- Email: [email protected]
Descent theory of locally internal categories
Journal of Pure and Applied Algebra 154 (2000) 15–26
www.elsevier.com/locate/jpaa
Descent theory of locally internal categories Renato Betti Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy Communicated by I. Moerdijk Dedicated to Bill Lawvere
Abstract We prove that, in a category with ÿnite limits, known characterizations of descent and e ective descent morphisms along a map p depend only on the adjointness of p and p◦ in a Span category. Therefore, they can be easily transferred to other situations, e.g. to locally internal c 2000 Elsevier Science B.V. All rights reserved. categories and to existential hyperdoctrines. MSC: 18D30; 18D20; 18D05
Introduction This paper deals with descent theory from the point of view of Span-enriched categories. This perspective is here applied to locally internal categories and to existential hyperdoctrines, mainly with the aim of describing the relationships between di erent approaches. Our general reference for terminology and basic results in descent theory is provided by Janelidze and Tholen [8,9]. Here however we adopt the new notation coming from Betti and Walters [5], where locally internal categories over C are regarded as suitable categories enriched in the bicategory Span C (for an early version of this approach, see Lawvere’s Perugia Notes [11]). These categories are the same as locally small categories indexed by C in the sense of ParÃe and Schumacher [13]. Hence they are presentations of (locally small) ÿbrations in the sense of BÃenabou [2]. If C is a ÿnitely complete category with stable coequalizers and X is a locally internal category over C, any universal regular epimorphism is an e ective X-descent if and only if the locally internal category X is “p-cocomplete”. This means that E-mail address: [email protected] (R. Betti). c 2000 Elsevier Science B.V. All rights reserved. 0022-4049/00/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 9 9 ) 0 0 1 8 9 - 9
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suitable right Kan extensions along p exist and are representable in the bicategory of (Span C)-enriched categories. As a consequence, universal regular epimorphisms in C are e ective X-descent morphisms for any cocomplete, locally internal category X over C. If X is the “principal” category over the locally cartesian closed category C, i.e. C regarded as locally internal over itself, then one recovers the original case. The point of view of enriched categories allows a common setting for internal and locally internal categories and we believe that it gives intuitive insights. Moreover, it provides a natural meaning to the diagrams which arise when dealing with such categories; thus, simplifying many proofs: internal categories are just one-object (Span C)-enriched categories, i.e. monads in Span C, and diagrams over the internal category A are just A-modules. Among other things, this allows to give to the whole subject a certain formal analogy with the original formulation of descent theory as in Grothendieck [7]. The ÿrst section recalls notations and results from [5], with the aim of rendering the paper self-contained. More details on internal categories regarded as one object enriched categories and their module calculus can be found in Betti and Walters [6].
1. Locally internal categories as (Span C) enriched categories 1.1. Internal categories and internal functors Let C be a category with pullbacks and consider the bicategory Span C. Recall that C can be embedded in the bicategory Span C by means of a homomorphism which is the identity on objects. Arrows of C, when regarded in Span C are called maps and are characterized (up to isomorphism) by the fact of having a right adjoint. Namely, the right adjoint of the map f is a span denoted by f◦ . If A is a category internal to C, with domain and codomain arrows @0 and @1 : −→ A1 −→ A0 then it becomes a span = @1 @◦0 : A0 9 A0 with a monad structure in Span C: the monad structure 2 → and 1A0 → is provided exactly by the operations of the internal category. We thus regard internal categories as monads (A; ) in Span C, or also as one-object categories enriched in Span C. Functors between internal categories are monad mappings. More precisely, let A = (A; ) and B = (B; ) be internal categories. A functor F = (f; ’) : A → B amounts to a map f : A → B (which provides the correspondence on objects) and a 2-cell ’ : → f◦ f (which provides the correspondence on arrows) compatible with compositions and identities in A e B (this compatibility is expressed by commutative diagrams of 2-cells in Span C). Composition of internal functors is easily deÿned and the category Cat C of internal categories and internal functors can be regarded as the category of monads and their mappings in Span C.
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Objects of Span C, when regarded as trivial monads become discrete internal categories and, provided natural transformations of internal functors are deÿned, one obtains an embedding of Span C into the bicategory Cat C. 1.2. Modules of internal categories Let A = (A; ) and B = (B; ) be internal categories. The notion of a module A 9 B was introduced by BÃenabou as profunctor or distributeur [1] and also by Lawvere [12] under the name of bimodule. Deÿnition. A module A 9 B is given by an arrow : A → B in Span C (the component of the module), endowed with a left action of the monad and a right action of the monad . In other words, there are given two 2-cells: −→ −→ which preserve identity and composition of A and B and moreover satisfy a mixed associativity law, expressed by the commutativity of the following diagram of 2-cells in Span C: −→ y y −→ Without ambiguity, we denote by both a module A 9 B and its component. The notion of a morphism of modules → consists of a 2-cell between the components, which is compatible with the category actions. In this way, one has a category Mod (A; B) which is ÿnitely complete and cocomplete if C is. Suppose now that C has universal coequalizers. Then, one can deÿne composition of modules ⊗B : A 9 C in the following situation:
(A; ) 9 (B; ) 9 (C; ): The composite module ⊗B is given by the coequalizer of the two actions of on and in the middle: −→ −→ → ⊗ : B
Composition of modules allows to deÿne the bicategory Mod C, whose objects are internal categories and whose arrows are modules. Observe that identities in Mod C are given by the category (i.e. monad) structures. In the following, we use the adjoint pair of modules F∗ a F ∗ associated to a functor F = (f; ’) : (A; ) → (B; ). The component of F∗ and of F ∗ can be easily described, respectively, as f and f◦ . Their actions are provided by the composition in B, as well as the unit and counit of the adjunction f a f◦ .
