Theory and Applications of Categories, Vol. 7, No. 7, 2000, pp. 148–170.

ON THE MONADICITY OF CATEGORIES WITH CHOSEN COLIMITS

G. M. KELLY AND STEPHEN LACK

Transmitted by Walter Tholen

ABSTRACT. There is a 2-category J -Colim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2-functor from J -Colim to the 2-category Cat of small categories is known to be monadic. We extend this result by considering not just conical colimits, but general weighted colimits; not just ordinary categories but enriched ones; and not just small classes of colimits but large ones; in this last case we are forced to move from the 2-category V-Cat of small V-categories to V-categories with object-set in some larger universe. In each case, the functors preserving the colimits in the usual “upto-isomorphism” sense are recovered as the pseudomorphisms between algebras for the 2-monad in question.

1. Introduction

An important structure on a category A is that of admitting a colimit for each diagram J → A with domain J in some small class J of small categories. Consider the 2-category J -Colim, an object of which is a small category A (necessarily a J -cocomplete one) together with, for each J ∈ J, a choice of a colimit for each diagram J → A; with the arrows being functors which preserve the chosen colimits strictly, and the 2-cells being arbitrary natural transformations. It is well known that the resulting forgetful 2-functor U : J -Colim → Cat is monadic, where Cat is the 2-category of small categories. A recent proof of this fact, in the dual case of limits rather than colimits, is found in [12]: the argument rests upon the fact that to give to a category A the structure of an object of J -Colim is precisely to give, for each J ∈ J, a left adjoint to the diagonal functor from A to the functor category [J, A], and that the structure describing such an adjoint consists of everywhere-defined operations and equational axioms, and so is monadic. Note that, in our use here of functors that preserve the colimits strictly, we have not abandoned those that preserve the colimits only to within a canonical isomorphism: for these reappear as the pseudomorphisms between algebras for the 2-monad in question. In fact the 2-category of algebras, pseudomorphisms, and algebra 2-cells is equivalent to the 2-category J -Cocts of small J -cocomplete categories, J -cocontinuous functors, and natural transformations.

It is of course well-known that J -Cocts can be seen as the 2-category of algebras for a “pseudomonad” or “doctrine” on Cat of the Kock-Zöberlein kind. (The epithet “lax idempotent” was suggested in [11] as a replacement for “Kock-Zöberlein”.) The point of the present paper, however, is precisely the generalization, to the most general “class of colimits” context, of the stronger result above about the (strict) 2-monadicity of J -Colim, which carries J -Cocts along with it when we turn from strict morphisms of algebras to “pseudo” ones. For it is this 2-monadicity — raised as an open question by Kock on page 43 of [14] — that exhibits the giving of chosen colimits of the given class as endowing the category with a purely algebraic structure, and opens the way to applying the results of [3] on 2-monads.

Both authors gratefully acknowledge the support of the Australian Research Council and DETYA.

Received by the editors 1999 October 22 and, in revised form, 2000 May 19.

Published on 2000 June 12.

2000 Mathematics Subject Classification: 18A35, 18C15, 18D20.

Key words and phrases: monadicity, categories with limits, weighted limits, enriched categories.

\begin{matrix}c \\ ◯ \end{matrix} G. M. Kelly and Stephen Lack, 2000. Permission to copy for private use granted.

Even in the case of unenriched categories, it has long been clear that a more versatile theory of colimits is obtained by considering not just the classical “conical” colimits, but also the more general weighted colimits of [9]. A weight is simply a functor ϕ : D_{ϕ}^{op} → Set with small domain, and the ϕ-weighted colimit of a functor s : D_{ϕ}^{op} → A is a representation of [D_{ϕ}^{op}, Set](ϕ, A(s, a)) : A^{op} → Set, as in

