObA_{X}}. A morphism from x̄ = (X, x) to ȳ = (Y, y) is a pair (f, ξ), where f is a morphism X → Y, ξ a morphism x → f^{∗}(y). A composition is defined by

(f, ξ) ∘ (g, μ) := c_{f,g}(y) ∘ g^{∗}(ξ) ∘ μ.  (1)

Set h_{f}(x̄, ȳ) := Hom_{A_{X}} (x, f^{∗}(y)) and A(x̄, ȳ) = ∐_{f∈E(X,Y)} h_{f}(x̄, ȳ). The composition (1) defines the composition on A. The projection functor, F : A ⟶ E, is given by the maps (X, x) ⟼ X, (f, ξ) ⟼ f.

The E-category F : A ⟶ E is fibered iff all morphisms c_{f,g} are isomorphisms.

A2.8. Limits, Colimits, and pseudo-functors. Let A = (A, F) be a prefibered category corresponding to a pseudo-functor E^{op} ⟶ Cat,

ObE ∋ X ⟼ A_{X}, HomE ∋ f ⟼ f^{∗}, HomE ×_{ObE} HomE ∋ (f, g) ⟼ c_{f,g}. (1)

A2.8.1. Colimits of pseudo-functors. The composition of inclusion functors A_{X} ↪ A and the canonical functor A ⟶ ColimA/E provide for any X ∈ ObE a functor q_{X} : A_{X} ⟶ ColimA/E, and for any morphism f : X → Y of E a diagram commutative up to isomorphism:

A_{Y} ⟶^{f^{∗}} A_{X}

q_{Y} ↘ ↙ q_{X} ColimA/E

Thus ColimA/E is a colimit in the sense of pseudo-functors of the pseudo-functor E^{op} ⟶ Cat.

Note that if X ⟼ A_{X}, f ⟼ f^{∗} is a functor, the category ColimA/E is not, in general, the colimit of this functor.

A2.8.2. Limits of pseudo-functors. Fix a pseudo-functor

E^{op} ⟶ Cat, X ⟼ A_{X}, f ⟼ f^{∗}, (f, g) ⟼ c_{f,g}.

For any X ∈ ObE, denote by p_{X} the functor LimA/E ⟶ A_{X} of evaluation at X. For any morphism f : X → Y of E, there is a diagram

A_{Y} ⟶^{f^{∗}} A_{X}

p_{Y} ↖ ↗ p_{X} LimA/E

commutative up to isomorphism. This means that LimA/E is a limit in the sense of pseudo-functors of the pseudo-functor E^{op} ⟶ Cat.

If X ⟼ A_{X}, f ⟼ f^{∗} is a functor, the category LimA/E is not, in general, the limit of this functor.

A2.9. Cofibered and bifibered categories. Fix a category A = (A, A →^{F} E), over E. A morphism ξ : x → y of A is called cocartesian if it is a cartesian morphism of the category A^{op} := (A^{op}, F^{op}) over E^{op}. This means that for any x′ ∈ ObA_{Y}, the map A_{X}(y, y′) ⟶ Hom_{f}(x, y′), u ⟼ u ∘ ξ, is bijective. In this case, (y, ξ) is called a direct image of x by f. If it exists for any x ∈ A_{X}, then there exists a direct image functor f_{∗} : A_{X} ⟶ A_{Y}. It is defined (uniquely up to isomorphism) by an isomorphism of bifunctors

A_{Y} (f_{∗}(x), y) ⟶^{∼} Hom_{f}(x, y).

A2.9.1. Suppose f_{∗} exists. Then f^{∗} exists iff f_{∗} has a right adjoint.

In fact, the functor f^{∗} is defined (uniquely up to isomorphism) by a functorial isomorphism A_{X}(x, f^{∗}(y)) ⟶^{∼} Hom_{f}(x, y). Therefore we have a functorial isomorphism A_{X}(x, f^{∗}(y)) ≃ A_{Y} (f_{∗}(x), y).

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