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Just recall that, for pairs of composable functors: (FG)∗ (FG)∗
∼ = F∗ G∗ ∼ = G∗ F ∗ :
1.3. Locally internal categories The general theory of categories enriched in a bicategory B appears in Betti et al. [4], the particular case B = Span C, which regards locally internal categories, can be found in Betti and Walters [5,6]. Let X be a category enriched in the bicategory Span C. We denote by ex the underlying object of x. Deÿnition. Let X be a (Span C)-category, x an object and f a map A → ex . A substitution xf of x along f is an object over A with isomorphisms: X(x; y)f ∼ = X(xf ; y); ◦ f X(y; x) ∼ = X(y; xf ); for each object y. We say that X has substitutions when for each x and each f a substitution xf exists. A locally internal category over C is a (Span C)-enriched category with substitution along maps. When it exists, the substitution xf is unique up to isomorphism and one can prove that the two substitution laws of the above deÿnition are equivalent (see [5]). Moreover any functor F : X → Y of (Span C)-categories preserves substitutions: F(x)h ∼ = F(xh ). Again from [5] one has: Theorem. The bicategory of locally small ÿbrations over C and of cartesian functors is equivalent to the bicategory of (Span C)-enriched categories with substitution along maps and of (Span C)-enriched functors. Observe that the basic ÿbration “codomain” : C 2 → C is a locally small ÿbration if and only if C is locally cartesian closed. In this case, C is itself a locally internal category, substitutions are given by pullback functors and the hom is computed by their right adjoints . Precisely: C(f; g) = f×B (X × g) is an arrow with codomain A × B, i.e. a span A 9 B, where f : X → A and g : Y → B. 1.4. Functor categories When A = (A; ) is an internal category and X is a locally internal one, a functor F : A → X is deÿned to be an object x of X over A endowed with a 2-cell ’ : → X(x; x) satisfying the usual axioms of functors. It is easy to see that X(x; x) is a
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monad in Span C; hence it is an internal category (it is the internal full subcategory generated by x). The functor F thus turns out to be given by an object x with a monad mapping ’: we write simply F = (x; ’). One deÿnes an ordinary category XA by taking as objects the functors A → X and as arrows (x; ’) → (y; ) those 2-cells : 1A → X(x; y) such that the following diagram commutes in (Span C) (A; A): · y
·
−−−−−→
X(x;y)·’
· X(x; y)
·X(x;y)
−−−−−→ X(y; y)· X(x; y) y
X(x; y) −−−−−→ X(x; y) · X(x; x) −−−−−→
X(x; y)
In particular, an internal functor F : A → B induces as usual by composition an ordinary functor: XF : XB → XA : When C is locally cartesian closed, one can consider the category CA for any internal category A, and prove that it holds the following natural bijection: B → CA : B9 A An object of CA can thus be regarded as a module 1 9 A and is called an internal diagram of type A (here 1 denotes the trivial internal category).
2. The descent property Suppose a morphism p : E → B is given in the ÿnitely complete category C. Then, p gives rise to an internal category Ep , namely the equivalence relation induced by p, and to a fully faithful (internal) functor: p : Ep → B where B is a discrete internal category. We regard Ep as having the category structure provided by the monad p◦ p in Span C. Explicitly, the identity is the unit : 1E → p◦ p and composition in Ep is given by p◦ p : p◦ pp◦ p → p◦ p, where : pp◦ → 1B is the counit of the adjointness p a p◦ . The correspondence on objects of the functor p is p : E → B, and the identity 2-cell of p◦ p provides the correspondence on arrows. Deÿnition. A descent datum for p is a module ( ; ) : 1 9 Ep . In other words, it is constituted by an arrow : C → E (the component of the datum) endowed with an
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(associative and unitary) action: : p◦ p → of the monad p◦ p. A morphism of descent data ( ; ) → ( 0 ; 0 ) for p is a morphism of actions, i.e. it is constituted by a component h : → 0 in C=E which preserves the actions, in the sense that the following diagram of 2-cells commutes:
p◦ p −−−−−→ ◦ p ph h y y 0
p◦ p 0 −−−−−→ 0
By deÿnition, descent data for p and their morphisms constitute the category Mod(1; Ep ), which can be described as the category of algebras for the monad induced by the adjoint pair p! a p∗ (here p! denotes composition with p and p∗ is the pullback along p): p∗
C=B C=E: p!