A(ϕ ∗ s, a) ≅ [D_{ϕ}^{op}, Set](ϕ, A(s, a));

often one speaks loosely of the representing object ϕ ∗ s as the “ϕ-weighted colimit of s”, but strictly speaking the colimit includes the isomorphism above, or equivalently the unit ϕ → K(s, ϕ ∗ s). As an example of the importance of weighted colimits, (pointwise) left Kan extensions are directly expressible as such colimits: given k : D → B and s : D → A we have (Lan_{k}s)b = B(k, b) ∗ s. One recovers the conical colimits within this framework as those with weights Δ1 : J^{op} → Set, where Δ1 is the functor constant at 1; accordingly one often writes colim s for Δ1 ∗ s. In fact to give the weighted colimit ϕ ∗ s is just to give the conical colimit of sd_{ϕ}^{op} : el(ϕ)^{op} → K, where el(ϕ) is the category of elements of ϕ and d_{ϕ} : el(ϕ) → D_{ϕ}^{op} is the projection; and so the existence of particular weighted colimits reduces to the existence of particular conical colimits. It is not however the case that the existence of all colimits weighted by some class Φ of weights reduces to the existence of all conical colimits of functors with domain in some class J of small categories. If Φ is a small class of weights we can define, as we did for J -Colim, a 2-category Φ-Colim of small categories with chosen Φ-colimits, functors preserving these strictly, and natural transformations, together with a forgetful 2-functor U : Φ-Colim → Cat; but the monadicity of this U will not follow from the known results of the first paragraph. It is however a special case of the monadicity established in Theorem 6.1 below.

Of course there is an elegant and well-developed theory of colimits in the context of categories enriched in a symmetric monoidal closed category V whose underlying ordinary category V_{0} is complete and cocomplete. A weight is now a V-functor ϕ : D_{ϕ}^{op} → V with small domain, and the ϕ-weighted colimit of a V-functor s : D_{ϕ} → A is now defined by a representation

A(ϕ ∗ s, a) ≅ [D_{ϕ}^{op}, V](ϕ, A(s, a)),

this of course now being an isomorphism not in Set but in V. We can again express left Kan extensions in terms of weighted colimits, but there is no longer a reduction of ϕ ∗ s to an ordinary colimit like colim(sd_{ϕ}^{op}). If Φ is a small collection of such weights, we have once again a forgetful 2-functor U : Φ-Colim → V-Cat, which will be shown in Theorem 6.1 below to be monadic; here of course V-Cat is the 2-category of small V-categories.

In the various situations above we have always taken the class Φ of weights to be small, which prevents us from dealing with such important cases as that of all (small) weights, and so of dealing with cocomplete categories. In the case of a large class of colimits we can no longer restrict attention to small categories — it is well-known, for instance, that the only cocomplete categories which are small are preorders — and so we adopt the following approach. We suppose given once and for all an inaccessible cardinal ∞, whereupon a set is said to be small if its cardinality is less than ∞; similarly by a small V-category we mean one whose object-set is small, and by a cocomplete V-category we mean a Vcategory admitting ϕ-weighted colimits for all small weights ϕ — that is, all ϕ : D_{ϕ}^{op} → V with D_{ϕ} small. In Section 7 we moreover suppose given another inaccessible cardinal ∞′ such that the collection of isomorphism classes of small weights, seen as objects of V-Cat/V, has cardinality less than ∞′. We say that a set is big if its cardinality is less than ∞′, and that a V-category is big if its set of objects is big; and we write V-CAT for the 2-category of big V-categories. Given a class Φ of small weights we now have the 2-category Φ-COLIM of big V-categories with chosen Φ-limits, and the forgetful 2functor U : Φ-COLIM → V-CAT. In Theorem 7.1 we prove that this forgetful 2-functor U is monadic. In particular, cocomplete ordinary categories (with chosen colimits) are monadic over the 2-category CAT (= Set-CAT) of big but locally-small categories.

