Representations of Internal Categories

Kyushu J. Math. Vol . 62, 2008, pp. 139–169

REPRESENTATIONS OF INTERNAL CATEGORIES

Atsushi YAMAGUCHI (Received 28 December 2006)

Abstract. Adams gave the notion of a Hopf algebroid generalizing the notion of a Hopf algebra and showed that certain generalized homology theories take values in the category of comodules over the Hopf algebroid associated with each homology theory. A Hopf algebra represents an affine group scheme which is a group in the category of a scheme and the notion of comodules over a Hopf algebra is equivalent to the notion of representations of the affine group scheme represented by a Hopf algebra. On the other hand, a Hopf algebroid represents a groupoid in the category of schemes. Therefore, it is natural to consider the notion of comodules over a Hopf algebroid as representations of the groupoid represented by a Hopf algebroid. This motivates the study of representations of groupoids, and more generally categories, for topologists. The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category, using the notion of a fibered category.

0. Introduction

In [1], Adams generalized the notion of Hopf algebras in the study of generalized homology theories satisfying certain conditions and showed that such a generalized homology theory, say E∗, takes values in the category of comodules over the ‘generalized Hopf algebra’ associated with E∗. The notion introduced by Adams is now called a Hopf algebroid which represents a functor taking values in the category of groupoids. Here ‘a groupoid’ means a special category whose morphisms are all isomorphisms. A comodule over a Hopf algebroid can be regarded as a representation of the groupoid represented by . The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category. We begin by reviewing the notion of a fibered category following [7] and an internal category in Section 1, we give a detailed description on the relationship between the notions of a fibered category and a 2-category in Section 2, which is originally observed in [7, Section 8]. There, we show that the 2-category of a fibered category over a given category E is equivalent to the 2-category of ‘lax functors’ from the opposite category of E to the 2-category of categories. Our construction of fibered categories from lax functors allows us to give the notion of fibered categories represented by internal categories (Example 2.18) and a short definition (Example 2.19) of Grothendieck topoi over a simplicial object in the given site.

2000 Mathematics Subject Classiﬁcation: Primary 18B40, 18D05, 18D30, 18D99, 55N20, 55U99. Keywords and Phrases: representation; groupoid; internal category; ﬁbered category; generalized homology. 140 A. Yamaguchi

By making use of the notion of fibered category, we give a definition of the representations of internal categories in Section 3 which generalizes the definition given by Deligne in [3]. We give the definition of ‘trivial representation’ and several examples of representations and show that the category of representations of an internal category G on objects of a fibered category represented by an internal category C is isomorphic to the category of internal functors from G to C and internal natural transformations between them (Theorem 3.17). In Section 4, we reformulate the notion of descent theory [6] in terms of representations of special groupoids, namely equivalence relations. We end this note by giving the definition of regular representations and restrictions of representations.

1. Recollections

For a category C and objects X, Y of C, we denote by C(X, Y ) the set of all morphisms from X to Y in C. Let p : F → E be a functor. For an object X of E, we denote by FX the subcategory of F consisting of objects M of F satisfying p(M) = X and morphisms ϕ satisfying p(ϕ) = idX. For a morphism f : X → Y in E and M ∈ Ob FX, N ∈ Ob FY , we denote by Ff (M, N) the set of morphisms ϕ ∈ F(M, N) satisfying p(ϕ) = f . Definition 1.1. (Grothendieck [7,Définition 5.1, p. 161]) Let α : M → N be a morphism in F and set X = p(M), Y = p(N), f = p(α). We call α a Cartesian morphism if, for any M ∈ Ob FX,themapFX(M ,M)→ Ff (M ,N)defined by ϕ → αϕ is bijective. The following assertion is immediate from the definition.

PROPOSITION 1.2. Let αi : Mi → Ni (i = 1, 2) be morphisms in F such that p(M1) = p(M2), p(N1) = p(N2), p(α1) = p(α2) and λ : N1 → N2 a morphism in F such = : → that p(λ) idp(N1).Ifα2 is Cartesian, there is a unique morphism µ M1 M2 such that = = p(µ) idp(M1) and α2µ λα1.

COROLLARY 1.3. If αi : Mi → N(i= 1, 2) are Cartesian morphisms in F such that p(M1) = p(M2) and p(α1) = p(α2), there is a unique morphism µ : M1 → M2 such = = that α1 α2µ and p(µ) idp(M1). Moreover, µ is an isomorphism. Definition 1.4. (Grothendieck [7,Définition 5.1, p. 162]) Let f : X → Y be a morphism in E and N ∈ Ob FY . If there exists a Cartesian morphism α : M → N such that p(α) = f , M is ∗ ∗ called an inverse image of N by f . We denote M by f (N) and α by αf (N) : f (N) → N. By (1.3), f ∗(N) is unique up to isomorphism.

Remark 1.5. For an identity morphism idX of X ∈ Ob E and N ∈ Ob FX, the identity morphism idN of N is obviously Cartesian. Hence, the inverse image of N by the : ∗ → identity morphism of X always exists and αidX (N) idX(N) N can be chosen as ∗ : the identity morphism of N. By the uniqueness of idX(N) up to isomorphism, αidX (N) ∗ → ∗ idX(N) N is an isomorphism for any choice of idX(N). The following assertion is also immediate.

PROPOSITION 1.6. Let f : X → Y be a morphism in E. If, for any N ∈ Ob FY , there exists ∗ ∗ ∗ a Cartesian morphism αf (N) : f (N) → N, N → f (N) deﬁnes a functor f : FY → FX Representations of internal categories 141

such that, for any morphism ϕ : N → N in FY , the following square commutes:

α (N) ∗ f / f (N) N

f ∗(ϕ) ϕ

α (N ) ∗ f / f (N ) N

Definition 1.7. (Grothendieck [7,Définition 5.1, p. 162]) If the assumption of (1.6) is satisfied, we say that the functor of the inverse image by f exists.

Definition 1.8. (Grothendieck [7,Définition 6.1, p. 164]) If a functor p : F → E satisfies the following condition (i), p is called a prefibered category and if p satisfies both (i) and (ii), p is called a fibered category or p is fibrant. (i) For any morphism f in E, the functor of the inverse image by f exists. (ii) The composition of Cartesian morphisms is Cartesian.

: F → E Definition 1.9.(Grothedieck [7,Définition 7.1, p. 170]) Let p be a functor. A map : E → F F κ Mor X,Y ∈Ob E Funct( Y , X) is called a cleavage if κ(f) gives an inverse image ∗ functor f : FY → FX for (f : X → Y)∈ Mor E. A cleavage κ is said to be normalized if = ∈ E F E κ(idX) idFX for any X Ob . A category over is called a cloven prefibered category (respectively normalized cloven prefibered category) if a cleavage (respectively normalized cleavage) is given.

The functor p : F → E has a cleavage if and only if p is preﬁbered. If p is preﬁbered, p has a normalized cleavage by Remark 1.5.

Let f : X → Y , g : Z → X be morphisms in E and N an object of FY .Ifp : F → E is ∗ ∗ ∗ a preﬁbered category, there is a unique morphism cf,g(N) : g f (N) → (fg) (N) such that the following square commutes and p(cf,g(N)) = idZ:

∗ αg(f (N)) g∗f ∗(N) /f ∗(N)

cf,g(N) αf (N)

α (N) ∗ fg / (fg) (N) N

Then, we see the following. 142 A. Yamaguchi

PROPOSITION 1.10. For a morphism ϕ : M → N in FY , the following square commutes:

cf,g(M) g∗f ∗(M) /(fg)∗(M)

g∗f ∗(ϕ) (fg)∗(ϕ) cf,g(N) g∗f ∗(N) /(fg)∗(N)

∗ ∗ ∗ In other words, cf,g gives a natural transformation g f → (fg) of functors from FY to FZ.

PROPOSITION 1.11. (Grothendieck [7, Proposition 7.2, p. 172]) Let p : F → E be a cloven preﬁbered category. Then, p is a ﬁbered category if and only if cf,g(N) is an isomorphism g f for any Z −→ X −→ Y and N ∈ Ob FX.

PROPOSITION 1.12. (Grothendieck [7, Proposition 7.4, p. 172]) Let p : F → E be a cloven f g h preﬁbered category. For a diagram X −→ Y −→ Z −→ W in E and an object M of FW ,we = ∗ ∗ = ∗ have ch,idZ (M) αidZ (idZh (M)), cidW ,h(M) h (αidW (M)) and the following diagram commutes:

∗ cg,f (h (M)) f ∗(g∗h∗)(M) (f ∗g∗)h∗(M) /(gf )∗h∗(M)

∗ f (ch,g(M)) ch,gf (M) chg,f (M) f ∗(hg)∗(M) /((hg)f )∗(M) (h(gf ))∗(M)

We give two examples below which are referred to in the later sections. Example 1.13. (Grothendieck [7, p. 182, a]) Let E be a category with pull-backs and 1 a 1 1 (2) 1 category such that Ob ={0, 1} and Mor ={id0, id1, 0 → 1}.SetE = Funct( , E) : E(2) → E ∈ E E(2) = E and let p be the evaluation functor at 1. Then, for X Ob , X /X and, for (f : X → Y)∈ Mor E, the pull-backs along f define the functor of the inverse image ∗ : E(2) → E(2) : → : → E : → f Y X . For morphisms f X Y , g Z X in and an object π E Y of (2) ∗ ∗ ∗ E , cf,g(π : E → Y): (fg) (π : E → Y)→ g f (π : E → Y)is the isomorphism induced × : × → × × : E(2) → E by (1 g, pr2) E Y Z (E Y X) X Z. Hence, p is a fibered category. Example 1.14. Let Sch be the category of schemes. We define a category Qmod as follows. Ob Qmod consists of pairs (X, M) of a scheme X and a quasi-coherent OX-module M. A morphism (X, M) → (Y, N ) in Qmod is a pair (f, ϕ) of morphisms f : X → Y in Sch and ϕ : N → f∗M in the category of OY -modules. The composition of morphisms (f, ϕ) : (X, M) → (Y, N ) and (g, ψ) : (Z, L) → (X, M) is defined to be (fg,f∗(ψ)ϕ) : (Z, L) → (Y, N ). Define a functor p : Qmod → Sch by p(X, M) = X and p(f, ϕ) = f . For a morphism f : X → Y of schemes and an OY -module N , we denote by ηf (N ) : N → ∗ ∗ ∗ f∗f N the unit of the adjunction of f and f∗. Then, (f, ηf (N )) : (X, f N ) → (Y, N ) a ∗ is a Cartesian morphism. In fact, for (f, ϕ) ∈ Qmodf ((X, M), (Y, N )), ϕ : f N → a ∗ M denotes the adjoint of ϕ : N → f∗M,then(idX,ϕ ) : (X, M) → (X, f N ) is the a unique morphism in QmodX such that (f, ϕ) = (f, ηf (N ))(idX,ϕ ). Thus, we define Representations of internal categories 143

∗ ∗ ∗ ∗ ∗ f : QmodY → QmodX by f (Y, N ) = (X, f N ) and f (idY ,ϕ)= (idX,f ϕ).More- ∗ over, αf (Y, N ) : f (Y, N ) → (Y, N ) is given by αf (Y, N ) = (f, ηf (N )). For morphisms ∗ f : X → Y , g : Z → X in Sch and an object (Y, N ) of Qmod,letc˜f,g(N ) : (fg) N → g∗f ∗N be the adjoint of composition

∗ ηf (N ) ∗ f∗(ηg(f N )) ∗ ∗ ∗ ∗ N −−−−→ f∗f N −−−−−−−→ f∗g∗g f N = (fg)∗g f N .

Then, c˜f,g(N ) is an isomorphism of OZ-modules and ∗ ∗ ∗ cf,g(Y, N ) = (idZ, c˜f,g(N )) : (fg) (Y, N ) → g f (Y, N ) is an isomorphism in QmodZ. Hence, p : Qmod → Sch is a fibered category. Definition 1.15. (Grothendieck [7, p. 152]) Let p : F → E and F : D → E be functors. Define a subcategory D ×E F of D × F as follows. An object (X, M) of D × F belongs to D ×E F if and only if F(X)= p(M). A morphism (f, ϕ) : (X, M) → (Y, N) in D × F belongs to D ×E F if and only if F(f)= p(ϕ). Define functors pF : D ×E F → D and F : D ×E F → F by pF (X, M) = X, pF (f, ϕ) = f and F(X,M)= M, F(f,ϕ)= ϕ. We call pF : D ×E F → D the pull-back of p along F .

PROPOSITION 1.16. (Grothendieck [7, Proposition 6.6, p. 167]) A morphism (f, α):(X, M) → (Y, N) in D ×E F is Cartesian if and only if α : M → N is Cartesian.

PROPOSITION 1.17. (Grothendieck [7, Corollaire 6.9, p. 168]) If p : F → E is a prefibered (respectively fibered) category, so is the pull-back pF : D ×E F → D of p along F : D → E. Moreover, F : D ×E F → F maps Cartesian morphisms to Cartesian morphisms. We need to introduce the notion of a ‘Cartesian section’ in order to define the notion of trivial representation. Definition 1.18. (Grothendieck [7,Définition 5.5, p. 164]) Let p : F → E be a functor. We call a functor s : E → F a Cartesian section if ps = idE and s(f) is Cartesian for any f ∈ Mor E. The subcategory of Funct(E, F) consisting of Cartesian sections and morphisms ϕ : s → s satisfying p(ϕ ) = id for any X ∈ Ob E is denoted by Lim(F/E). X X ←−

PROPOSITION 1.19. (Giraud [5, Lemme 5.7]) If E has a terminal object 1, the functor e : Lim(F/E) → F given by e(s) = s(1) and e(ϕ) = ϕ is fully faithful. Moreover, if p : F → E ←− 1 1 is a fibered category, e is an equivalence of categories. Next, we briefly review the notion of a bifibered category following [7, Section 10]. Definition 1.20. Let p : F → E be a functor and α : M → N a morphism in F.SetX = p(M), Y = p(N), f = p(α). We call α a co-Cartesian morphism if, for any N ∈ Ob FY ,the map FX(N, N ) → Ff (M, N ) defined by ϕ → ϕα is bijective. The following assertion is the dual of Proposition 1.2.