It is easy to see that any module ( ; 1) : 1 9 B becomes a descent datum in a canonical way by pulling it back along p. Here is just an arrow D → B with the trivial action. One has that p◦ : 1 9 E admits the canonical action: p◦
p◦ pp◦ → p◦ : Moreover, any morphism → in Mod (1; B) gives rise to a morphism of descent data p◦ h : p◦ → p◦ . This correspondence describes the comparison functor p ∗ ⊗B −, through which p∗ factors: p ∗ ⊗B − C=B ∼ = Alg(p◦ p): = Mod (1; B)−−−−−→Mod (1; Ep ) ∼ Deÿnition. The arrow p : E → B is said to be (an e ective) descent if the functor p ∗ ⊗B − is (an equivalence) fully faithful. Lemma. A morphism k : p◦ → p◦ in Mod (1; E) ∼ = C=E is the component of a morphism in Mod (1; Ep ) with respect to the canonical actions of p◦ and p◦ , if and only if its adjoint kˆ : pp◦ → coequalizes the pair: pp◦
−−→ pp◦ pp pp − −−−→ ◦ ◦
◦
(1)
pp
in Mod (1; B) ∼ = C=B. Proof. The proof depends only on the properties of the adjointness p a p◦ . One has to show that the square of 2-cells:
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p◦
◦ ◦ ◦ p pp −−−−−→ p p◦ pk k y y ◦ p
p◦ pp◦ −−−−−→ p◦
commutes if and only if kˆ : pp◦ → coequalizes the pair (1). Theorem. The functor p ∗ ⊗B − : Mod(1; B) → Mod(1; Ep ) is fully faithful if and only if p is a universal regular epimorphism. Proof. Just observe that p is a universal regular epimorphism if and only if : pp◦ → is the coequalizer of the pair (1). By the previous lemma, exactly in this case one has a natural bijection between arrows → in Mod(1; B) and arrows p◦ → p◦ , endowed with the canonical actions in Mod(1; Ep ). Suppose now that ( ; ) is an object in Mod(1; Ep ). Lemma. The object ( ; ) in Mod(1; Ep ) is isomorphic to an object of the type p◦ , endowed with the canonical action, if and only if there exists an isomorphism h : → p◦ in Mod(1; E), such that its adjoint hˆ : p → is a pullback stable coequalizer of the pair: p
−→
pp◦ p −→ p : p
(2)
Proof. The properties of the adjointness p a p◦ show that the square p◦ ph
◦ ◦ ◦ p p −−−−−→ p pp p◦ y y
h
−−−−−→ p◦
commutes if and only if hˆ : p → coequalizes the pair (2). As in Reiterman–Sobral–Tholen [14] and in Sobral–Tholen [15], one can prove Theorem. If p : E → B is a descent morphism, then it is an e ective descent if and only if for any ( ; ) in Mod(1; Ep ) there exists a stable coequalizer of the pair (2). Proof. By assumption, the functor p ∗ ⊗B − : Mod(1; B) → Mod(1; Ep ) is fully faithful. Now, one has to show that it is representative (i.e. essentially surjective) on objects exactly when the pair (2) has a stable coequalizer.
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Suppose ÿrst that q : p → is such a stable coequalizer. Then also p◦ q is a coequalizer and factors through it and a unique t : p◦ → :
It is not dicult to see that t is an isomorphism, having as inverse the morphism h given by the composition: p◦ q
−→ p◦ p −→ p◦ : Moreover, by the previous lemma, one has that h is an isomorphism in Mod(1; Ep ) ˆ because q = h. Conversely, suppose that h is an isomorphism in Mod(1; Ep ) between ( ; ) and p◦ endowed with the canonical action. By the previous lemma, one knows that hˆ coequalizes the pair (2). Consider now any r : p → that coequalizes the pair (2) and recall that hˆ is the composite morphism: ph hˆ : p −→ pp◦ −→
with ph invertible. Then one has the morphism: (ph)−1
r
pp◦ −→ p −→ and, by adjointness, a morphism: p◦ → p◦ . Because p ∗ ⊗B − is fully faithful, one ˆ obtains a unique arrow k : → for which it is not dicult to see that r = k · h. ˆ Hence h is the (necessarily stable) coequalizer of the pair (2). Suppose now that C admits universal coequalizers. Then, the comparison functor p ∗ ⊗B − has a left adjoint: p ∗ ⊗Ep − Mod(1; Ep )−−−−−→Mod(1; B)
deÿned on the object ( ; ) of Mod(1; Ep ) by the coequalizer p
−−→ pp◦ p − −−−→ p −→ p ∗ ⊗ : p
Ep
Beck’s monadicity theorem ensures that the comparison functor is an equivalence of categories: Theorem. If C has ÿnite limits and universal coequalizers, a morphism is an e ective descent morphism if and only if it is a descent morphism, i.e. an universal regular epimorphism.
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Proof. The counit of the adjointness p ∗ ⊗Ep − a p ∗ ⊗B − is an isomorphism ∼ =
p ∗ ⊗Ep p ∗ → 1B if and only if p is an universal regular epimorphism. The unit is always ∼ =
an isomorphism 1Ep −→ p ∗ ⊗B p ∗ .