We shall shortly outline the method of proof of the monadicity of U : Φ-Colim → V-Cat for a small class Φ of weights. First, however, we make a comment about this method. The reader may wonder why our proof is not more direct. Can we not describe an object of Φ-Colim as an object of V-Cat provided with a structure — the chosen colimits — given by a small family of operations and equations, each having some small “arity”, as was done in the classical case of J -Colim? One would have to assert that the appropriate morphism A(ϕ ∗ s, a) → [D^{op}, V](ϕ, A(s, a)) is invertible in V, or equivalently that the induced function V_{0}(G, A(ϕ ∗ s, a)) → V_{0}(G, [D^{op}, V](ϕ, A(s, a))) is bijective for each element G of a strongly-generating family G of objects of the ordinary category V_{0} underlying V. Only when V_{0} is locally presentable have we a small family G of this kind giving suitable “arities” (in that each V_{0}(G, −) preserves α-filtered colimits for some α), opening the way to a direct proof as above. The 2-monadicity of U : Φ-Colim → V-Cat does, however, hold for every symmetric monoidal closed category V with the usual properties of being locally small, complete, and cocomplete. Furthermore, even in the case where a direct proof is possible, to describe all the operations and equations, as is required in such a direct proof, is extremely complicated and rather tedious. Finally, the technique we develop to construct a left adjoint will clearly be applicable to other problems: one such application is sketched in Section 8 below. It is for all three of these reasons — the greater generality of the theorem, the reduced technical complexity of the proof, and the applicability of the new technique — that we have adopted the method of proof we now sketch.

The main step is to construct a left adjoint to U; the monadicity of U is then deduced using the well-known theorem of Beck. The construction of the left adjoint involves consideration, alongside Φ-Colim, of the 2-category Φ-Cocts of Φ-cocomplete V-categories (those that admit Φ-colimits), Φ-cocontinuous V-functors (those that preserve Φ-colimits in the usual non-strict sense, which merely requires the canonical comparison morphism to be invertible), and V-natural transformations. We write U′ : Φ-Cocts → V-Cat for the forgetful 2-functor, and henceforth use U_{s} : Φ-Colim → V-Cat for the 2-functor previously called U, so as to free the letter U for another purpose — it will turn out that U_{s} denotes a “strict-case” analogue of U. Forgetting the choice of colimits gives, of course, a 2-functor L : Φ-Colim → Φ-Cocts with U′L = U_{s}. It is convenient to factorize L further into a bijective-on-objects 2-functor J and a fully-faithful one M, as in

\xymatrix{Φ-Colim \ar[r]^{J} & Φ-Cocts_{c} \ar[r]^{M} & Φ-Cocts;}

here the objects of Φ-Cocts_{c} are, like those of Φ-Colim, V-categories with a choice of Φcolimits (the subscript c standing for “choice”), while the morphisms are, as in Φ-Cocts, merely the Φ-cocontinuous V-functors, with the 2-cells as before. The 2-functor M is an equivalence in the weaker sense of being fully faithful and essentially surjective on objects, since it is actually surjective on objects; it is therefore an equivalence in the stronger sense of satisfying MN ≅ 1 and NM ≅ 1 for some N if we suppose the axiom of choice to hold. In fact our argument does not need the axiom of choice; it suffices to suppose that a definite choice of Φ-colimits has been made in the cocomplete V-category V. We shall now write U for U′M : Φ-Cocts_{c} → V-Cat, so that UJ = U_{s}; it is this last relation between U_{s} and U that is central to our argument. Note that the 2-functor J, besides being bijective on objects, is faithful and locally fully faithful.

Recall from [9, Section 5.7] that U′ has a left biadjoint Φ′, whose value Φ′C at a small V-category C is the closure of the representables in [C^{op}, V] under Φ-colimits, the unit y : C → Φ′C being the Yoneda embedding seen as landing in Φ′C. That is to say, composition with y gives an equivalence of categories Φ-Cocts(Φ′C, A) → V-Cat(C, U′A). We shall use the existence and properties of Φ′C to construct a left adjoint to U_{s}. Of course the more usual name for Φ′C is ΦC; but here we want to use ΦC for the object of Φ-Cocts_{c} consisting of the Φ-cocomplete V-category Φ′C together with some definite choice of Φ-colimits. This is no problem: we are supposing definite Φ-colimits to be chosen in V, and we now choose Φ-colimits in [C^{op}, V] to be the pointwise ones; then Φ′C is, being replete by definition, closed under these in [C^{op}, V]. So now y : C → ΦC is the unit for a left biadjoint to U, composition with y giving an equivalence of categories Φ-Cocts_{c}(ΦC, A) → V-Cat(C, UA).