PROPOSITION 1.21. If αi : M → Ni (i = 1, 2) are co-Cartesian morphisms in F such that p(N1) = p(N2) and p(α1) = p(α2), there is a unique morphism ψ : N1 → N2 such = = that α1 α2ψ and p(ψ) idp(N1). Moreover, ψ is an isomorphism. 144 A. Yamaguchi

Deﬁnition 1.22. Let f : X → Y be a morphism in E and M ∈ Ob FX. If there exists a Cartesian morphism α : M → N such that p(α) = f , N is called a direct image of M by f . We denote M by f∗(N) and α by αf (M) : M → f∗(M). By Proposition 1.21, f∗(N) is unique up to isomorphism. PROPOSITION 1.23. Let α : M → N, α : M → N be morphisms in F such that p(M) = p(M), p(N) = p(N), p(α) = p(α)(=f) and λ : M → M a morphism in F such that p(λ) = idp(M).Ifα is co-Cartesian, there is a unique morphism µ : N → N such that p(µ) = idp(N) and α µ = λα.

COROLLARY 1.24. Let f : X → Y be a morphism in E. If, for any M ∈ Ob FX, there exists f a co-Cartesian morphism α (M) : M → f∗(M), M → f∗(M) defines a functor f∗ : FX → FY . Definition 1.25. If the assumption of Corollary 1.24 is satisfied, we say that the functor of the direct image by f exists. Definition 1.26. If a functor p : F → E satisfies the following condition (i), p is called a precofibered category and if p satisfies both (i) and (ii), p is called a cofibered category or p is cofibrant. (i) For any morphism f in E, the functor of the direct image by f exists. (ii) The composition of co-Cartesian morphisms is co-Cartesian. In other words, p : F → E is a precofibered (respectively cofibered) category if and only : F op → Eop if p is a prefibered (respectively fibered) category. : F → E : E → F F Let p be a functor. A map κ Mor X,Y ∈Ob E Funct( X, Y ) is called acocleavageifκ(f) is a direct image functor f∗ : FX → FY for (f : X → Y)∈ Mor E. = ∈ E F Acocleavageκ is said to be normalized if κ(idX) idFX for any X Ob . A category over E is called a cloven precofibered category (respectively normalized cloven precofibered category) if a cocleavage (respectively normalized cocleavage) is given. The functor p : F → E has a cocleavage if and only if p is precofibered. If p is precofibered, p has a normalized cocleavage. Let f : X → Y , g : Z → X be morphisms in E and M an object of FZ.Ifp : F → E is a precofibered category, there is a unique morphism cf,g(M) : (fg)∗(M) → f∗g∗(M) such that the following square commutes and p(cf,g(M)) = idZ:

fg α (M) / M (fg)∗(M)

αg(M) cf,g(M)

f α (g∗(M))/ g∗(M) f∗g∗(M)

The following is the dual of Deﬁnition 1.9.

PROPOSITION 1.27. Let p : F → E be a cloven precoﬁbered category. Then, p is a coﬁbered f,g g f category if and only if c (M) is an isomorphism for any Z → X → Y and M ∈ Ob FZ.

PROPOSITION 1.28. Let p : F → E be a functor and f : X → Y a morphism in E. Representations of internal categories 145

(1) Suppose that the functor of the inverse image by f exists. Then, the inverse image ∗ f : FY → FX by f has a left adjoint if and only if the functor of the direct image by f exists. (2) Suppose that the functor of the direct image by f exists. Then, the direct image f∗ : FX → FY by f has a right adjoint if and only if the functor of the inverse image by f exists. Proof. (1) Suppose that the functor of the inverse image by f exists and that it has a ∗ left adjoint f∗ : FX → FY .ForM ∈ Ob FX, we denote by ηM : M → f f∗(M) the unit of ∗ f the adjunction f∗ f .Setα (M) = αf (f∗(M))ηM : M → f∗(M). By the assumption, the following composition is bijective for any M ∈ Ob FX, N ∈ Ob FY :

∗ ∗ f ∗ ∗ ηM ∗ αf (N)∗ FY (f∗(M), N) −→ FX(f f∗(M), f (N)) −→ FX(M, f (N)) −−−−→ Ff (M, N). ∗ We note that, since αf (N)f (ϕ) = ϕαf (f∗(M)) for ϕ ∈ FY (f∗(M), N),theabove f ∗ composition coincides with the map α (M) : FY (f∗(M), N) → Ff (M, N) induced by αf (M). This shows that the functor of the direct image by f exists. Conversely, assume that the functor of the direct image by f exists. For M ∈ Ob FX, let us denote by αf (M) : M → f∗(M) the canonical co-Cartesian morphism. Then, we f ∗ ∗ have bijections α (M) : FY (f∗(M), N) → Ff (M, N) and αf (M)∗ : FX(M, f (N)) → f Ff (M, N) given by ψ → ψα (M) and ϕ → αf (M)ϕ, which are natural in M ∈ Ob FX ∗ and N ∈ Ob FY . Thus, we have a natural bijection FY (f∗(M), N) → FX(M, f (N)). Proof of (2) is the dual of the above. 2 We remark that, if p : E → F is a preﬁbered and precoﬁbered category, the unit η : → ∗ ∗ idFX f f∗ of the adjunction f∗ f is the unique natural transformation satisfying = f ∈ F : ∗ → αf (f∗(M))ηM α (M) for any M Ob X. Dually, the counit ε f∗f idFY is the f ∗ unique natural transformation satisfying εN α (f (N)) = αf (N) for any N ∈ Ob FY .

PROPOSITION 1.29. (Grothendieck [7, Proposition 10.1, p. 182]) Let p : E → F be a prefibered and precofibered category. Then, it is a fibered category if and only if it is a cofibered category. Definition 1.30. We call a functor p : F → E a bifibered category if it is a fibered and cofibered category. Example 1.31. Let p : E(2) → E be the fibered category given in Example 1.13. For a : → E : E(2) → E(2) : → = morphism f X Y in , consider the functor f∗ X Y given by f∗(π E X) ∗ (f π : E → Y). It is easily seen that f∗ is a left adjoint of f . Example 1.32. Let p : Qmod → Sch be the fibered category given in Example 1.14. If f : X → Y is a quasi-separated and quasi-compact morphism, then for any object (X, M) of Qmod, the direct image f∗M of M by f is also quasi-coherent. Hence, the inverse image ∗ functor f : QmodY → QmodX has a left adjoint f∗ : QmodX → QmodY . We end this section by recalling the definition of internal category and related notions. Definition 1.33. (Johnstone [11]) Let E be a category with finite limits. An internal category C in E consists of the following objects and morphisms. 146 A. Yamaguchi

(1) A pair of objects C0 (the object-of-objects) and C1 (the object-of-morphisms). (2) Four morphisms σ : C1 → C0 (domain), τ : C1 → C0 (codomain), ε : C0 → C1 : × → ←pr−1 × −pr→2 (identity), µ C1 C0 C1 C1 (composition), where C1 C1 C0 C1 C1 is a −→τ ←−σ = = limit of diagram C1 C0 C1, such that σε τε idC0 and the following diagrams commute:

pr pr µ×idC o 1 × 2 / × × 1/ × C1 C1 C0 C1 C1 C1 C0 C1 C0 C1 C1 C0 C1

× σ µ τ idC1 µ µ µ o σ τ / × / C0 C1 C0 C1 C0 C1 C1

idC ×ε ε×idC × 1 / × o 1 × C1 C0 C0 C1 C0 C1 C0 C0 C1

pr1 µ pr2 C1 C1 C1

We denote by (C0,C1; σ, τ, ε, µ) an internal category C whose object-of-objects and object- of-morphisms are C0 and C1, respectively, with structure maps σ, τ, ε, µ. An internal functor f : C → D of internal categories consists of two morphisms f0 : C0 → D0 and f1 : C1 → D1 in E such that the following diagrams commute:

µ o σ τ / × / o ε C0 C1 C0 C1 C0 C1 C1 C0

f0 f1 f0 f1×f1 f1 f0 σ τ µ o / × / o ε D0 D1 D0 D1 D0 D1 D1 D0

The above internal functor f is denoted by (f0,f1). If both f0 and f1 are monomorphisms, D is regarded as an internal subcategory of C. An internal natural transformation ϕ : f → g of internal functors f, g : C → D is a morphism ϕ : C0 → D1 in E making the following diagrams commute:

f g (f ,ϕτ) o 0 0 / 1 / × D0 C0 D0 C1 D1 D0 D1

ϕ (ϕσ,g1) µ σ τ µ o / × / D0 D1 D0 D1 D0 D1 D1

We denote by cat(E) the category of internal categories in E.

If f = (f0,f1) : C → D and g = (g0,g1) : D → E are internal functors, the composi- tionisdeﬁnedtobegf = (g0f0,g1f1) : C → E. Thus, we have the category cat(E) of internal categories and internal functors in E. Remark 1.34. (1) Let f, g, h : C → D be internal functors and ϕ : f → g, ψ : g → h internal natural transformations. The composition ψ·ϕ : f → h is the morphism C0 → D1 Representations of internal categories 147

E = = : → × in given as follows. Since τϕ g0 σψ, there is a morphism (ϕ, ψ) C0 D1 D0 D1 = = · = : → satisfying pr1(ϕ, ψ) ϕ and pr2(ϕ, ψ) ψ.Wesetψ ϕ µ(ϕ, ψ) C0 D1. Then, it is easy to verify that ψ·ϕ is an internal natural transformation from f to h. (2) For an internal functor f : C → D,wesetidf = εf0 : C0 → D1. Then, idf is an internal natural transformation from f to f and it can be veriﬁed that, for internal natural transformations ϕ : f → g and ψ : g → f ,idf ·ψ = ψ and ϕ·idf = ϕ hold. (3) Let f, f : C → D, g, g : D → E be internal functors and ϕ : f → f , ψ : g → = g internal natural transformations. Then, τg1ϕ σψf0 and there exists a morphism : → × = = (g1ϕ, ψf0) C0 E1 E0 E1 satisfying pr1(g1ϕ, ψf0) g1ϕ and pr2(g1ϕ, ψf0) ψf0. ∗ = ∗ We put ϕ ψ µ(g1ϕ, ψf0). It is routine to check that ϕ ψ is an internal natural transformation from gf to gf .

Deﬁnition 1.35. Let C = (C0,C1; σ, τ, ε, µ) be an internal category in E. We call a pair of : → : × → E C morphisms (π X C0,α X C0 C1 X) of an internal diagram on if it makes the × : → following diagrams commute, where X C0 C1 is the ﬁbered product of π X C0 and σ : C1 → C0: × α α idC1 α X × C / X × C × C /X × C X × C / C0 1 X C0 1 C0 1 C0 1 CO0 1 X

pr2 π idX×µ α idX×ε pr τ / × α / × 1 / C1 C0 X C0 C1 X X C0 C0 X

Let (π : X → C0,α)and (ρ : Y → C0,β)be internal diagrams on C. A morphism f : X → E = × = Y in is called a morphism of internal diagrams if it satisﬁes ρf π and β(f idC0 ) fα. We denote by EC the category of internal diagrams on C.

2. 2-categories and lax functors

First we give the deﬁnitions of a 2-category and a lax functor (see [2, 12, 13]). Deﬁnition 2.1. A 2-category C is determined by the following data. (1) A set Ob C called a set of objects. (2) For each pair of objects (X, Y ), a category C(X, Y ). An object of C(X, Y ) is called a 1-arrow and a morphism of C(X, Y ) is called a 2-arrow. The composition of 2-arrows f g S → T , T → U in C(X, Y ) is denoted by gf . (3) For each triple (X, Y, Z) of objects of C, a functor µX,Y,Z : C(X, Y ) × C(Y, Z) → C(X, Z) called a composition functor (here C(X, Y ) × C(Y, Z) is the product category of C(X, Y ) and C(Y, Z)) such that the following diagram commutes:

µX,Y,Z ×1 C(X, Y ) × C(Y, Z) × C(Z, W) /C(X, Z) × C(Z, W)

1×µY,Z,W µX,Z,W µX,Y,W C(X, Y ) × C(Y, W) /C(X, W)

We usually denote µX,Y,Z(S, T ) by T ◦S if (S, T ) ∈ Ob C(X, Y ) × C(Y, Z) and denote µX,Y,Z(f, g) by g∗f if (f, g) ∈ Mor C(X, Y ) × C(Y, Z). 148 A. Yamaguchi