Remark. Observe that, under the given assumptions, the previous theorem says that the adjoint pair of modules p ∗ a p ∗ induced by p is an equivalence in the bicategory ModC of internal categories and internal modules. 3. Descent theory of existential hyperdoctrines We now transfer the previous characterization of descent and e ective descent morphisms to existential hyperdoctrines. For this, it is enough to observe that the proofs of the previous lemmas depend only on the adjointness p a p◦ . Recall that an existential hyperdoctrine over C [10] can be regarded as a ÿbration X → C with small colimits. This means that (i) any ÿber XA is ÿnitely cocomplete, (ii) for any f : A → B in C, the substitution functor f∗ : XB → XA has a left adjoint: f a f ∗ ;
(3)
(iii) the Beck–Chevalley condition is satisÿed. Now, according to BÃenabou and Roubaud [3], one has that the category of X-descent data is equivalent with the category of algebras for the monad Tf generated by the adjointness (3). More precisely, the substitution functor f∗ : XB → XA lands in Alg Tf , i.e. f∗ x has the canonical structure of a Tf -algebra: f∗ (f f∗ x) → f∗ x provided by the counit of (3), for any x in XB . Deÿnition. The morphism p : E → B in C is an X-descent morphism if the functor p∗ : XB → Alg Tp is fully faithful. It is an X-e ective descent if p∗ is an equivalence. Theorem. Suppose that C has ÿnite limits and X is an existential hyperdoctrine over C. Then, given an arrow p : E → B in C: (i) p is an X-descent morphism if and only if, for any x in XB , the counit x : p p∗ x → x is a stable coequalizer of the pair: ∗ −→ p p∗ p p∗ x − −−→ p p x:
(ii) p is an e ective X-descent morphism if and only if it is an X-descent and, for any ( ; ) in Alg Tp , there exists the coequalizer of the pair: p
−−→ p (p∗ (p )) − −−−→ p : p
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4. Descent data of locally internal categories Suppose again that C is a ÿnitely complete category, and denote by X a locally internal category over C. It is easy to see that, given a map p : E → B in C and an object x in X over B, the substitution xp comes equipped with a 2-cell: p◦ p
x 1 Ep ∼ = X(xp ; xp ) = p◦ p−−−−−→p◦ X(x; x)p ∼
in such a way to deÿne a functor Ep → X, i.e. an object of XEp . So, one has a functor ( )p : XB → XEp that factors the substitution functor ( )p : XB → XE . The following property is not dicult to prove, taking into account that a locally internal category can be regarded as a locally small ÿbration (Theorem of Section 1): Theorem. (i) A X-descent datum for p is a functor Ep → X, i.e. it is given by an object x over E, together with a monad map x : p◦ p → X(x; x). (ii) A morphism of X-descent data (x; x ) → (y; y ) for p is a 2-cell f : 1E → X(x; y) such that the following diagram commutes: ◦ p p y f y
fx
−−−−−→ X(x; y)X(x; x) y
X(y; y)X(x; y) −−−−−→
X(x; y)
(iii) The morphism p : E → B is an (e ective) X-descent morphism if and only if the substitution functor: ( )p
XB −→ XEp is (an equivalence) fully faithful. In other words, in the context of locally internal categories, the category of X-descent data for p can be described as the category XEp and the descent problem regards the functor “substitution along p”. Moreover, when X = C the notion of X-descent data and the descent problem reduce to the original ones. In [6] it was proved that a locally internal category, regarded as a (Span C)-category is internally cocomplete when it admits all colimits indexed by modules. Recall that the colimit of the functor F : A → X, indexed by the module ’ : A 9 B, when it exists, is an object ’ ∗ F of X, uniquely deÿned up to isomorphism, which represents the right Kan extension [’; F] of F through ’. In other words, for any module : B 9 C
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and for any G : C → X one has the natural bijection of 2-cells: → X(’ ∗ F; G) : ⊗B ’ → X(F; G) Theorem. Let C be a ÿnitely complete category with stable coequalizers and X a locally internal category over C. The regular epimorphism p : E → B is an Xe ective descent morphism if and only if X is p-cocomplete. Proof. Recall that the adjoint pair of modules p ∗ a p ∗ is an equivalence in ModC (Remark at the end of Section 2). This means in particular that the right Kan extension [p ∗ ; x] can be written as [p ∗ ; x] ∼ = x⊗Ep p ∗ for any descent datum x. Observe moreover that the substitution functor ( )p can be described as the composition with p ∗ . Namely: ( )p ∼ = −⊗B p ∗ . Now, X is p-cocomplete if and only if x⊗Ep p ∗ is representable, for any descent datum x : Ep → X. Exactly in this case one has a functor: − ⊗ p ∗ : XEp → XB Ep
which is left adjoint to −⊗p ∗ and with unit and counit invertible. Remark. Recall that a stack over C is a ÿbration F over C such that, for any regular epimorphism p : E → B, composition with p : Ep → B is an equivalence between the category of cartesian functors Fib(B; F) and that of cartesian functors Fib(Ep ; F). For locally small ÿbrations this means exactly that a stack is a p-cocomplete category. Hence, for a locally small stack the descent and e ective descent morphism agree with the regular epimorphisms. References [1] J. BÃenabou, Les distributeurs, Inst. de Math. Pure et Appliquee, Univ. Cath. de Louvain, Rapport No. 33, 1973. [2] J. BÃenabou, Fibered categories and the foundations of naive category theory, J. Symbolic Logic 50 (1985) 1–37. [3] J. BÃenabou, J. Roubaud, Monades et descente, C.R. Acad. Sci. Paris 270 (1970) A96–98. [4] R. Betti, A. Carboni, R. Street, R.F.C. Walters, Variation through enrichment, J. Pure Appl. Algebra 29 (1983) 109–127. [5] R. Betti, R.F.C. Walters, Closed categories and variable category theory, Ist. Mat. Univ. di Milano, Quad. No. 5, 1985. [6] R. Betti, R.F.C. Walters, On completeness of locally internal categories, J. Pure Appl. Algebra 47 (1987) 105–117. [7] A. Grothendieck, CatÃegories ÿbrÃees et descente, Lecture Notes in Mathematics, Vol. 224, Springer, Berlin, 1971, pp. 145 –194. [8] G. Janelidze, W. Tholen, Facets of descent I, Appl. Cat. Struct. 2 (1994) 245–281. [9] G. Janelidze, W. Tholen, Facets of descent II, Appl. Cat. Struct. 5 (1997) 229–248. [10] F.W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics, Vol. XVII, 1970, pp. 1–14. [11] F.W. Lawvere, Category theory over a base topos, Lect. Notes Univ. di Perugia 1972–73.