Given a 2-category K, a (fixed) class M of arrows in K, and an object A of K, we write K ↓ A for the 2-category whose objects are arrows m : B → A in M with codomain A, whose arrows are commutative triangles, as in

\xymatrix{B \ar[rr]^{f} \ar[dr]_{m} &  & B′ \ar[dl]^{m′} \\  & A & ,}

and whose 2-cells from f : (B, m) → (B′, m′) to g : (B, m) → (B′, m′) are the 2-cells α : f → g in K satisfying m′α = id_{m}; thus K ↓ A is a full sub-2-category of the slice 2-category K/A.

Now consider the 2-categories V-Cat ↓ ΦC and Φ-Colim ↓ ΦC where in each case the class M of arrows consists of those which are fully faithful as V-functors. We shall see that the evident forgetful 2-functor U_{ΦC} : Φ-Colim ↓ ΦC → V-Cat ↓ ΦC has a left adjoint; the value at (y : C → ΦC) of this left adjoint has the form (w : F C → ΦC), where F C has chosen Φ-colimits, strictly preserved by w. It is this F C which turns out to be the value at C of the left adjoint F to U_{s} : Φ-Colim → V-Cat.

In fact this technique clearly applies more generally than to the study of the 2-functor U_{s} : Φ-Colim → V-Cat. We consider the general context given by a diagram

\xymatrix{𝔸_{s} \ar[rr]^{J} \ar[dr]_{U_{s}} &  & 𝔸 \ar[dl]^{U} \\  & ℂ}

of 2-categories and 2-functors, with J bijective on objects, faithful, and locally fully faithful, and describe conditions under which a left biadjoint to U may be used to construct a left adjoint to U_{s}.

The outline of the paper is as follows. We start by recalling the basic facts about the free cocompletions ΦC — we shall now drop the notation Φ′C — and about the pseudolimit of an arrow. We then prove Lemma 4.1 reducing, under suitable hypotheses, the problem of finding a left adjoint to U_{s} in the abstract context above to that of finding a left adjoint to U_{ΦC}. In Section 5, we show that these hypotheses are satisfied in our case of 𝔸_{s} = Φ-Colim, concluding in Theorem 5.1 that our U_{s} has a left adjoint. In Section 6 we prove that this U_{s} is actually monadic, and furthermore that the pseudomorphisms for the resulting 2-monad are precisely the Φ-cocontinuous V-functors. In Section 7 we turn to the case of a large class Φ of weights; and in Section 8 we give some further applications of our main abstract result, Lemma 4.1.

2. Background material on free cocompletions

In this section we recall without proof the main facts about free cocompletions; all can be found in [9, Section 5.7], except the last two, which appear in [2, Section 4].

Let C be an arbitrary V-category. If C is small, we can of course form the V-category [C^{op}, V] of V-functors from C^{op} to V; its hom-objects are formed as certain (small) limits in the underlying ordinary category V_{0} of V. Specifically, for V-functors f and g from C^{op} to V (henceforth called presheaves on C), we define

[C^{op}, V](f, g) = ∫_{c∈C}[fc, gc]

where [fa, ga] denotes the internal hom in V.

If C is not small, the above limit will not exist in general, but it will exist provided that f is small, in the sense that it may be formed as the left Kan extension of its restriction to some small full subcategory of C^{op}; small functors have also been called accessible [9]. We may now, following Lindner [15], define the V-category PC, whose objects are the small presheaves on C, and whose hom-objects are defined by the limit above. Representable presheaves are easily seen to be small, and so we have a (fully faithful) Yoneda embedding y_{C} : C → PC; we often abbreviate y_{C} to y. Of course PC is just [C^{op}, V] if C is small.

2.1. Theorem. PC is the free cocompletion of C; that is, PC is cocomplete, and for any cocomplete V-category A, composition with y : C → PC induces an equivalence of categories between V-Cat(C, A) and the full subcategory of V-Cat(PC, A) consisting of the cocontinuous V-functors. Furthermore, a V-functor f : C → A corresponds under this equivalence to the left Kan extension Lan_{y}f of f along y.

We can now use PC to form free cocompletions with respect to a general class of colimits Φ. Let Φ be a class of colimits; we do not assume Φ to be small, but we do as always assume that for each ϕ : D_{ϕ}^{op} → V in Φ, the domain D_{ϕ} is small.