(4) For each object X of C, a 1-arrow 1X in C(X, X) called an identity 1-arrow of X satisfying S◦1X = S and 1X◦T = T for any S ∈ Ob C(X, Y ) and T ∈ Ob C(Y, X). Example 2.2. (1) Let C be a category. For any pair of objects (X, Y ) of C,weregardthe set C(X, Y ) of morphisms from X to Y as a discrete category, that is, C(X, Y ) is a category whose set of morphisms consists of only identity morphisms. The composition of morphisms µX,Y,Z : C(X, Y ) × C(Y, Z) → C(X, Z) gives a composition functor; that is, idf ∗idg = idgf . Hence, C can be regarded as a 2-category. (2) Let us denote by cat the category of categories. For a pair (C, D) of categories, cat(C, D) is the functor category Funct(C, D). Define the composition functor µC,D,E : cat(C, D) × cat(D, E) → cat(C, E) by µC,D,E (F, G) = G◦F (the composition of functors), µC,D,E (f, g)X = gF (X)G(fX) = G (fX)gF(X) for an object X ∈ Ob C and natural transformations f : F → F , g : G → G. For a category C, the identity functor of C is the identity 1-arrow of C. Hence, cat has a structure of a 2-category. (3) Let us denote by pf i b (E) the 2-category of cloven prefibered category over a category E defined as follows. Objects of pf i b (E) are cloven prefibered categories over E. For cloven prefibered categories p : F → E and q : D → E, a 1-arrow of pf i b (E)(p, q) is a functor F : F → D satisfying qF = p. A 2-arrow ϕ : F → G in pf i b (E)(p, q) is a natural transformation of functors such that, for each M ∈ Ob F, q(ϕM) = idp(M). Let r : C → E be another cloven prefibered category. The composition functor µp,q,r : pf i b (E)(p, q) × pf i b (E)(q, r) → pf i b (E)(p, r) is defined in similar way as (2) above. That is, the composition of 1-arrows is just the composition of functors and composition of 2- arrows is given by µp,q,r(f, g)M = gF (M)G(fM ) = G (fM )gF(M) for an object M ∈ Ob F and natural transformations f : F → F , g : G → G of functors F, F : F → D, G, G : D → C. (4) We define 2-categories fib(E), pf i b c(E), fibc(E) as follows. Here fib(E) is a full subcategory of pf i b (E) consisting of fibered categories, pf i b c(E) is a subcategory of pf i b (E) having the same objects as those of pf i b (E) and morphisms which map Cartesian morphisms to Cartesian morphisms and fibc(E) is a full subcategory of pf i b c(E) consisting of fibered categories. (5) Let E be a category with finite limits. Using the compositions of internal natural transformations given in Remark 1.34, the category of internal category cat(E) has a structure of 2-category whose 2-arrows are internal natural transformations. Definition 2.3. Let D and C be 2-categories. A lax functor (, γ ) : C → D consists of the following data. (1) A map : Ob C → Ob D. (2) For each pair (X, Y ) of objects of C, a functor X,Y : C(X, Y ) → D((X), (Y )). (3) For each object X of C, a 2-arrow γX : 1(X) → X,X(1X). (4) For each triple (X, Y, Z) of objects of C, there is a natural transformation

γX,Y,Z : µ(X),(Y ),(Z)(X,Y × Y,Z) → X,ZµX,Y,Z

(namely, there is a 2-arrow (γX,Y,Z)(S,T ) : Y,Z(T )◦X,Y (S) → X,Z(T ◦S) in D for composable 1-arrows S : X → Y , T : Y → Z) making the following diagrams in the categories D((X), (Y )) and D((X), (W)), respectively, commute for 1-arrows Representations of internal categories 149

S : X → Y , T : Y → Z and U : Z → W in C:

X,Y (S)◦1(X) X,Y (S) 1(Y)◦X,Y (S)

∗ ∗ 1(S)X,Y γX 1X,Y (S) γY 1X,Y (S) (γ ) (γ ) X,X,Y (1X,S) / o X,Y,Y (S,1Y ) X,Y (S)◦X,X(1X) X,Y (S) Y,Y (1Y )◦X,Y (S)

1 ∗(γ ) Z,W (U) X,Y,Z (S,T ) / Z,W (U)◦Y,Z(T )◦X,Y (S) Z,W(U)◦X,Z(T ◦S)

∗ (γY,Z,W )(T ,U) 1 (S) (γX,Z,W )(T ◦S,U) X,Y (γX,Y,W )(S,U◦T) / Y,W (U◦T)◦X,Y (S) X,W (U◦T ◦S)

A lax functor (, γ ) : C → D is a genuine functor if, for any X, Y, Z ∈ Ob C, γX is the identity 2-arrow in D and γX,Y,Z is the identity natural transformation. Deﬁnition 2.4. Let (, γ ) : C → D and (, δ) : D → E be lax functors. We deﬁne a lax functor (, π) : C → E as follows. Put (X) = ((X)) and X,Y = (X),(Y)X,Y : C(X, Y ) → D(((X)), ((Y ))) for X, Y ∈ Ob C. πX : 1(X) → X,X(1X) is a composition

δ(X) 1(X) = 1((X)) −−−→ (X),(X)(1(X))

(X),(X)(γX) −−−−−−−−−→ (X),(X)X,X(1X) = X,X(1X). For 1-arrows S : X → Y , T : Y → Z in C,

(πX,Y,Z)(S,T ) : µ(X),(Y ),(Z)(X,Y × Y,Z)(S, T ) → X,ZµX,Y,Z(S, T ) is deﬁned to be the composition

µ((X)),((Y )),((Z))((X),(Y)X,Y × (Y),(Z)Y,Z)(S, T )

(δ(X),(Y ),(Z))(X,Y (S),Y,Z(T )) −−−−−−−−−−−−−−−−−−−→ (X),(Z)µ(X),(Y ),(Z)(X,Y × Y,Z)(S, T )

(X),(Z)((γX,Y,Z)(S,T )) −−−−−−−−−−−−−−→ (X),(Z)X,ZµX,Y,Z(S, T ). We call (, π) the composition of (, γ ) and (, δ). We denote by IC = (I, ι) : C → C the identity lax functor, that is, I is the identity map of Ob C, IX,Y : C(X, Y ) → C(I (X), I (Y )) is the identity functor, ιX : 1I(X) → IX,X(1X) is the identity 2-arrow and ιX,Y,Z : µI (X),I (Y ),I (Z)(IX,Y × IY,Z) → IX,ZµX,Y,Z is the identity natural transformation. Deﬁnition 2.5. (1) A lax functor (, γ ) : C → D is called a 2-functor if the 2- arrow γX : 1(X) → X,X(1X) is an isomorphism for every X ∈ Ob C and γX,Y,Z : µ(X),(Y ),(Z)(X,Y × Y,Z) → X,ZµX,Y,Z is a natural equivalence for every X, Y, Z ∈ Ob C. (2) If C is a category regarded as a 2-category as in (1) of Example 2.2, we call a lax functor (, γ ) : C → D a lax diagram. (3) A lax diagram which is also a 2-functor is called a pseudo-functor. 150 A. Yamaguchi

∗ Example 2.6. For a functor F : D → E, we define a 2-functor pf i b (F ) = (F ,γF ) : ∗ pf i b (E) → pf i b (D) as follows. For an object p : F → E of pf i b (E),letF (p) = pF : D ×E F → D be the pull-back of p along F .Ifκ is a cleavage of p, the cleavage κF of pF is given by (κF (f ))(Y, N) = (X, κ(F (f ))(N)) and (κF (f ))(idY ,ϕ)= (idX, κ(F (f ))(ϕ)) for a morphism f : X → Y in D and N ∈ Ob FF(Y), ϕ ∈ Mor FF(Y). For a 1-arrow ϕ : p → q : F → E : C → E pf i b E ∗ : → from an object p to an object q of ( ),letFp,q(ϕ) pF qF be the 1-arrow in pf i b (D) induced by idD × ϕ : D × F → D × C. It follows from Proposition 1.16 pf i b c E ∗ pf i b c D : → that if ϕ is a 1-arrow in ( ), Fp,q(ϕ) is a 1-arrow in ( ).Letϕ, ψ p q pf i b E : → ∗ : ∗ → be 1-arrows in ( ) and χ ϕ ψ a 2-arrow. Define a 2-arrow Fp,q(χ) Fp,q(ϕ) ∗ ∗ = : ∗ → ∗ ∈ Fp,q(ψ) by Fp,q(χ)(X,M) (idX,χM ) Fp,q(ϕ)(X, M) Fp,q(ψ)(X, M) for (X, M) D × F ∗ : pf i b E → pf i b D Ob( E ). Thus, we have a functor Fp,q ( )(p, q) ( )(F (p), F (q)).For ∗ ∗ ∈ pf i b E ∗ = = : → : ∗ → p Ob ( ),since1F (p) idpF Fp,p(1p) pF pF ,let(γF )p 1F (p) Fp,p(1p) be the identity 2-arrow. For 1-arrows ϕ : p → q, ψ : q → r in pf i b (E), composition of ∗ : → ∗ : → ∗ 1-arrows Fp,q(ϕ) pF qF and Fq,r(ψ) qF rF coincides with Fp,r(ψϕ).Wedefinea : ∗ ∗ → ∗ pf i b D 2-arrow ((γF )p,q,r)(ϕ,ψ) Fq,r(ψ)Fp,q(ϕ) Fp,r(ψϕ) in ( ) as the identity 2-arrow. In fact, pf i b (F ) is a functor.

Deﬁnition 2.7. For 2-categories C and D, we deﬁne the 2-category Lax(C, D) of lax functors as follows. Objects of Lax(C, D) are lax functors from C to D.Let(, γ ), (, δ) : C → D be lax functors. A 1-arrow (, λ) : (, γ ) → (, δ) consists of the following data. (1) For each X ∈ Ob C, a 1-arrow X : (X) → (X) in D. (2) For each 1-arrow S : X → Y in C, a 2-arrow λS : Y ◦X,Y (S) → X,Y (S)◦X in D, making the following diagrams commute for every X ∈ Ob C and 1-arrows S : X → Y , T : Y → Z in C: id X / X◦1(X) 1(X)◦X

∗ ∗ idX γX δX idX λ 1X / X◦X,X(1X) X,X(1X)◦X

id ∗(γ ) Z X,Y,Z (S,T ) / Z◦Y,Z(T )◦X,Y (S) Z◦X,Z(T ◦S)

∗ λT id (S) λT ◦S X,Y Y,Z(T )◦Y ◦X,Y (S) X,Z(T ◦S)◦X

∗ idY,Z(T ) λS (δ ) ∗id X,Y,Z (S,T ) X / Y,Z(T )◦X,Y (S)◦X X,Z(T ◦S)◦X

For 1-arrows (, λ) : (, γ ) → (, δ) and (, ϕ) : (, δ) → (E, ),wedeﬁnea 1-arrow (, ψ) : (, γ ) → (E, ) by

X = X◦X : (X) → E(X), = ∗ ∗ : ◦ ◦ → ◦ ◦ ψS (ϕS idX )(idY λS) Y Y X,Y (S) EX,Y (S) X X Representations of internal categories 151 for X ∈ Ob C and a 1-arrow S : X → Y in C. The composition (, ϕ)◦(, λ) is deﬁned to be (, ψ). The identity 1-arrow 1(,γ ) of (, γ ) ∈ Ob Lax(C, D) is a pair (I, ι) such that IX is the identity 1-arrow of (X) for X ∈ Ob C and ιS is the identity 2-arrow of X,Y (S) for a 1-arrow S : X → Y in C. Let (, λ), (, ϕ) : (, γ ) → (, δ) be 1-arrows in Lax(C, D). A 2-arrow χ : (, λ) → (, ϕ) consists of 2-arrows χX : X → X in D for X ∈ Ob C such that, for every 1-arrow S : X → Y in C, the following diagram commutes:

λS / Y ◦X,Y (S) X,Y (S)◦X

∗ ∗ χY idX,Y (S) idX,Y (S) χX ϕS / Y ◦X,Y (S) X,Y (S)◦X

For 2-arrows χ : (, λ) → (, ϕ) and ω : (, ϕ) → (, ψ) ((, λ), (, ϕ), (, ψ) : (, γ ) → (, δ)), the composition ωχ : (, λ) → (, ψ) in Lax(C, D)((, γ ), (, δ)) is defined by (ωχ)X = ωXχX.Ifχ : (, λ) → (, ϕ) and ω : (, ψ) → (ϒ, υ)((, λ), (, ϕ) : (, γ ) → (, δ), (, ψ), (ϒ, υ) : (, δ) → (E, )) are 2-arrows, the composition ω∗χ : (, ψ)◦(, λ) → (ϒ, υ)◦(, ϕ) is defined by (ω∗χ)X = ωX∗χX : X◦X → ϒX◦X. The identity 2-arrow ι(,λ) of a 1-arrow (, λ) in Lax(C, D) is given by (ι(,λ))X = (the identity 2-arrow of X) for any X ∈ Ob C. Definition 2.8. For later use, we denote by Laxs(C, D) the full subcategory of Lax(C, D) consisting of lax functors (, γ ) : C → D such that γX : 1(X) → X,X(1X) is an isomorphism for each X ∈ Ob C. Moreover, 2-Funct(C, D) denotes the full subcategory of Lax(C, D) consisting of 2-functors. We also consider a subcategory Laxc(C, D) of Laxs(C, D) and a subcategory 2-Functc(C, D) of 2-Funct(C, D) given as follows. Here Laxc(C, D) (respectively 2-Functc(C, D)) has the same objects as Laxs (C, D) (respectively 2-Funct(C, D)). A 1-arrow (, λ) in Laxs(C, D) (respectively 2-Funct(C, D)) belongs to c c Lax (C, D) (respectively 2-Funct (C, D)) if and only if a 2-arrow λS in D is an isomorphism for every 1-arrow S in C. A 2-arrow in Laxs (C, D) (respectively 2-Funct(C, D)) belongs to Laxc(C, D) (respectively 2-Functc(C, D)) if and only if its domain and codomain are 1-arrows in Laxc(C, D) (respectively 2-Functc(C, D)). Note that 2-Funct(C, D) is also a full subcategory of Laxs(C, D). Example 2.9. Let (A, α) : C → C be a lax functor. Define a lax functor (A, α)∗ : ∗ ∗ Lax(C, D) → Lax(C , D) as follows. Put (A, α) = (A ,γA). For a lax functor (, γ ) : C → D,wesetA∗((, γ )) = (, γ )◦(A, α). For a 1-arrow (, λ) : (, γ ) → (, δ) in Lax(C, D), let us define a 1-arrow (A,λA) : (, γ )◦(A, α) → (, δ)◦(A, α) in Lax(C , D) by (A)X = A(X) for X ∈ ObC and (λA)S = λA(S) : A(Y )◦A(X),A(Y )(AX,Y (S)) → ◦ : → C ∗ = A(X),A(Y )(AX,Y (S)) A(X) for a 1-arrow S X Y in .WesetA(,γ ),(,δ)((, λ)) (A,λA). For a 2-arrow χ : (, λ) → (, ϕ) in Lax(C, D),letχA : (A,λA) → (A,ϕA) be the 2-arrow given by (χA)X = χA(X) : (A)X = A(X) → A(X) = (A)X for X ∈ ObC . ∗ = We set A(,γ ),(,δ)(χ) χA. 152 A. Yamaguchi