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[12] F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. e Fis. di Milano 43 (1973) 135–166. [13] R. ParÃe, D. Schumacher, Abstract families and the adjoint functor theorem, Lecture Notes in Mathematics, Vol. 66, Springer, Berlin, 1978, pp. 1–125. [14] J. Reiterman, M. Sobral, W. Tholen, Coposites of e ective descent maps, Cahiers Top. GÃeom. Di . Categ. XXXIV (1993) 193–207. [15] M. Sobral, W. Tholen, E ective descent morphisms and e ective equivalence relations, Can. Math. Soc. 13 (1992) 421–433.
www.elsevier.com/locate/jpaa
Descent theory of locally internal categories Renato Betti Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy Communicated by I. Moerdijk Dedicated to Bill Lawvere
Abstract We prove that, in a category with ÿnite limits, known characterizations of descent and e ective descent morphisms along a map p depend only on the adjointness of p and p◦ in a Span category. Therefore, they can be easily transferred to other situations, e.g. to locally internal c 2000 Elsevier Science B.V. All rights reserved. categories and to existential hyperdoctrines. MSC: 18D30; 18D20; 18D05
Introduction This paper deals with descent theory from the point of view of Span-enriched categories. This perspective is here applied to locally internal categories and to existential hyperdoctrines, mainly with the aim of describing the relationships between di erent approaches. Our general reference for terminology and basic results in descent theory is provided by Janelidze and Tholen [8,9]. Here however we adopt the new notation coming from Betti and Walters [5], where locally internal categories over C are regarded as suitable categories enriched in the bicategory Span C (for an early version of this approach, see Lawvere’s Perugia Notes [11]). These categories are the same as locally small categories indexed by C in the sense of ParÃe and Schumacher [13]. Hence they are presentations of (locally small) ÿbrations in the sense of BÃenabou [2]. If C is a ÿnitely complete category with stable coequalizers and X is a locally internal category over C, any universal regular epimorphism is an e ective X-descent if and only if the locally internal category X is “p-cocomplete”. This means that E-mail address: [email protected] (R. Betti). c 2000 Elsevier Science B.V. All rights reserved. 0022-4049/00/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 9 9 ) 0 0 1 8 9 - 9
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suitable right Kan extensions along p exist and are representable in the bicategory of (Span C)-enriched categories. As a consequence, universal regular epimorphisms in C are e ective X-descent morphisms for any cocomplete, locally internal category X over C. If X is the “principal” category over the locally cartesian closed category C, i.e. C regarded as locally internal over itself, then one recovers the original case. The point of view of enriched categories allows a common setting for internal and locally internal categories and we believe that it gives intuitive insights. Moreover, it provides a natural meaning to the diagrams which arise when dealing with such categories; thus, simplifying many proofs: internal categories are just one-object (Span C)-enriched categories, i.e. monads in Span C, and diagrams over the internal category A are just A-modules. Among other things, this allows to give to the whole subject a certain formal analogy with the original formulation of descent theory as in Grothendieck [7]. The ÿrst section recalls notations and results from [5], with the aim of rendering the paper self-contained. More details on internal categories regarded as one object enriched categories and their module calculus can be found in Betti and Walters [6].
1. Locally internal categories as (Span C) enriched categories 1.1. Internal categories and internal functors Let C be a category with pullbacks and consider the bicategory Span C. Recall that C can be embedded in the bicategory Span C by means of a homomorphism which is the identity on objects. Arrows of C, when regarded in Span C are called maps and are characterized (up to isomorphism) by the fact of having a right adjoint. Namely, the right adjoint of the map f is a span denoted by f◦ . If A is a category internal to C, with domain and codomain arrows @0 and @1 : −→ A1 −→ A0 then it becomes a span = @1 @◦0 : A0 9 A0 with a monad structure in Span C: the monad structure 2 → and 1A0 → is provided exactly by the operations of the internal category. We thus regard internal categories as monads (A; ) in Span C, or also as one-object categories enriched in Span C. Functors between internal categories are monad mappings. More precisely, let A = (A; ) and B = (B; ) be internal categories. A functor F = (f; ’) : A → B amounts to a map f : A → B (which provides the correspondence on objects) and a 2-cell ’ : → f◦ f (which provides the correspondence on arrows) compatible with compositions and identities in A e B (this compatibility is expressed by commutative diagrams of 2-cells in Span C). Composition of internal functors is easily deÿned and the category Cat C of internal categories and internal functors can be regarded as the category of monads and their mappings in Span C.
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Objects of Span C, when regarded as trivial monads become discrete internal categories and, provided natural transformations of internal functors are deÿned, one obtains an embedding of Span C into the bicategory Cat C. 1.2. Modules of internal categories Let A = (A; ) and B = (B; ) be internal categories. The notion of a module A 9 B was introduced by BÃenabou as profunctor or distributeur [1] and also by Lawvere [12] under the name of bimodule. Deÿnition. A module A 9 B is given by an arrow : A → B in Span C (the component of the module), endowed with a left action of the monad and a right action of the monad . In other words, there are given two 2-cells: −→ −→ which preserve identity and composition of A and B and moreover satisfy a mixed associativity law, expressed by the commutativity of the following diagram of 2-cells in Span C: −→ y y −→ Without ambiguity, we denote by both a module A 9 B and its component. The notion of a morphism of modules → consists of a 2-cell between the components, which is compatible with the category actions. In this way, one has a category Mod (A; B) which is ÿnitely complete and cocomplete if C is. Suppose now that C has universal coequalizers. Then, one can deÿne composition of modules ⊗B : A 9 C in the following situation:
(A; ) 9 (B; ) 9 (C; ): The composite module ⊗B is given by the coequalizer of the two actions of on and in the middle: −→ −→ → ⊗ : B
Composition of modules allows to deÿne the bicategory Mod C, whose objects are internal categories and whose arrows are modules. Observe that identities in Mod C are given by the category (i.e. monad) structures. In the following, we use the adjoint pair of modules F∗ a F ∗ associated to a functor F = (f; ’) : (A; ) → (B; ). The component of F∗ and of F ∗ can be easily described, respectively, as f and f◦ . Their actions are provided by the composition in B, as well as the unit and counit of the adjunction f a f◦ .