: C → D ∗ For a lax functor (, γ ) ,sinceA(,γ ),(,γ )(1(,γ )) is the identity 1-arrow ∗ ∗ : ∗ → of 1A ((,γ )),let(γA)(,γ ) 1A ((,γ )) A(,γ ),(,γ )(1(,γ )) be the identity 2-arrow of 1A∗((,γ )). For lax functors (, γ ), (, δ), (E, ) : C → D,let(γA)(,γ ),(,δ),(E,) be the identity natural transformation ∗ ∗ ∗ ∗ ∗ ∗ × → µA ((,γ )),A ((,δ)),A ((E,))(A(,γ ),(,δ) A(,δ),(E,)) A(,γ ),(E,)µ(,γ ),(,δ),(E,). In fact, for composable 1-arrows (, λ) : (, γ ) → (, δ) and (, ϕ) : (, δ) → (E, ) in Lax(C, D), it is easy to verify ∗ ◦ ∗ = ∗ ◦ A(,δ),(E,)((, ϕ)) A(,γ ),(,δ)((, λ)) A(,γ ),(E,)((, ϕ) (, λ)). Let

: ∗ ◦ ∗ ((γA)(,γ ),(,δ),(E,))((,λ),(,ϕ)) A(,δ),(E,)((, ϕ)) A(,γ ),(,δ)((, λ)) → ∗ ◦ A(,γ ),(E,)((, ϕ) (, λ)) be the identity 2-arrow in Lax(C, D). It is routine to verify that (A, α)∗ : Lax(C, D) → Lax(C, D) is a lax functor. Remark 2.10. Note that the lax functor (A, α)∗ : Lax(C, D) → Lax(C, D) defined above maps Laxs(C, D) to Laxs(C, D), 2-Funct(C, D) to 2-Funct(C, D) and Laxc(C, D) to Laxc(C, D). Therefore, we have the following lax functors: ∗ ∗ (A, α) : Laxs(C, D) → Laxs (C , D), (A, α) : Laxc(C, D) → Laxc(C , D), ∗ ∗ (A, α) : 2-Funct(C, D) → 2-Funct(C , D), (A, α) : 2-Functc(C, D) → 2-Functc(C , D). Example 2.11. Let (B, β): D → D be a 2-functor. Define a lax functor (B, β)∗ : Lax(C, D)→Lax(C, D ) as follows. Put (B, β)∗ = (B∗,γB). For a lax functor (, γ ) : C → D,wesetB∗((, γ )) = (B, β)◦(, γ ). For a 1-arrow (, λ) : (, γ ) → (, δ) in Lax(C, D), let us define a 1-arrow (B,λ(B,β)) : (B, β)◦(, γ ) → (B, β)◦(, δ) in Lax(C, D ) by (B)X = B(X),(X)(X) for X ∈ ObC and (λ(B,β))S isgivenbythe following composition for a 1-arrow S : X → Y in C:

B(Y),(Y)(Y )◦B(X),(Y)(X,Y (S))

(β(X),(Y ),(Y ))(Y ,X,Y (S)) −−−−−−−−−−−−−−−−→ B(X),(Y)(Y ◦X,Y (S))

B(X),(Y)(λS) −−−−−−−−→ B(X),(Y)(X,Y (S)◦X)

−1 (β(X),(X),(Y )) (X,Y (S),X) −−−−−−−−−−−−−−−−−→ B(X),(Y )(X,Y (S))◦B(X),(X)(X).

We set (B∗)(,γ ),(,δ)((, λ)) = (B ,λ(B,β)). For a 2-arrow χ : (, λ) → (, ϕ) in Lax(C, D),letχB : (B,λ(B,β)) → (B ,ϕ(B,β)) be the 2-arrow given by (χB)X = B(X),(X)(χX) : (B )X = B(X),(X)(X) → B(X),(X)(X) = (B)X for X ∈ ObC. We set (B∗)(,γ ),(,δ)(χ) = χB.

For a lax functor (, γ ) : C → D,let(γB)(,γ ) : 1B∗(,γ ) → (B∗)(,γ ),(,γ )(1(,γ )) be the 2-arrow in Lax(C, D ) given by ((γB )(,γ ))X = β(X) for each X ∈ Ob C. For lax functors Representations of internal categories 153

(, γ ), (, δ), (E, ) : C → D,let(γB)(,γ ),(,δ),(E,) be the natural transformation

µB∗((,γ )),B∗((,δ)),B∗((E,))((B∗)(,γ ),(,δ) × (B∗)(,δ),(E,))

→ (B∗)(,γ ),(E,)µ(,γ ),(,δ),(E,) deﬁned as follows. For composable 1-arrows (, λ) : (, γ ) → (, δ) and (, ϕ) : (, δ) → (E, ) in Lax(C, D),let

((γB)(,γ ),(,δ),(E,))((,λ),(,ϕ)) : (B∗)(,δ),(E,)((, ϕ))◦(B∗)(,γ ),(,δ)((, λ))

→ (B∗)(,γ ),(E,)((, ϕ)◦(, λ)) be the 2-arrow in Lax(C, D) given by = (((γB)(,γ ),(,δ),(E,))((,λ),(,ϕ)))X (β(X),(X),E(X))(X,X) for X ∈ Ob C. Again, it is routine to verify that (B, β)∗ : Lax(C, D) → Lax(C, D ) is a lax functor. Remark 2.12. The lax functor (B, β)∗ : Lax(C, D) → Lax(C, D ) deﬁned above maps Laxs(C, D) to Laxs (C, D), 2-Funct(C, D) to 2-Funct(C, D) and Laxc(C, D) to Laxc(C, D). Therefore, we have the following lax functors: s s c c (B, β)∗ : Lax (C, D) → Lax (C, D ), (B, β)∗ : Lax (C, D) → Lax (C, D ), c c (B, β)∗ : 2-Funct(C, D) → 2-Funct(C, D ), (B, β)∗ : 2-Funct (C, D) → 2-Funct (C, D ). Let p : F → E be a cloven preﬁbered category with cleavage κ. We associate a lax diagram ((p), γ (p)) : Eop → cat as follows.

Construction 2.13. We set (p)(X) = FX for X ∈ Ob E. For a morphism f : X → Y in E, ∗ we define (p)Y,X(f ) : FY → FX to be the inverse image functor f = κ(f) : FY → FX. ∈ E : ∗ → For X Ob ,sinceαidX (N) idX(N) N is an isomorphism by Proposition 1.5, a natural : → ∗ = −1 : → ∗ ∈ transformation γ(p)X 1FX idX is given by (γ (p)X)N αidX (N) N idX(N) (N ∗ ∗ Ob FX). For each pair (g, f ) ∈ E(Z, X) × E(X, Y ), a 2-arrow (γ (p)Y,X,Z)(f,g) : g f → ∗ (fg) is defined to be cf,g. It follows from Proposition 1.12 that ((p), γ (p)) is a lax diagram. We call ((p), γ (p)) the lax diagram associated with p : F → E. Moreover, ((p), γ (p)) is a pseudo-functor if and only if p : F → E is a fibered category. Let p : F → E, q : D → E be objects of pf i b (E) and F : p → q a morphism in pf i b (E). We define a morphism ((F ), λ(F )) : ((p), γ (p)) → ((q), γ (q)) as follows. For X ∈ Ob E, (F )X : FX → DX is the restriction FX of F . For a morphism f : ∗ ∗ X → Y in E and an object N of FY ,let(λ(F )f )N : FXf (N) → f (FY (N)) be the unique morphism in DX such that αf (FY (N))(λ(F )f )N = F(αf (N)). Then, (λ(F )f )N is natural in N and we have a natural transformation λ(F )f : (F )X◦(p)Y,X(f ) → (q)Y,X(f )◦(F )Y . We note that, if F is the identity morphism of p, (F )X is the identity functor of FX for every X ∈ Ob E and λ(F )f = idf ∗ for every morphism f in E. Also note that λ(F )f is an equivalence for every morphism f if and only if F preserves Cartesian morphisms. Let F, G : p → q be a morphism in pf i b (E) and ϕ : F → G a 2-arrow in pf i b (E).For each object X of E, the natural transformation ϕX : FX → GX induced by ϕ defines a 2-arrow χ(ϕ): ((F ), λ(F )) → ((G), λ(G)) by χ(ϕ)X = ϕX : FX → GX. 154 A. Yamaguchi

For a category E, we deﬁne a functor = E : pf i b (E) → Laxs (Eop , cat) as follows. Let p : F → E be a cloven preﬁbered category with cleavage κ.Put(p) = ((p), γ (p)). If F : p → q (p : F → E, q : D → E) is a morphism in pf i b (E), put (F) = ((F ), λ(F )). Let G : q → r (r : C → E) be a morphism in pf i b (E). Then,

(GF )X = GXFX = (G)X(F )X and = ∗ ◦ ∗ αf (GY FY (N))(λ(G)f )FY (N)GX((λ(F )f )N ) ((λ(G)f id(F )Y ) (id(G)X λ(F )f ))N . Hence, ((GF ), λ(GF )) = ((G), λ(G))◦((F ), λ(F )).

For a 2-arrow ϕ : F → G (F, G : p → q)inpf i b (E),weset(ϕ) = χ(ϕ), namely (ϕ)X = ϕX : FX → DX for X ∈ Ob E. Moreover, if ξ : G → H (H : p → q)andψ : K → L (K, L : q → r) are 2-arrows in pf i b (E), then it is straightforward to verify (ξϕ) = (ξ)(ϕ) and (ψ∗ϕ) = (ψ)∗(ϕ). Thus, is a functor. In other words, we have a lax functor s op (E ,θE ) : pf i b (E) → Lax (E , cat),where(θE )p is the identity 2-arrow of 1(p) and (θE )p,q,r is the identity natural transformation for any object p, q, r of pf i b (E). Let (, γ ) : C → D be an object of Laxs (C, D). For a 1-arrow S : X → Y in C, = : ◦ → = : we put Rγ (S) (γX,Y,Y )(S,1Y ) Y,Y (1Y ) X,Y (S) X,Y (S), Lγ (S) (γX,X,Y )(1X,S) X,Y (S)◦X,X(1X) → X,Y (S).SinceγY : 1(Y) → Y,Y (1Y ) is an isomorphism, the commutativity of the upper diagram of (4) in Deﬁnition 2.3 implies the following assertion.

PROPOSITION 2.14. The 2-arrows Rγ (f ) and Lγ (f ) in D are isomorphisms. Let (, γ ) : Eop → cat be an object of Laxs (Eop , cat). We construct a cloven prefibered category p() : F() → E as follows. Construction 2.15. Set Ob F() ={(X, x) | X ∈ Ob E, x ∈ Ob (X)}.For(X, x), (Y, y) ∈ Ob F(), we put F()((X, x), (Y, y)) ={(f, u) | f ∈ E(X, Y ), u ∈ (X) (x, Y,X(f )(y))}. Composition of morphisms (f, u) : (X, x) → (Y, y) and (g, v) : (Y, y) → (Z, z) is defined to be (gf, ((γZ,Y,X)(g,f ))zY,X(f )(v)u). Note that (idX,(γX)x) is the identity morphism of (X, x). Define a functor p() : F() → E by p()(X, x) = X, p()(f, u) = f . For each X ∈ Ob E, there is an isomorphism fX : F()X → (X) of f = f = −1 : → categories given by X(X, x) x and X(idX,u) (γX)y u ((idX,u) (X, x) (X, y)). We claim that a morphism (f, u) : (X, x) → (Y, y) is Cartesian if and only if u : x → Y,X(f )(y) is an isomorphism in (X). In fact, since

(f, u)(idX,v)= (f, Rγ (f )y X,X(idX)(u)v) = −1 = −1 (f, (Rγ (f ))y (γX)Y,X(y)u(γX)x v) (f, u(γX)x v) by the assumption and the commutativity of the upper diagram of (4) of (2.3), the map

F()X((X, z), (X, x)) → F()f ((X, z), (Y, y)) given by (idX,v)→ (f, u)(idX,v)is bijective for every z ∈ Ob (X) if and only if u is an isomorphism. Representations of internal categories 155

In particular, (idX,u): (X, x) → (X, y) is an isomorphism if and only if u is an −1 isomorphism. The inverse of (idX,u)is given by (idX,(γX)xu (γX)y). ∗ For a morphism f : X → Y in E,setf (Y, y) = (X, Y,X(f )(y)) and the : ∗ → canonical morphism αf (Y, y) f (Y, y) (Y, y) is deﬁned to be (f, idY,X(f )(y)). Then αf (Y, y) is Cartesian by the above fact, hence the inverse image functor ∗ ∗ ∗ f : F()Y → F()X of f is given by f (Y, y) = (X, Y,X(f )(y)) and f (idY ,v)= −1 : → E (idX,Rγ (f )z Lγ (f )zY,X(f )(v)). Note that, for a morphism f X Y in ,

∗ f / F()X F()Y

fX fY Y,X(f ) (X) /(Y) commutes. For morphisms f : X → Y and g : Z → X in E and (Y, y) ∈ Ob F(),wedeﬁne : ∗ ∗ → ∗ = −1 cf,g(Y, y) g f (Y, y) (fg) (Y, y) by cf,g(Y, y) (idZ,(Rγ (fg))y ((γY,X,Z)(f,g))y). Then, the following square commutes:

∗ αg(f (Y,y)) g∗f ∗(Y, y) /f ∗(Y, y)

cf,g(Y,y) αf (Y,y) αfg(Y,y) (fg)∗(Y, y) /(Y, y)

It follows from (1.11) that p() : F() → E is a ﬁbered category if and only if (, γ ) is a pseudo-functor. Let (, γ ), (, δ) : Eop → cat be objects of Laxs(Eop , cat). For 1-arrow (, λ) : (, γ ) → (, δ) of lax diagrams, we construct a functor F : F() → F() as follows. For (X, x) ∈ Ob F(), F(X, x) = (X, X(x)) and, for a morphism (f, u) : (X, x) → (Y, y) in F(), F(f, u) = (f, (λf )y X(u)). It is clear that F preserves ﬁbers. Since = = = F(αf (Y, y)) F(f, idY,X(f )(y)) (f, (λf )y X(idY,X(f )(y))) (f, (λf )y ),

F preserves Cartesian morphism if and only if λf : XY,X(f ) → Y,X(f )Y is a natural equivalence of functors from (Y) to (X) for every morphism f : X → Y in E. Let (, λ), (, ϕ) : (, γ ) → (, δ) be 1-arrows in Laxs(Eop , cat). For a 2-arrow χ : (, λ) → (, ϕ), we define a natural transformation χ˜ : F → F by χ˜(X,x) = ∈ F (idX,(δX)X(x)(χX)x) for (X, x) Ob (). Remark 2.16. Let (, γ ) be an object of Laxs (Eop , cat) and f : X → Y a morphism in E. ∗ Then, the pull-back functor f : F()Y → F()X has a left adjoint if and only if Y,X(f ) : (Y) → (X) has a left adjoint. If (, γ ) : Eop → cat is a 2-functor, the fibered category p() : F() → E is a bifibered category if and only if Y,X(f ) : (Y) → (X) has a left adjoint for every morphism f : X → Y in E.

The following examples are applications of the above construction. 156 A. Yamaguchi

Example 2.17. Let C be a category and F a presheaf of sets on C, that is a functor from Cop to the category of sets. Here CF denotes a category with objects (X, x) for X ∈ Ob C, x ∈ F(X) and morphisms CF ((X, x), (Y, y)) ={α ∈ C(X, Y ) | F (α)(y) = x}. We call CF the category of F -models. Note that there is a functor UF : CF → C given by UF (X, x) = X. For a morphism u : F → G of presheaves, we define a functor u : CF → CG by u(X, x) = (X, uX(x)) and u(α) = α. Let us denote by C the category of presheaves of ∗ ∗ sets on C. Then, u induces a functor u : CG → CF by u (S) = Su. We note that, for ∗ morphisms u : F → G and v : E → F of presheaves, since (uv) = uv,wehave(uv) = ∗ ∗ op op ∗ v u . Define a functor : C → cat by (F) = CF and (u) = u . The fibered category p() : F() → Cassociated with is called the fibered category of models on C.

Example 2.18. Let E be a category with ﬁnite limits and C = (C0,C1; σ, τ, ε, µ) an internal category in E. We denote by C : Eop → cat the functor represented by C.Thatis,C is described as follows. For X ∈ Ob E, put

Ob C(X) = E(X, C0) and C(X)(u, v) ={ϕ ∈ E(X, C1) | σϕ = u, τϕ = v}. The composition of morphisms ϕ : u → v and ψ : v → w is deﬁned to be a composition −(ϕ,ψ)−−→ × →µ : → X C1 C0 C1 C1. For an object u of C(X), εu X C1 is the identity morphism 1u : u → u. For a morphism f : X → Y in E, C(f ) : C(Y ) → C(X) is deﬁned by C (f )(u) = uf for an object u : Y → C0 of C(Y ) and C(f )(ϕ) = ϕf for a morphism

ϕ : Y → C1 in C(Y ). We call pC : F(C) → E the fibered category represented by C and we simply denote this by pC : F(C) → E. Example 2.19. Let (E,J) beasite(see[8, Section 1]). For each object X of E,wegive E/X the topology induced by the functor ΣX : E/X → E given by ΣX(p : E → X) → E (see [9, Section 3]). If f : X → Y is a morphism in E, then the functor Σf : E/X → E/Y defined by Σf (p : E → X) = (fp : E → Y) is continuous and cocontinuous by [9, ∗ : E → E ∗ = Proposition 5.2]. Define a functor Σf /Y /X by Σf (S) SΣf for a presheaf S on E ∗ : E → E ∗ : E → E /Y. Then, Σf /Y /X induces Σf /Y /X which is naturally equivalent to a ∗ i Σf a composition E/Y → E/Y −→ E/X → E/X. Here we denote by Cthe category of sheaves on C,byi : C→ C the inclusion functor and by a : C→ C the associated sheaf functor ∗ (see [8,Théorème 3.4]). It follows from [9, Proposition 2.3] that Σf is left exact and it has ˜ : E → E ˜ ∗ : E → E a right adjoint f∗ /X /Y. Thus, (f∗, Σf ) /X /Y is a geometric morphism : Eop → cat = E = ∗ of Grothendieck topoi. Define a functor by (X) /X and (f ) Σf . Applying the construction given in Construction 2.15 to , we have a bifibered category p() : F() → E which is an example of fibered topos [10,Définition 7.1.1.]. Let X. be a simplicial object in E, namely a functor op → E. Consider the pull-back op pX. : F(X.) → of p() : F() → E along X.. This fibered category pX. is called the Grothendieck topos over X.. In the case E = Sch and J is theetale ´ topology, pX. is the category of sheaves on simplicial scheme X (cf. [4]).

We deﬁne a functor = E : Laxs(Eop , cat) → pf i b (E) as follows. For an object (, γ ) : Eop → cat of Laxs(Eop , cat), put (, γ ) = (p() : F() → E) (see Construction 2.15). If (, λ) : (, γ ) → (, δ) is a 1-arrow in Laxs(Eop , cat),we put (, λ) = (F : F() → F()). For 1-arrows (, λ) : (, γ ) → (, δ) and (, ϕ) : Representations of internal categories 157

(, δ) → (, π), put (, ψ) = (, ϕ)◦(, λ) : (, γ ) → (, π).Let(X, x) be an object of F() and (f, u) : (X, x) → (Y, y) a morphism in F(). Then, it is routine to verify FF(X, x) = F (f, u). It follows that ((, ϕ)◦(, λ)) = (, ϕ)◦ (, λ).Let (, λ), (, ϕ) : (, γ ) → (, δ) be 1-arrows in Laxs (Eop , cat). For a 2-arrow χ : (, λ) → (, ϕ),weset (χ) =˜χ.Ifω : (, ϕ) → (, ψ) is a 2-arrow in Laxs (Eop , cat),then we have ( (ω) (χ))(X,x) = (ωχ)(X,x), hence (ωχ) = (ω) (χ).Let(, λ), (, ϕ) : (, γ ) → (, δ) and (, ψ), (ϒ, υ) : (, δ) → (, π) be 1-arrows in Laxs (Eop , cat). If χ : (, λ) → (, ϕ) and ω : (, ϕ) → (, ψ) are 2-arrows in Laxs (Eop , cat),then ( (ω)∗ (χ))(X,x) = (ω∗χ)(X,x). Hence, (ω∗χ)= (ω)∗ (χ). Thus, we have a lax s op functor ( E ,ξE ) : Lax (E , cat) → pf i b (E),where(ξE )(,γ ) is the identity 2-arrow of 1 (,γ ) and ξ(,γ ),(,δ),(.ϕ) is the identity natural transformation for any object (, γ ), (, δ), (, ϕ) of Laxs(Eop , cat).

THEOREM 2.20. There is an equivalence : pf i b (E) → Laxs(Eop , cat) of 2-categories. This induces the following equivalences: pf i b c(E) → Laxc(Eop , cat), fib(E) → 2-Funct(Eop , cat), fibc(E) → 2-Functc(Eop , cat). Proof. Put (Z, ζ ) = ( , ξ)(, θ) : pf i b (E) → pf i b (E). For an object p : F → E of pf i b (E), Z(p) = ((p)) : F((p)) → E is a cloven preﬁbered category such that

Ob F((p)) ={(X, M) | X ∈ Ob E,M∈ Ob FX}, ∗ F((p))((X, M), (Y, N)) ={(f, u) | f ∈ E(X, Y ), u ∈ F(M, f (N))}.

Define a 1-arrow Ep : Z(p) → p in pf i b (E) as follows. If (X, M), (X, N) ∈ Ob F((p)) and (f, u) : (X, M) → (X, N) is a morphism in F((p)), we put Ep(X, M) = M and Ep(f, u) = αf (N)u. Then, Ep is an isomorphism and −1 : → −1 = −1 = its inverse Ep p (p) is given by Ep (M) (p(M), M) and Ep (ρ) (p(ρ), u), ∗ where u : M → p(ρ) (N) is the unique morphism satisfying αp(ρ)(N)u = ρ. If F : p → q is a 1-arrow in pf i b (E),for(X, M) ∈ Ob F((p)) and a morphism (f, u) : (X, M) → (Y, N) in F((p)), then it is easy to verify Z(F )(X, M) = (X, F (M)). Hence, EqZ(F )(X, M) = FEp(X, M) and F : Eq Z(F) → FEp is defined to be the identity 2-arrow in pf i b (E). Therefore, we have a 1-arrow (E, ) : (Z, ζ ) → Ipf i b (E) in Lax(pf i b (E), pf i b (E)) whichisanisomorphism. Put (", ω) = (, θ)( , ξ) : Laxs (Eop , cat) → Laxs(Eop , cat) and let (, γ ) be an object of Laxs(Eop , cat). By the definitions of and ,wehave"(,γ) = ((p()), γ (p())) and Ob (p())(X) ={(X, x) | x ∈ Ob(X)},

Mor(p())(X) ={(idX,u)| u ∈ Mor (X)}.

For X ∈ Ob E,letH(,γ)X : (X) → (p())(X) be a functor given by

H(,γ)X(x) = (X, x) and H(,γ)X(u) = (idX,(γX)y u), where x, y ∈ Ob (X), (u : x → y) ∈ Mor (X). Then, H(,γ)X is an isomorphism. In −1 : → −1 = fact, the inverse H(,γ)X (p())(X) (X) is given by H(,γ)X (X, x) x, 158 A. Yamaguchi

−1 = −1 ∈ : → ∈ H(,γ)X (idX,v) (γX)y v for x, y Ob (X) and (v x X,X(idX)(y)) Mor (X). For a morphism f : X → Y in E and y ∈ Ob (Y), it is routine to verify that

(p())(f )Y,XH(,γ)Y (y) = H(,γ)XY,X(f )(y), that is, H(,γ)X is natural in X.Letη(, γ)f : H(,γ)XY,X(f ) → (p())Y,X(f ) H(,γ)Y be the identity natural transformation. Deﬁne a 1-arrow H(,γ ) : (, γ ) → s op "(,γ) in Lax (E , cat) by H(,γ ) = (H (, γ ), η(, γ )). For a 1-arrow (, λ) : (, γ ) → (, δ) in Laxs(Eop , cat) and x, y ∈ Ob (X), (u : x → y) ∈ Mor (X),wehave (F)XH(,γ)X(x) = H(,δ)XX(x) and (F)XH(,γ)X(u) = H(,δ)XX(u), which show the naturality of H(,γ) in (, γ ). We denote by η(,λ) : H(,δ)(, λ) → s op "(, λ)H(,γ ) the identity 2-arrow in Lax (E , cat). Now we have a 1-arrow (H, η) : s op s op ILaxs (Eop ,cat) → " in Lax(Lax (E , cat), Lax (E , cat)) whichisanisomorphism. 2 For a functor F : D → E, we denote by F op : Dop → Eop the functor induced by F . Regarding (F op , id) as a lax functor, we have a lax functor (F op , id)∗ : Lax(Eop , cat) → Lax(Dop , cat).

PROPOSITION 2.21. The following diagrams commutes up to natural equivalence:

pf i b (F ) (F op ,id)∗ pf i b (E) /pf i b (D) Laxs (Eop , cat) /Laxs (Dop , cat)

(E ,θE ) (D,θD) ( E ,ξE ) ( D,ξD) (F op ,id)∗ pf i b (F ) Laxs (Eop , cat) /Laxs (Dop , cat) pf i b (E) /pf i b (D)

Proof. Let (,γ) : pf i b (E) → Laxs(Dop , cat) be the composition of pf i b (F ) : pf i b (E) → pf i b (D) and (D,θD) : pf i b (D) → Laxs(Dop , cat). For an object p : F → E op of pf i b (E), (p) : D → cat is given by (p)(X) = (D ×E F)X for X ∈ Ob D = : → ∈ D : and X,Y (f ) κF (f ) for (f X Y) Mor . On the other hand, let ( ,δ) pf i b (E) → Laxs(Dop , cat) be the composition of (E ,θE ) : pf i b (E) → Laxs(Eop , cat) and (F op , id)∗ : Laxs(Eop , cat) → Laxs(Dop , cat). Then, for an object p : F → E of pf i b E : Dop → cat = F ∈ D = ( ), (p) is given by (p)(X) F(X) for X Ob and X,Y (f ) κ(F(f)) for (f : X → Y)∈ Mor D. Since the projection functor F : D ×E F → F induces an isomorphism from (D ×E F)X to FF(X) for each object X of D, and are naturally equivalent. This shows the commutativity of the ﬁrst diagram. The commutativity of the second diagram can be veriﬁed similarly. Details are left to readers. 2

If we identify Laxs(Eop , cat) with Laxs (E, catop ), then we can say that the functor ‘pf i b ’ from cat to the category of preﬁbered categories is ‘represented’ by catop by Theorem 2.20 and the above result.