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Just recall that, for pairs of composable functors: (FG)∗ (FG)∗
∼ = F∗ G∗ ∼ = G∗ F ∗ :
1.3. Locally internal categories The general theory of categories enriched in a bicategory B appears in Betti et al. [4], the particular case B = Span C, which regards locally internal categories, can be found in Betti and Walters [5,6]. Let X be a category enriched in the bicategory Span C. We denote by ex the underlying object of x. Deÿnition. Let X be a (Span C)-category, x an object and f a map A → ex . A substitution xf of x along f is an object over A with isomorphisms: X(x; y)f ∼ = X(xf ; y); ◦ f X(y; x) ∼ = X(y; xf ); for each object y. We say that X has substitutions when for each x and each f a substitution xf exists. A locally internal category over C is a (Span C)-enriched category with substitution along maps. When it exists, the substitution xf is unique up to isomorphism and one can prove that the two substitution laws of the above deÿnition are equivalent (see [5]). Moreover any functor F : X → Y of (Span C)-categories preserves substitutions: F(x)h ∼ = F(xh ). Again from [5] one has: Theorem. The bicategory of locally small ÿbrations over C and of cartesian functors is equivalent to the bicategory of (Span C)-enriched categories with substitution along maps and of (Span C)-enriched functors. Observe that the basic ÿbration “codomain” : C 2 → C is a locally small ÿbration if and only if C is locally cartesian closed. In this case, C is itself a locally internal category, substitutions are given by pullback functors and the hom is computed by their right adjoints . Precisely: C(f; g) = f×B (X × g) is an arrow with codomain A × B, i.e. a span A 9 B, where f : X → A and g : Y → B. 1.4. Functor categories When A = (A; ) is an internal category and X is a locally internal one, a functor F : A → X is deÿned to be an object x of X over A endowed with a 2-cell ’ : → X(x; x) satisfying the usual axioms of functors. It is easy to see that X(x; x) is a
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monad in Span C; hence it is an internal category (it is the internal full subcategory generated by x). The functor F thus turns out to be given by an object x with a monad mapping ’: we write simply F = (x; ’). One deÿnes an ordinary category XA by taking as objects the functors A → X and as arrows (x; ’) → (y; ) those 2-cells : 1A → X(x; y) such that the following diagram commutes in (Span C) (A; A): · y
·
−−−−−→
X(x;y)·’
· X(x; y)
·X(x;y)
−−−−−→ X(y; y)· X(x; y) y
X(x; y) −−−−−→ X(x; y) · X(x; x) −−−−−→
X(x; y)
In particular, an internal functor F : A → B induces as usual by composition an ordinary functor: XF : XB → XA : When C is locally cartesian closed, one can consider the category CA for any internal category A, and prove that it holds the following natural bijection: B → CA : B9 A An object of CA can thus be regarded as a module 1 9 A and is called an internal diagram of type A (here 1 denotes the trivial internal category).
2. The descent property Suppose a morphism p : E → B is given in the ÿnitely complete category C. Then, p gives rise to an internal category Ep , namely the equivalence relation induced by p, and to a fully faithful (internal) functor: p : Ep → B where B is a discrete internal category. We regard Ep as having the category structure provided by the monad p◦ p in Span C. Explicitly, the identity is the unit : 1E → p◦ p and composition in Ep is given by p◦ p : p◦ pp◦ p → p◦ p, where : pp◦ → 1B is the counit of the adjointness p a p◦ . The correspondence on objects of the functor p is p : E → B, and the identity 2-cell of p◦ p provides the correspondence on arrows. Deÿnition. A descent datum for p is a module ( ; ) : 1 9 Ep . In other words, it is constituted by an arrow : C → E (the component of the datum) endowed with an
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(associative and unitary) action: : p◦ p → of the monad p◦ p. A morphism of descent data ( ; ) → ( 0 ; 0 ) for p is a morphism of actions, i.e. it is constituted by a component h : → 0 in C=E which preserves the actions, in the sense that the following diagram of 2-cells commutes:
p◦ p −−−−−→ ◦ p ph h y y 0
p◦ p 0 −−−−−→ 0
By deÿnition, descent data for p and their morphisms constitute the category Mod(1; Ep ), which can be described as the category of algebras for the monad induced by the adjoint pair p! a p∗ (here p! denotes composition with p and p∗ is the pullback along p): p∗
C=B C=E: p!