3. Representations of internal categories

Let E be a category with ﬁnite limits and p : F → E a cloven ﬁbered category over E. Representations of internal categories 159

Definition 3.1. Let C = (C0,C1; σ, τ, ε, µ) be an internal category in E and M an object of F : ∗ → ∗ F C C0 . A morphism ξ σ (M) τ (M) in C1 is called a representation of on M if the following conditions are satisfied: ∗ −1 ∗ −1 = ∗ −1 (A) cτ,pr2 (M)pr2(ξ)cσ,pr2 (M) cτ,pr1 (M)pr1(ξ)cσ,pr1 (M) cτ,µ(M)µ (ξ)cσ,µ(M) , ←pr−1 × −pr→2 −→τ ←−σ where C1 C1 C0 C1 C1 is a limit of diagram C1 C0 C1; ∗ (U) cτ,ε(M)ε (ξ) = cσ,ε(M). Let ξ : σ ∗(M) → τ ∗(M) and ζ : σ ∗(N) → τ ∗(N) be representations of C on M and N, : → F respectively. A morphism ϕ M N in C0 is called a morphism of representations of C if τ ∗(ϕ)ξ = ζσ∗(ϕ). We denote by Rep(C; F) the category of the representations of C. We often denote the representation ξ : σ ∗(M) → τ ∗(M) by (M, ξ) and denote by : C; F → F → U Rep( ) C0 the forgetful functor (M, ξ) M. ∗ ∗ Let ξ : σ (M) → τ (M) be a representation of an internal category C = (C0,C1; σ, τ, ε, µ) in E on M and i : N → M a subobject of M. We call a representation ζ : σ ∗(N) → τ ∗(N) a subrepresentation if i : N → M is a morphism of representations from ζ to ξ. There exists at most one subrepresentation of ξ for each subobject of M if τ ∗ preserves monomorphisms, namely, it is easy to verify the following. ∗ : F → F PROPOSITION 3.2. Suppose that τ C0 C1 preserves monomorphisms. (For example, if p : F → E is a bifibered category, then τ ∗ has a left adjoint by (1) of Proposition 1.28, hence τ ∗ preserves monomorphisms.) Let ζ and ζ be representations of C on N and i : N → M a monomorphism. If i is the underlying morphism of morphisms ζ → ξ and ζ → ξ of representations, then ζ = ζ .

PROPOSITION 3.3. Let D : D → Rep(C; F) be a functor. : → : → F (1) Let (πi M UD(i))i∈Ob D be a limiting cone of UD D C0 . Suppose ∗ : ∗ → ∗ ∗ : D → F that (τ (πi ) τ (M) τ UD(i))i∈Ob D is a limiting cone of τ UD C1 and that ∗ ∗ ∗ ∗ ∗ ∗ (µ τ (π ) : µ τ (M) → µ τ UD(i)) ∈ D is a monomorphic family of F × . Then, i i Ob C1 C0 C1 there exists a unique morphism ξ : σ ∗(M) → τ ∗(M) such that ξ is a representation ∗ of C on M and (πi : (M, ξ) → D(i))i∈Ob D is a limiting cone of D. Hence, if τ : ∗ F → F preserves limits and µ : F → F × preserves monomorphic families, C0 C1 C1 C1 C0 C1 : C; F → F U Rep( ) C0 creates limits in the sense of MacLane [14,Ch.V]. : → : → F (2) Let (ιi UD(i) M)i∈Ob D be a colimiting cone of UD D C0 . Suppose that ∗ : ∗ → ∗ ∗ : → F (σ (ιi) σ UD(i) σ (M))i∈Ob D is a colimiting cone of σ UD D C1 ) and that ∗ ∗ ∗ ∗ ∗ ∗ (µ σ (ιi) : µ σ UD(i)→ µ σ (M))i∈Ob D is an epimorphic family. Then, there exists a unique morphism ξ : σ ∗(M) → τ ∗(M) such that ξ is a representation of C on M and : → ∗ : F → F (ιi D(i) (M, ξ))i∈Ob D is a colimiting cone of D. Hence, if σ C0 C1 preserves ∗ colimits and µ : F → F × preserves epimorphic families, U : Rep(C; F) → F C1 C1 C0 C1 C0 creates colimits. ∗ ∗ Proof. For i ∈ Ob D,wedenotebyξi : σ UD(i)→ τ UD(i) the structure morphism of the representation of C on UD(i). ∗ : ∗ → ∗ ∗ : D → F (1) Since (σ (πi) σ (M) σ UD(i))i∈Ob D is a cone of σ UD C1 , : ∗ → ∗ F ∗ = there exists a unique morphism ξ σ (M) τ (M) in C1 satisfying τ (πi)ξ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ξi σ (πi) for every i ∈ Ob D. Thus, we have f τ (πi)f (ξ) = f (ξi)f σ (πi) for = : × → f pr1, pr2,µ C1 C0 C1 C1.Sinceξi satisﬁes (A) of Deﬁnition 3.1, we can 160 A. Yamaguchi

∗ ∗ ∗ = −1 ∗ ∗ : ∗ ∗ → easily verify that µ τ (πi)µ (ξ) cσ,p1 (M) cσ,µ(M).Since(µ τ (πi) µ τ (M) ∗ ∗ µ τ UD(i))i∈Ob D is a monomorphic family, it follows that ξ satisfies (A) of Definition 3.1. ∗ ∗ ∗ ∗ ∗ ∗ Since ε τ (πi)ε (ξ) = ε (ξi )ε σ (πi) and ξi satisfies (U) of Definition 3.1, we have id∗ (π )c (M) = id∗ (π )c (M)ε∗(ξ).Sinceα (UD(i))id∗ (π ) = π α (M) by C0 i σ,ε C0 i τ,ε idC0 C0 i i idC0 Proposition 1.6, we have π α (M)c (M)ε∗(ξ) = π α (M)c (M) for every i ∈ i idC0 τ,ε i idC0 σ,ε Ob D. Thus, α (M)c (M)ε∗(ξ) = α (M)c (M).Sinceα (M) is an isomorphism, idC0 τ,ε idC0 σ,ε idC0 we see that ξ satisfies (U) of Definition 3.1. The proof of (2) is similar. 2 ∗ ∗ COROLLARY 3.4. Suppose that τ : F → F and µ : F → F × have left C0 C1 C1 C1 C0 C1 : C; F → F : F → E adjoints. Then, U Rep( ) C0 creates limits. In particular, if p is a : C; F → F bifibered category, U Rep( ) C0 creates limits. : C; F → F PROPOSITION 3.5. The forgetful functor U Rep( ) C0 reflects isomorphisms. Proof. Let ϕ : ξ → ζ be a morphism in Rep(C; F) such that U(ϕ) is an isomorphism. Since τ ∗(ϕ−1)ζ = τ ∗(ϕ−1)ζ σ ∗(ϕ)σ ∗(ϕ−1) = τ ∗(ϕ−1)τ ∗(ϕ)ξσ ∗(ϕ−1) = ξσ∗(ϕ−1), ϕ−1 is also a morphism in Rep(C; F). Hence, ϕ is an isomorphism in Rep(C; F). 2 : ∗ → ∗ F PROPOSITION 3.6. If a morphism ξ σ (M) τ (M) in C1 satisfies (A) and it is a monomorphism or an epimorphism, then ξ is a representation of C on M.IfC = (C0,C1; σ, τ, ε, µ) is an internal groupoid in E and ξ is a representation of C on M,thenξ is an isomorphism. = = : Proof. Assume that ξ satisfies (A). Consider morphisms ε1 (idC0 ,ετ),ε2 (εσ, idC0 ) → × = = C1 C1 C0 C1.Sinceµε1 µε2 idC1 , pulling back the equality of (A) along εi,we ∗ −1 ∗ −1 have ξcτ,εσ (M)(εσ) (ξ)cσ,εσ (M) = cτ,ετ(M)(ετ) (ξ)cσ,ετ (M) ξ = ξ. Hence, if ξ is a ∗ ∗ monomorphism, cτ,εσ (M)(εσ) (ξ) = cσ,εσ (M),ifξ is an epimorphism, cτ,ετ(M)(ετ) (ξ) = ∗ cσ,ετ (M). Apply ε to the above equalities, we see that (U) holds in the both cases. Assume that C is an internal groupoid and that ξ is an representation. We can define the −1 ∗ ∗ −1 ∗ −1 ∗ inverse ξ : τ (M) → σ (M) by ξ = cτ,ι(M)ι (ξ)cσ,ι(M) . A morphism ξ : σ (M) → ∗ F C τ (M) in C1 is a representation of on M if and only if ξ is an isomorphism and satisfies the = = : → × condition (A). In fact, consider morphisms ι1 (idC1 ,ι),ι2 (ι, idC1 ) C1 C1 C0 C1. Since µι1 = εσ and µι2 = ετ, pulling back the equality of (A) along ιi (i = 1, 2), we have −1 −1 ξ ξ = idσ(M) and ξξ = idτ(M) by (U). 2 The above notion of the representations of internal categories can be generalized as follows.

Deﬁnition 3.7. Let C = (C0,C1; σ, τ, ε, µ) be an internal category in E, (π : X → C0,α) C F : ∗ → ∗ an internal diagram on and M an object of X. A morphism ξ pX(M) α (M) (p : X × C → X is the projection) in F × is called a representation of C on M X C0 1 X C0 C1 over (X, α) if the following conditions are satisﬁed: ∗ −1 ∗ −1 (A) c × (M)(α × id ) (ξ)c × (M) c (M)p (ξ)c (M) α,α idC1 C1 pX,α idC1 α,p12 12 pX,p12 = × ∗ −1 cα,idX×µ(M)(idX µ) (ξ)cpX,idX×µ(M) , : × × → × where p12 X C0 C1 C0 C1 X C0 C1 denotes the projection; ∗ = (U) cpX,(idX,επ)(M)(idX,επ) (ξ) cα,(idX,επ)(M). Representations of internal categories 161

: ∗ → ∗ : ∗ → ∗ C Let ξ pX(M) α (M) and ζ pX(N) α (N) be representations of on M and N over (X, α), respectively. A morphism ϕ : M → N in FX is called a morphism of C ∗ = ∗ C ; F representations of over (X, α) if α (ϕ)ξ ζpX(ϕ). We denote by Rep( ,X ) the C : ∗ → ∗ category of the representations of over (X, α). We denote an object ξ pX(M) α (M) of Rep(C,X; F) by (M, ξ).

For an internal diagram (π : X → C0,α) on C, we deﬁne an internal category C = ; = × = = α (Cα,X σα,τα,εα,µα) associated with (X, α) by Cα X C0 C1, σα pX,τα : → = : → = × × : × → α Cα X, εα (idX,επ) X Cα and µα (idX µ)(idCα pr2) Cα X Cα × → : = × → Cα X C1 Cα.Herepr2 Cα X C0 C1 C1 denotes the projection. F : ∗ → ∗ F Let M be an object of X and ξ pX(M) α (M) a morphism in Cα . Then, ξ is a representation of Cα if and only if it is a representation of C over (X, α). Thus, we see the following result.

PROPOSITION 3.8. Let C be an internal category and (X, α) an internal diagram on C. Then, the category Rep(C,X; F) is isomorphic to Rep(Cα; F). Example 3.9. Let p : E(2) → E be the ﬁbered category given in Example 1.13 and C an internal category in E. For an object M = (π : M → C ) of E(2), consider the following pull- 0 C0 back diagrams in E:

σ τ ∗ π / ∗ π / σ (M) C1 τ (M) C1

σ¯ σ τ¯ τ π / π / M C0 M C0

∗ ∗ ∗ (2) ∗ ∗ Here τ¯∗ : E(σ (M), τ (M)) → E(σ (M), M) gives a bijection τ¯∗ : E (σ (M), τ (M)) → C1 {α ∈ E(σ ∗(M), M) | πζ = τπσ }. It is easy to verify that a morphism ξ : σ ∗(M) → τ ∗(M) in E(2) is a representation of C if and only if τξ¯ : σ ∗(M) = M × C → M is a structure C1 C0 1 morphism of an internal diagram on C. In fact, ξ satisﬁes the condition (A) (respectively (U)) of Deﬁnition 3.1 if and only if the following diagram on the left (respectively right) commutes, where we put α =¯τξ:

α× idC1 α M × C × C /M × C M × C / C0 1 C0 1 C0 1 CO0 1 M

idM ×µ α idM ×ε pr × α / × 1 / M C0 C1 M M C0 C0 M

Hence, Rep(C; E(2)) is regarded as the category of internal diagrams EC on C. Example 3.10. Let p : Qmod → Sch be the ﬁbered category given in Example 1.14 C S M Q and an internal category in ch. For an object (C0, ) of modC0 , a morphism : ∗ M = ∗M → ∗M = ∗ M C (idC1 ,ξ) σ (C0, ) (C1,σ ) (C1,τ ) τ (C0, ) is an representation of M : ∗M → ∗M O on (C0, ) if and only if a morphism ξ τ σ of C1 -modules satisﬁes ˜ M −1 ∗ ˜ M ˜ M −1 ∗ ˜ M =˜ M −1 ∗ ˜ M cσ,p1 ( ) p1 (ξ)cτ,p1 ( )cσ,p2 ( ) p2 (ξ)cτ,p2 ( ) cσ,µ( ) µ (ξ)cτ,µ( ) 162 A. Yamaguchi

∗ and ε (ξ)c˜σ,ε(M) =˜cτ,ε(M). Suppose that C is the internal category in Sch associated with a Hopf algebroid (A, H ),thatis,C0 = Spec A, C1 = Spec H ,andthatM is the quasi- O coherent C0 -module associated with an A-module M. There is a natural bijection ∗ ∗ : homO (τ M,σ M) → homA(M, M ⊗A H), C1 : → O where H is regarded as a left A-module by the left unit ηL A H inducing σ.An C1 - ∗ ∗ module homomorphism ξ : τ (M) → σ (M) deﬁnes a representation of C on (C0, M) if and only if (ξ) is a structure map of a H -comodule. For a morphism f : X → Y in E and a Cartesian section s : E → F of a ﬁbered ∗ category p : F → E, let us denote by sf : s(X) → f (s(Y )) the unique morphism in FX satisfying αf (s(Y ))sf = s(f). Since both s(f) and αf (s(Y )) are Cartesian morphisms, sf = −1 : ∗ → ∗ is necessarily an isomorphism. We put sf,g sgsf f (s(Y )) g (s(Z)) for morphisms f : X → Y and g : X → Z in E.

PROPOSITION 3.11. Let C = (C0,C1; σ, τ, ε, µ) be an internal category in E and s : E → ∗ ∗ F a Cartesian section. Then, sσ,τ : σ s(C0) → τ s(C0) is a representation of C on s(C0). Proof. By Proposition 3.6, we only have to verify the condition (A). Since we assumed that E has ﬁnite limits, in particular, E has a terminal object 1, we may assume that s = sT for some T ∈ Ob F by Proposition 1.19. Then, s = c (T )−1, s = c (T )−1 and we have the 1 σ oC0 ,σ τ oC0 ,τ → following equalities by Proposition 1.10 (here oC0 denotes the unique morphism C0 1): c (s(C ))p∗(s ) = c (o∗ (T ))p∗(c (T )−1) = c (T )−1c (T ), τ,p2 0 2 τ τ,p2 C0 2 oC0 ,τ oC0 ,τp2 oC0 τ,p2 p∗(s−1)c (s(C ))−1 = p∗(c (T ))c (o∗ (T ))−1 = c (T )−1c (T ), 2 σ σ,p2 0 2 oC0 ,σ σ,p2 C0 oC0 σ,p2 oC0 ,σp2 c (s(C ))p∗(s ) = c (o∗ (T ))p∗(c (T )−1) = c (T )−1c (T ), τ,p1 0 1 τ τ,p1 C0 1 oC0 ,τ oC0 ,τp1 oC0 τ,p1 c (s(C ))µ∗(s ) = c (o∗ (T ))µ∗(c (T )−1) = c (T )−1c (T ), τ,µ 0 τ τ,µ C0 oC0 ,τ oC0 ,τµ oC0 τ,µ µ∗(s−1)c (s(C ))−1 = µ∗(c (T ))c (o∗ (T ))−1 = c (T )−1c (T ), σ σ,µ 0 oC0 ,σ σ,µ C0 oC0 σ,µ oC0 ,σµ c (s(C ))p∗(s ) = c (o∗ (T ))p∗(c (T )−1) = c (T )−1c (T ). σ,p1 0 1 σ σ,p1 C0 1 oC0 ,σ oC0 ,σp1 oC0 σ,p1 Hence, it is straightforward to verify

∗ ∗ −1 −1 ∗ cτ,p2 (s(C0))p2 (sτ )p2 (sσ )cσ,p2 (s(C0)) cτ,p1 (s(C0))p1 (sτ ) = ∗ ∗ −1 −1 ∗ cτ,µ(s(C0))µ (sτ )µ (sσ )cσ,µ(s(C0)) cσ,p1 (s(C0))p1 (sσ ). Therefore, we have

∗ −1 ∗ −1 cτ,p2 (s(C0))p2 (sσ,τ )cσ,p2 (s(C0)) cτ,p1 (s(C0))p1 (sσ,τ )cσ,p1 (s(C0)) ∗ −1 = cτ,µ(s(C0))µ (sσ,τ )cσ,µ(s(C0)) . 2 ∗ ∗ Deﬁnition 3.12. (1) For a Cartesian section s, we call sσ,τ : σ s(C0) → τ s(C0) the trivial representation associated with s. In the case s = sT for some T ∈ Ob F1, we also call sσ,τ the trivial representation associated with T . ∗ ∗ (2) Let ξ : σ (M) → τ (M) be a representation of C on M and T an object of F1.We call a morphism ϕ : (sT )σ,τ → ξ a primitive element of ξ with respect to T . Representations of internal categories 163

Example 3.13. Let p : E(2) → E be the fibered category given in Example 1.13 and C E E(2) E an internal category in . We note that 1 is identified with by an isomorphism of E(2) → E → → E(2) = E categories 1 (X 1) X. For an object T of 1 , the Cartesian section : E → E(2) = : × → ∗ sT associated with T is given by sT (X) (pr1 X T X).Hereσ sT (C0) ∗ : × → and τ sT (C0) are both identified with (pr1 C1 T C1). Hence, the trivial representation ∗ ∗ (sT )σ,τ : σ sT (C0) → τ sT (C0) associated with T can be regarded as the identity morphism of C1 × T . Example 3.14. Let p : Qmod → Sch be the fibered category given in Example 1.14 and C an internal category in Sch. In this case, since the terminal object 1 in Sch is SpecZ, Qmod1 is identified with the category of abelian groups. For an abelian group G, the Cartesian section sG : Sch → Qmod associated with G is given by sG(X) = (X, o∗ G) (o : X → 1). The isomorphisms c˜ : o∗ G → σ ∗o∗ G, c˜ : o∗ G → X X σ,oC0 C1 C0 τ,oC0 C1 τ ∗o∗ G define isomorphisms c : σ ∗o∗ (1, G) → o∗ (1, G) , c : τ ∗o∗ (1, G) → C0 σ,oC0 C0 C1 τ,oC0 C0 o∗ (1, G) in Qmod . The trivial representation (s ) : σ ∗s (C ) → τ ∗s (C ) associated C1 C0 G σ,τ G 0 G 0 −1 with G is cσ,oC c . 0 τ,oC0 We describe the notion of representation of internal categories in terms of 2-categories and lax diagrams. Let C = (C0,C1; σ, τ, ε, µ) be an internal category in E and M an object of (C0). ∗ = ∗ = Recall that σ (C0,M) (C1,C0,C1 (σ)(M)), τ (C0,M) (C1,C0,C1 (τ)(M)).Fora : → : morphism ζ C0,C1 (σ)(M) C0,C1 (τ)(M) in (C1), we define a morphism ξ(ζ) ∗ → ∗ F = −1 σ (C0,M) τ (C0,M)in ()C1 by ξ(ζ) (idC1 ,Rγ (τ)M ζ). PROPOSITION 3.15. We have that: (1) ξ(ζ) satisfies (A) of Definition 3.1 if and only if ζ satisfies the equality

−1 ((γ × ) ) × (p )(ζ )((γ × ) ) C0,C1,C1 C0 C1 (τ,p2) M C1,C1 C0 C1 2 C0,C1,C1 C0 C1 (σ,p2) M −1 ◦((γ × ) ) × (p )(ζ )((γ × ) ) C0,C1,C1 C0 C1 (τ,p1) M C1,C1 C0 C1 1 C0,C1,C1 C0 C1 (σ,p1) M −1 = ((γ × ) ) × (µ)(ζ )((γ × ) ) ; C0,C1,C1 C0 C1 (τ,µ) M C1,C1 C0 C1 C0,C1,C1 C0 C1 (σ,µ) M (2) ξ(ζ) satisfies (U) of Definition 3.1 if and only if ζ satisfies = ((γC0,C1,C0 )(τ,ε))M C1,C0 (ε)(ζ ) ((γC0,C1,C0 )(σ,ε))M .

Let C and G be internal categories in E. Consider the ﬁbered category pC : F(C) → E represented by C given in Example 2.18. The following is immediate from deﬁnitions.

LEMMA 3.16. We have the following. (1) Let f0 : G0 → C0 and f1 : G1 → C1 be morphisms in E.Here(f0,f1) : G → C is : ∗ = → an internal functor if and only if a morphism (idG1 ,f1) σ (G0,f0) (G1,f0σ) = ∗ G ∈ F C (G1,f0τ) τ (G0,f0) is a representation of on (G0,f0) Ob ( )G0 . (2) Let f = (f0,f1), g = (g0,g1) : G → C be internal functors and ϕ : G0 → C1 a morphism in E.Hereϕ is an internal natural transformation from f to g if and only if : ∗ → ∗ (idG0 ,ϕ)is a morphism of representations from (idG1 ,f1) σ (G0,f0) τ (G0,f0) : ∗ → ∗ to (idG1 ,g1) σ (G0,g0) τ (G0,g0). Thus, we have the following result. 164 A. Yamaguchi

THEOREM 3.17. Deﬁne a functor F : cat(E)(G, C) → Rep(G; F(C)) = = : G → C = by F(f) (idG1 ,f1) for an internal functor f (f0,f1) and F(ϕ) (idG0 ,ϕ). Then, F is an isomorphism of categories.

4. Descent formalism

Definition 4.1. (Grothendieck [6,Définition 1.3]) Let p : F → E beaclovenfibered p1 f category. We say that a diagram R ⇒ X → Y in E is F-exact if fp1 = fp2 and, for any p2 ∈ F = = M, N Ob Y , the following diagram is an equalizer, where we put g fp1 fp2 and pi = → ∗ −1 (i 1, 2) are maps defined by ϕ cf,pi (N)pi (ϕ)cf,pi (M) : ∗ p f ∗ ∗ 1 ∗ ∗ FY (M, N) → FX(f (M), f (N)) ⇒ FR(g (M), g (N)). p2

p1 f Example 4.2. AdiagramR ⇒ X → Y in E is E(2)-exact if and only if, for any π : M → Y , p2 idM × f : M ×Y X → M ×Y Y = M is a coequalizer of idM × p1, idM × p2 : M ×Y R → M ×Y X,inotherwords,f is a universal strict epimorphism.

Definition 4.3. (Grothendieck [6,Définition 1.4]) Let p1,p2 : R → X be morphisms in E F : ∗ → ∗ and M is an object of X. An isomorphism ξ p1(M) p2 (M) is called a gluing morphism : ∗ → ∗ : ∗ → ∗ on M with respect to a pair (p1,p2).Ifξ p1 (M) p2 (M) and ζ p1(N) p2 (N) are gluing morphisms on M, N ∈ Ob FX, a morphism ϕ : M → N in FX is said to be compatible with ξ and ζ if the following square commutes:

∗ ξ / ∗ p1 (M) p2(M)

p∗(ϕ) p∗(ϕ) 1 2 ∗ ζ / ∗ p1 (N) p2 (N)

Thus, we can consider the category of gluing morphisms.

p1 f Definition 4.4. (Grothendieck [6,Définition 1.5]) Let R ⇒ X → Y be a diagram in E such p2 = : ∗ → ∗ ∈ F that fp1 fp2. We say that a gluing morphism ξ p1(M) p2 (M) on M Ob X is ∗ effective with respect to f if there exists an isomorphism κ : M → f (N) in FX for some N ∈ Ob FY such that the following diagram commutes:

∗ p (κ) cf,p (N) ∗ 1 / ∗ ∗ 1 / ∗ p1 (M) p1f (N) (fp1) (N)

ξ ∗ p (κ) cf,p (N) ∗ 2 / ∗ ∗ 2 / ∗ p2 (M) p2f (N) (fp2) (N) Representations of internal categories 165

Assume that E is a category with ﬁnite limits below.

Deﬁnition 4.5. Let C = (C0,C1; σ, τ, ε, µ) be an internal category in E. (1) If (σ, τ) : C1 → C0 × C0 is a monomorphism, we call C an internal poset. (2) If C is an internal groupoid and an internal poset, we call C an equivalence relation on C0.

p1 For a morphism f : X → Y in E, the kernel pair X ×Y X ⇒ X of f is an equivalence p2 relation on X with the following structure maps: the domain σ = p1, the codomain τ = p2, the identity ε = (the diagonal morphism), the composition µ = p1 × p2 : (X ×Y X) ×X (X ×Y X) → X ×Y X and the inverse ι = (p2,p1) : X ×Y X → X ×Y X. We denote this internal groupoid by Ef = (X ×Y X, X; p1,p2,,p1 × p2). The notion of descent data is given in terms of representation of groupoids as follows.

Definition 4.6. (Grothendieck [6,Définition 1.6]) For an object M of FX, a representation : ∗ → ∗ E : → ξ p1 (M) p2(M) of f on M is called a descent data on M for a morphism f X Y in E.

Let f : X → Y be a morphism in E. We denote by pi : X ×Y X → X (i = 1, 2), qi : X ×Y X ×Y X → X (i = 1, 2, 3) the projections onto the ith component. : X → X ×Y X denotes the diagonal morphism. Deﬁne pij : X ×Y X ×Y X → X ×Y X (1 ≤ i

PROPOSITION 4.7. Let E be a category with finite limits and p : F → E a cloven fibered : ∗ → ∗ ∈ F category. A gluing morphism ξ p1 (M) p2(M) on M Ob X with respect to a pair (p1,p2) is a descent data on M for a morphism f : X → Y in E if and only if ξ satisfies the following equalities: ∗ = cp1,(M) (ξ) cp2,(M),

∗ −1 cp2,p13 (M)p13(ξ)cp1,p13 (M) = ∗ −1 ∗ −1 cp2,p23 (M)p23(ξ)cp1,p23 (M) cp2,p12 (M)p12(ξ)cp1,p12 (M) . Definition 4.8. (Grothendieck [6,Définition 1.7]) A morphism f : X → Y in E is called a p1 f morphism of F-descent if X ×Y X ⇒ X → Y is F-exact. Moreover, if every descent data on p2 arbitrary object of FX is effective, we say that f is a morphism of effective F-descent.