It is easy to see that any module ( ; 1) : 1 9 B becomes a descent datum in a canonical way by pulling it back along p. Here is just an arrow D → B with the trivial action. One has that p◦ : 1 9 E admits the canonical action: p◦
p◦ pp◦ → p◦ : Moreover, any morphism → in Mod (1; B) gives rise to a morphism of descent data p◦ h : p◦ → p◦ . This correspondence describes the comparison functor p ∗ ⊗B −, through which p∗ factors: p ∗ ⊗B − C=B ∼ = Alg(p◦ p): = Mod (1; B)−−−−−→Mod (1; Ep ) ∼ Deÿnition. The arrow p : E → B is said to be (an e ective) descent if the functor p ∗ ⊗B − is (an equivalence) fully faithful. Lemma. A morphism k : p◦ → p◦ in Mod (1; E) ∼ = C=E is the component of a morphism in Mod (1; Ep ) with respect to the canonical actions of p◦ and p◦ , if and only if its adjoint kˆ : pp◦ → coequalizes the pair: pp◦
−−→ pp◦ pp pp − −−−→ ◦ ◦
◦
(1)
pp
in Mod (1; B) ∼ = C=B. Proof. The proof depends only on the properties of the adjointness p a p◦ . One has to show that the square of 2-cells:
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p◦
◦ ◦ ◦ p pp −−−−−→ p p◦ pk k y y ◦ p
p◦ pp◦ −−−−−→ p◦
commutes if and only if kˆ : pp◦ → coequalizes the pair (1). Theorem. The functor p ∗ ⊗B − : Mod(1; B) → Mod(1; Ep ) is fully faithful if and only if p is a universal regular epimorphism. Proof. Just observe that p is a universal regular epimorphism if and only if : pp◦ → is the coequalizer of the pair (1). By the previous lemma, exactly in this case one has a natural bijection between arrows → in Mod(1; B) and arrows p◦ → p◦ , endowed with the canonical actions in Mod(1; Ep ). Suppose now that ( ; ) is an object in Mod(1; Ep ). Lemma. The object ( ; ) in Mod(1; Ep ) is isomorphic to an object of the type p◦ , endowed with the canonical action, if and only if there exists an isomorphism h : → p◦ in Mod(1; E), such that its adjoint hˆ : p → is a pullback stable coequalizer of the pair: p
−→
pp◦ p −→ p : p
(2)
Proof. The properties of the adjointness p a p◦ show that the square p◦ ph
◦ ◦ ◦ p p −−−−−→ p pp p◦ y y
h
−−−−−→ p◦
commutes if and only if hˆ : p → coequalizes the pair (2). As in Reiterman–Sobral–Tholen [14] and in Sobral–Tholen [15], one can prove Theorem. If p : E → B is a descent morphism, then it is an e ective descent if and only if for any ( ; ) in Mod(1; Ep ) there exists a stable coequalizer of the pair (2). Proof. By assumption, the functor p ∗ ⊗B − : Mod(1; B) → Mod(1; Ep ) is fully faithful. Now, one has to show that it is representative (i.e. essentially surjective) on objects exactly when the pair (2) has a stable coequalizer.
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Suppose ÿrst that q : p → is such a stable coequalizer. Then also p◦ q is a coequalizer and factors through it and a unique t : p◦ → :
It is not dicult to see that t is an isomorphism, having as inverse the morphism h given by the composition: p◦ q
−→ p◦ p −→ p◦ : Moreover, by the previous lemma, one has that h is an isomorphism in Mod(1; Ep ) ˆ because q = h. Conversely, suppose that h is an isomorphism in Mod(1; Ep ) between ( ; ) and p◦ endowed with the canonical action. By the previous lemma, one knows that hˆ coequalizes the pair (2). Consider now any r : p → that coequalizes the pair (2) and recall that hˆ is the composite morphism: ph hˆ : p −→ pp◦ −→
with ph invertible. Then one has the morphism: (ph)−1
r
pp◦ −→ p −→ and, by adjointness, a morphism: p◦ → p◦ . Because p ∗ ⊗B − is fully faithful, one ˆ obtains a unique arrow k : → for which it is not dicult to see that r = k · h. ˆ Hence h is the (necessarily stable) coequalizer of the pair (2). Suppose now that C admits universal coequalizers. Then, the comparison functor p ∗ ⊗B − has a left adjoint: p ∗ ⊗Ep − Mod(1; Ep )−−−−−→Mod(1; B)
deÿned on the object ( ; ) of Mod(1; Ep ) by the coequalizer p
−−→ pp◦ p − −−−→ p −→ p ∗ ⊗ : p
Ep
Beck’s monadicity theorem ensures that the comparison functor is an equivalence of categories: Theorem. If C has ÿnite limits and universal coequalizers, a morphism is an e ective descent morphism if and only if it is a descent morphism, i.e. an universal regular epimorphism.
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Proof. The counit of the adjointness p ∗ ⊗Ep − a p ∗ ⊗B − is an isomorphism ∼ =
p ∗ ⊗Ep p ∗ → 1B if and only if p is an universal regular epimorphism. The unit is always ∼ =
an isomorphism 1Ep −→ p ∗ ⊗B p ∗ .
Remark. Observe that, under the given assumptions, the previous theorem says that the adjoint pair of modules p ∗ a p ∗ induced by p is an equivalence in the bicategory ModC of internal categories and internal modules. 3. Descent theory of existential hyperdoctrines We now transfer the previous characterization of descent and e ective descent morphisms to existential hyperdoctrines. For this, it is enough to observe that the proofs of the previous lemmas depend only on the adjointness p a p◦ . Recall that an existential hyperdoctrine over C [10] can be regarded as a ÿbration X → C with small colimits. This means that (i) any ÿber XA is ÿnitely cocomplete, (ii) for any f : A → B in C, the substitution functor f∗ : XB → XA has a left adjoint: f a f ∗ ;
(3)
(iii) the Beck–Chevalley condition is satisÿed. Now, according to BÃenabou and Roubaud [3], one has that the category of X-descent data is equivalent with the category of algebras for the monad Tf generated by the adjointness (3). More precisely, the substitution functor f∗ : XB → XA lands in Alg Tf , i.e. f∗ x has the canonical structure of a Tf -algebra: f∗ (f f∗ x) → f∗ x provided by the counit of (3), for any x in XB . Deÿnition. The morphism p : E → B in C is an X-descent morphism if the functor p∗ : XB → Alg Tp is fully faithful. It is an X-e ective descent if p∗ is an equivalence. Theorem. Suppose that C has ÿnite limits and X is an existential hyperdoctrine over C. Then, given an arrow p : E → B in C: (i) p is an X-descent morphism if and only if, for any x in XB , the counit x : p p∗ x → x is a stable coequalizer of the pair: ∗ −→ p p∗ p p∗ x − −−→ p p x:
(ii) p is an e ective X-descent morphism if and only if it is an X-descent and, for any ( ; ) in Alg Tp , there exists the coequalizer of the pair: p
−−→ p (p∗ (p )) − −−−→ p : p
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4. Descent data of locally internal categories Suppose again that C is a ÿnitely complete category, and denote by X a locally internal category over C. It is easy to see that, given a map p : E → B in C and an object x in X over B, the substitution xp comes equipped with a 2-cell: p◦ p
x 1 Ep ∼ = X(xp ; xp ) = p◦ p−−−−−→p◦ X(x; x)p ∼
in such a way to deÿne a functor Ep → X, i.e. an object of XEp . So, one has a functor ( )p : XB → XEp that factors the substitution functor ( )p : XB → XE . The following property is not dicult to prove, taking into account that a locally internal category can be regarded as a locally small ÿbration (Theorem of Section 1): Theorem. (i) A X-descent datum for p is a functor Ep → X, i.e. it is given by an object x over E, together with a monad map x : p◦ p → X(x; x). (ii) A morphism of X-descent data (x; x ) → (y; y ) for p is a 2-cell f : 1E → X(x; y) such that the following diagram commutes: ◦ p p y f y
fx
−−−−−→ X(x; y)X(x; x) y
X(y; y)X(x; y) −−−−−→
X(x; y)
(iii) The morphism p : E → B is an (e ective) X-descent morphism if and only if the substitution functor: ( )p
XB −→ XEp is (an equivalence) fully faithful. In other words, in the context of locally internal categories, the category of X-descent data for p can be described as the category XEp and the descent problem regards the functor “substitution along p”. Moreover, when X = C the notion of X-descent data and the descent problem reduce to the original ones. In [6] it was proved that a locally internal category, regarded as a (Span C)-category is internally cocomplete when it admits all colimits indexed by modules. Recall that the colimit of the functor F : A → X, indexed by the module ’ : A 9 B, when it exists, is an object ’ ∗ F of X, uniquely deÿned up to isomorphism, which represents the right Kan extension [’; F] of F through ’. In other words, for any module : B 9 C
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and for any G : C → X one has the natural bijection of 2-cells: → X(’ ∗ F; G) : ⊗B ’ → X(F; G) Theorem. Let C be a ÿnitely complete category with stable coequalizers and X a locally internal category over C. The regular epimorphism p : E → B is an Xe ective descent morphism if and only if X is p-cocomplete. Proof. Recall that the adjoint pair of modules p ∗ a p ∗ is an equivalence in ModC (Remark at the end of Section 2). This means in particular that the right Kan extension [p ∗ ; x] can be written as [p ∗ ; x] ∼ = x⊗Ep p ∗ for any descent datum x. Observe moreover that the substitution functor ( )p can be described as the composition with p ∗ . Namely: ( )p ∼ = −⊗B p ∗ . Now, X is p-cocomplete if and only if x⊗Ep p ∗ is representable, for any descent datum x : Ep → X. Exactly in this case one has a functor: − ⊗ p ∗ : XEp → XB Ep
which is left adjoint to −⊗p ∗ and with unit and counit invertible. Remark. Recall that a stack over C is a ÿbration F over C such that, for any regular epimorphism p : E → B, composition with p : Ep → B is an equivalence between the category of cartesian functors Fib(B; F) and that of cartesian functors Fib(Ep ; F). For locally small ÿbrations this means exactly that a stack is a p-cocomplete category. Hence, for a locally small stack the descent and e ective descent morphism agree with the regular epimorphisms. References [1] J. BÃenabou, Les distributeurs, Inst. de Math. Pure et Appliquee, Univ. Cath. de Louvain, Rapport No. 33, 1973. [2] J. BÃenabou, Fibered categories and the foundations of naive category theory, J. Symbolic Logic 50 (1985) 1–37. [3] J. BÃenabou, J. Roubaud, Monades et descente, C.R. Acad. Sci. Paris 270 (1970) A96–98. [4] R. Betti, A. Carboni, R. Street, R.F.C. Walters, Variation through enrichment, J. Pure Appl. Algebra 29 (1983) 109–127. [5] R. Betti, R.F.C. Walters, Closed categories and variable category theory, Ist. Mat. Univ. di Milano, Quad. No. 5, 1985. [6] R. Betti, R.F.C. Walters, On completeness of locally internal categories, J. Pure Appl. Algebra 47 (1987) 105–117. [7] A. Grothendieck, CatÃegories ÿbrÃees et descente, Lecture Notes in Mathematics, Vol. 224, Springer, Berlin, 1971, pp. 145 –194. [8] G. Janelidze, W. Tholen, Facets of descent I, Appl. Cat. Struct. 2 (1994) 245–281. [9] G. Janelidze, W. Tholen, Facets of descent II, Appl. Cat. Struct. 5 (1997) 229–248. [10] F.W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics, Vol. XVII, 1970, pp. 1–14. [11] F.W. Lawvere, Category theory over a base topos, Lect. Notes Univ. di Perugia 1972–73.
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[12] F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. e Fis. di Milano 43 (1973) 135–166. [13] R. ParÃe, D. Schumacher, Abstract families and the adjoint functor theorem, Lecture Notes in Mathematics, Vol. 66, Springer, Berlin, 1978, pp. 1–125. [14] J. Reiterman, M. Sobral, W. Tholen, Coposites of e ective descent maps, Cahiers Top. GÃeom. Di . Categ. XXXIV (1993) 193–207. [15] M. Sobral, W. Tholen, E ective descent morphisms and e ective equivalence relations, Can. Math. Soc. 13 (1992) 421–433.