We set Desc(F/E,f)= Rep(Ef ; F) and regard this as a full subcategory of ¯ Glue(F/E,f). We deﬁne a functor Df : FY → Glue(F/E,f)as follows. For N ∈ FY ,let ¯ : ∗ ∗ → ∗ ∗ Df (N) p1 f (N) p2 f (N) be the composition

c (N) c (N)−1 ∗ ∗ −−−−−f,p1 → ∗ = ∗ −−−−−−f,p2 → ∗ ∗ p1f (N) (fp1) (N) (fp2) (N) p2 f (N). ¯ ∗ ¯ For a morphism ϕ : N → N , Df (ϕ) = f (ϕ). Then, Df factors through the inclusion functor Desc(F/E,f)→ Glue(F/E,f)and we have a functor Df : FY → Desc(F/E,f). 166 A. Yamaguchi

: ∗ → ∗ ∈ F Moreover, a gluing morphism ξ p1 (M) p2(M) on M Ob X is effective with respect to f : X → Y if and only if ξ is isomorphic to an object in the image of Df : FY → Desc(F/E,f). The following fact is also immediate.

PROPOSITION 4.9. A morphism f : X → Y is of F-descent (respectively effective F- descent) if and only if Df : FY → Desc(F/E,f) is fully faithful (respectively an equivalence). Example 4.10. Let Top be the category of topological spaces and continuous maps. Consider : T (2) → T the ﬁbered category p op op (1.13). For a topological space B, suppose that an open = : → covering (Ui)i∈I of B is given. Put X i∈I Ui and let f X B be the map induced by → × = ∩ the inclusion maps Ui # B. Then, X B X i,j∈I Ui Uj and the following diagrams commute: oinc inc / inc / Ui Ui ∩ Uj Uj Ui Ui ∩ Ui Ui ∩ Uj ∩ Uk Ui ∩ Uj

ιi ιij ιj ιi ιii ιijk ιij op1 p2 / / p12 / X X ×B X X X X ×B X X ×B X ×B X X ×B X

inc / inc / Ui ∩ Uj ∩ Uk Uj ∩ Uk Ui ∩ Uj ∩ Uk Ui ∩ Uk

ιijk ιjk ιijk ιik p23 / p13 / X ×B X ×B X X ×B X X ×B X ×B X X ×B X : × → T (2) = T For a topological space F , consider an object pr1 X F X of opX op/X. Then, : × → : × → = : × the pull-back of pr1 X F X along pi X B X X (i 1, 2) is the map q (X B × → × = ∈ × ∈ : × × X) F X B X given by q(x, y) x (x X B X, y F ). For a map ξ (X B X) = ∩ × → ∩ × = × × F i,j∈I (Ui Uj ) F i,j∈I (Ui Uj ) F (X BX) F making the following diagram commute, we denote by ξij : (Ui ∩ Uj ) × F → (Ui ∩ Uj ) × F the restriction of ξ: ξ (Ui ∩ Uj ) × F / (Ui ∩ Uj ) × F i,j∈I i,j∈I pr1 pr1 i,j∈I i,j∈I Ui ∩ Uj Ui ∩ Uj i,j∈I i,j∈I

k : ∩ ∩ × → ∩ ∩ × We also denote by ξij (Ui Uj Uk) F (Ui Uj Uk) F the restriction of ξij . × ; × : × → Then, a descent data ξ of (X B X, X p1,p2,,p1 p2) on pr1 X F X is a : ∩ × → ∩ × homeomorphism ξ i,j∈I (Ui Uj ) F i,j∈I (Ui Uj ) F which makes the above i k = j diagram commute and satisﬁes ξjkξij ξik.

5. Regular representations and restrictions

Let C be an internal category in C and p : F → C a cloven ﬁbered category. Representations of internal categories 167

Deﬁnition 5.1. A representation ρ : σ ∗(R) → τ ∗(R) of C is called a regular representation F C : if there exist an object N of C0 and, for each representation ξ of , a bijection αξ C; F → F Rep( )(ρ, ξ) C0 (N, U(ξ)) which is natural in ξ. ∗ ∗ PROPOSITION 5.2. A representation ρ : σ (R) → τ (R) of C is a regular representation : → = F ∈ if and only if there exists a morphism η N R U(ρ) in C0 such that, for any ξ ∗ C; F C; F −→U F −→η F Ob Rep( ), composition Rep( )(ρ, ξ) C0 (U(ρ), U(ξ)) C0 (N, U(ξ)) is bijective. Proof. Suppose that ρ : σ ∗(R) → τ ∗(R) of C is a regular representation as in Deﬁnition 5.1. Put η = αρ (idρ) : N → U(ρ). For a morphism θ : ρ → ξ, U(θ)η= U(θ)αρ (idρ ) = αξ (θ) by ∗ : C; F → F the naturality. Hence, the composition η U Rep( )(ρ, ξ) C0 (U(ξ), N) coincides with αξ . The converse is obvious. 2 By the above result and Theorem 3.17, we have the following.

COROLLARY 5.3. Let C and G be internal categories in E. Consider the ﬁbered category : F C → E C : pC ( ) represented by given in Example 2.18. A representation (idG1 ,ρ1) ∗ ∗ σ (G0,ρ0) → τ (G0,ρ0) of G is a regular representation if and only if there exists a : → ∈ E F C morphism (idG0 ,η) (G0,u) (G0,ρ0)(η (G0,C1)) in ( )G0 such that, for any internal functor (f0,f1) : G → C, the map from the set of internal natural transformations (ρ0,ρ1) → (f0,f1) to C(G0)(u, f0) given by ϕ → µ(η, ϕ) is bijective: : C; F → F PROPOSITION 5.4. The forgetful functor U Rep( ) C0 has a left adjoint if and ∈ F C only if, for every N Ob C0 , there exist a representation ρN of and a morphism : → F ∈ C; F ηN N U(ρN ) in C0 such that, for any ξ Ob Rep( ), the following composition is bijective:

η∗ C; F −→U F −→N F Rep( )(ρN ,ξ) C0 (U(ρN ), U(ξ)) C0 (N, U(ξ)). : C; F → F : F → C; F Proof. Suppose that U Rep( ) C0 has left adjoint L C0 Rep( ).Let η : idF → UL be the unit of this adjunction. For N ∈ Ob FC , a representation L(N) C0 0 and a morphism ηN : N → UL(N) satisﬁes the condition. In fact, for ξ ∈ Ob Rep(C; F), η∗ C; F −→U F −→N F the composition Rep( )(L(N), ξ) C0 (UL(N), U(ξ)) C0 (N, U(ξ)) is the : F → C; F adjoint bijection. We show the converse. Deﬁne a functor L C0 Rep( ) as follows. F = : → F For an object N of C0 , put L(N) ρN . For a morphism ϕ N M in C0 ,let L(ϕ) : ρN → ρM be the morphism in Rep(C; F) which maps to ηM ϕ by the composition η∗ C; F −→U F −→N F Rep( )(ρN ,ρM ) C0 (U(ρN ), U(ρM )) C0 (N, U(ρM )). It is easy to verify that L is a functor and that it is a left adjoint of U. 2 : C; F → F PROPOSITION 5.5. Suppose that U Rep( ) C0 has a left adjoint L. Let us denote by η and the unit and the counit of this adjunction. Put T = UL and consider the monad T = (T,η,U(L)) associated with this adjunction. Then, the comparison functor K : Rep(C; F) → F T given by K(ξ) = U(ξ),U(ε ) is an isomorphism of categories. C0 ξ ϕ U(ϕ) Proof. Let ξ ⇒ ζ be parallel arrows in Rep(C; F) such that U(ξ) ⇒ U(ζ) has a split ψ U(ψ) F ∗ ∗ coequalizer in C0 .Sinceσ preserves split coequalizers and µ preserves split epimorphism, 168 A. Yamaguchi U(ϕ) U creates the coequalizer of U(ξ) ⇒ U(ζ) by (2) of Proposition 3.3. Hence, by Beck’s U(ψ) theorem (see [14, p. 151]) the assertion follows. 2

Let C = (C0,C1; σ, τ, ε, µ) and D = (D0,D1; σ, τ, ε, µ) be internal categories in E, f = (f0,f1) : D → C an internal functor and p : F → E a ﬁbered category. Suppose : ∗ → ∗ C ∈ F that a representation ξ σ (M) τ (M) of over M Ob C0 is given. We denote by : ∗ ∗ → ∗ ∗ F ξf σ f0 (M) τ f0 (M) the unique morphism in D1 such that the following diagram commutes:

−1 cσ,f (M) cf ,σ (M) ∗ ∗ 1 / ∗ = ∗ 0 / ∗ ∗ f1 σ (M) (σf1) (M) (f0σ) (M) σ f0 (M)

∗ f1 (ξ) ξf −1 cτ,f (M) cf ,τ (M) ∗ ∗ 1 / ∗ = ∗ 0 / ∗ ∗ f1 τ (M) (τf1) (M) (f0τ) (M) τ f0 (M) : ∗ ∗ → ∗ ∗ D Then, it is straightforward to verify that ξf σ f0 (M) τ f0 (M) is a representation of ∗ ∈ F over f0 (M) Ob D0 . : ∗ ∗ → ∗ ∗ Deﬁnition 5.6. We call ξf σ f0 (M) τ f0 (M) the restriction of ξ along f . : ∗ → ∗ C ∈ F : → If ζ σ (N) τ (N) is a representation of over N Ob C0 and ψ ξ ζ a ∗ : ∗ → ∗ ∗ : morphism of representations, f0 (U(ψ)) f0 (M) f0 (N) deﬁnes a morphism f0 (ψ) ξf → ζf of representations. If g = (g0,g1) : D → C is an internal functor and ϕ is an internal natural transformation : ∗ → ∗ from f to g,letϕξ f0 (M) g0 (M) be the following composition:

−1 ∗ ∗ = ∗ −c−−−−−σ,ϕ(M)→ ∗ ∗ −ϕ−−(ξ)→ ∗ ∗ −c−−−τ,ϕ(M)→ ∗ = ∗ f0 (M) (σϕ) (M) ϕ σ (M) ϕ τ (M) (τϕ) (M) f0 (M).

Then, ϕξ is a morphism of representations ξf → ξg. It can be shown that the following diagram in Rep(D; F) commutes:

f ∗(ψ) 0 / ξf ζf

ϕξ ϕζ

g∗(ψ) 0 / ξg ζg

Hence, we have a functor Res : cat(E)(D, C) × Rep(C; F) → Rep(D; F).IfF = F(G) for an internal category G, we remark that Res is identiﬁed with the composition of internal functors by the isomorphism of Theorem 3.17, that is, the following diagram commutes:

composition cat(E)(D, C) × cat(E)(C, G) /cat(E)(D, G)

id×F F Res cat(E)(D, C) × Rep(C; F(G)) /Rep(D; F(G)) Representations of internal categories 169

REFERENCES [1] J. F. Adams. Lectures on Generalised Cohomology (Lecture Notes in Mathematics, 99). Springer, Berlin, 1969, pp. 1–138. [2] J. Bénabou. Introduction to Bicategories (Lecture Notes in Mathematics, 47). Springer, Berlin, 1969, pp. 1–77. [3] P. Deligne. Catégorie tannakiennes (Progress in Mathematics, 87). Birkhäuser, Basel, 1990, pp. 111–194. [4] E. M. Friedlander. Etale Homotopy of Simplicial Schemes (Annals of Mathematics Studies, 109). Princeton University Press, Princeton, NJ, 1982. [5] J. Giraud. Métode de la descente. Bull. Soc. Math. France, Suppl. Mém. 2 (1964). [6] A. Grothendieck. Technique de descente et théoremès d’existence en géométrie algébrique I. Généralités. Descente par morphismes fidèlement plats, Séminaire Bourbaki 1957–62, Secrétariat Math., Paris, 1962. [7] A. Grothendieck. Catégorie fibrees et Descente (Lecture Notes in Mathematics, 224). Springer, Berlin, 1971, pp. 145–194. [8] J. L. Verdier. Topologies et Faisceaux (Lecture Notes in Mathematics, 269). Springer, Berlin, 1972, pp. 219–263. [9] J. L. Verdier. Fonctoialite des Categories de Faisceaux (Lecture Notes in Mathematics, 269). Springer, Berlin, 1972, pp. 265–297. [10] A. Grothendieck and J. L. Verdier. Condition de finitude. Topos et Sites fibrés. Applications aux questions de passagea ` la limite (Lecture Notes in Mathematics, 270). Springer, Berlin, 1972, pp. 163–340. [11] P. T. Johnstone. Topos Theory. Academic Press, New York, 1977. [12] G. M. Kelly and R. Street. Review of the elements of 2-categories (Lecture Notes in Mathematics, 420). Springer, Berlin, 1974, pp. 75–103. [13] M. Hakim. Topos annelés et schémas relatifs (Ergebisse der Mathematik, 64). Springer, Berlin, 1972. [14] S. MacLane. Categories for the Working Mathematician, 2nd edn (Graduate Texts in Mathematics, 5). Springer, Berlin, 1997.

Atsushi Yamaguchi Osaka Prefecture University Faculty of Liberal Arts and Sciences 1-1 Gakuen-cho, Sakai Osaka 599-8531, Japan (E-mail: [email protected])