For any V-category C, by a free Φ-cocompletion of C we mean a Φ-cocomplete Vcategory \overline{C} and a V-functor y : C → \overline{C} such that for any Φ-cocomplete V-category A, composition with y induces an equivalence of categories between V-Cat(C, A) and the full subcategory of V-Cat(\overline{C}, A) consisting of the Φ-cocontinuous V-functors.

2.2. Theorem. If ΦC is the closure in PC of the representables under Φ-colimits, then the restricted Yoneda embedding y : C → ΦC exhibits ΦC as the free Φ-cocompletion of C; furthermore the Φ-cocontinuous V-functor ΦC → A corresponding to an arbitrary V-functor f : C → A is the left Kan extension Lan_{y}f of f along y.

We also need:

2.3. Theorem. If Φ is a small class of weights, then ΦC is small if C is so.

Finally we record:

2.4. Theorem. If A admits Φ-colimits, then the essentially unique Φ-cocontinuous Vfunctor a : ΦA → A with ay_{A} ≅ 1 is left adjoint to y_{A}, with the isomorphism ay_{A} ≅ 1 as counit.

2.5. Theorem. Consider a V-functor f : A → B where A and B admit Φ-colimits. Write a : ΦA → A and b : ΦB → B for the essentially unique Φ-cocontinuous V-functors with ay_{A} ≅ 1 and by_{B} ≅ 1, and write g : ΦA → ΦB for the essentially unique Φcocontinuous V-functor satisfying gy_{A} ≅ y_{B}f. Then f is Φ-cocontinuous if and only if there is a V-natural isomorphism fa ≅ bg.

3. Background material on pseudolimits of an arrows

We recall the notion of the pseudolimit of an arrow [10]. Let K be a 2-category, and f : A → B an arrow in K. A pseudolimit of the arrow f is an invertible 2-cell

[figure: A; f; L; u; v; ⇑; λ; B]

in K which is universal: that is, given arrows u′ : L′ → A and v′ : L′ → B and an invertible 2-cell λ′ : v′ → fu′, there is a unique arrow x : L′ → L satisfying λx = λ′ (and so in particular ux = u′ and vx = v′); and furthermore given 2-cells β : vx → vy and α : ux → uy satisfying

[figure: L; u; A; f; L; u; A; f; ⇑; α; L′; y; x; y; L; u; v; ⇑; λ; =; L′; y; x; x; L; u; v; ⇑; λ; ⇑; β; L; v; B; L; v; B]

there is a unique 2-cell ξ : x → y satisfying uξ = α and vξ = β.

In the case K = V-Cat, the pseudolimit of an arrow f : A → B has a very simple description. An object of L consists of objects a and b of A and B, and an isomorphism φ : b → fa in B; while the V-valued-hom L((a, b, φ), (a′, b′, φ′)) is given by A(a, a′), and composition is defined as in A. We leave to the reader the (obvious) definitions of u, v, and λ; as well as the verification of the universal property.

It is clear from this description that the projection u in the pseudolimit of f is an equivalence, at least in the case K = V-Cat, but in fact this is true in any 2-category K. We could prove this by observing that it is true in Cat, and that the notions of equivalence and of the pseudolimit of an arrow are both defined representably, but we choose instead to give an abstract 2-categorical proof:

3.1. Lemma. If

[figure: A; f; L; u; v; ⇑; λ; B]

is the pseudolimit of f, then u is an equivalence.

Proof. The arrows 1_{A} : A → A and f : A → B and the identity 2-cell from f to f1_{A} induce a unique arrow s : A → L satisfying us = 1, vs = f, and λs = id_{f}. On the other hand the 2-cells id_{u} : u1 → usu and λ : v → fu = vsu induce a unique 2-cell σ : 1 → su satisfying uσ = 1 and vσ = λ, which is invertible since id_{u} and λ are so. Thus su ≅ 1 and us = 1, giving the desired equivalence.

4. The main lemma

We suppose given 2-categories 𝔸_{s}, 𝔸, and ℂ, 2-functors U : 𝔸 → ℂ and J : 𝔸_{s} → 𝔸 with J bijective on objects, faithful, and locally fully faithful, and a left biadjoint Φ to U with unit y : 1 → UΦ. Write U_{s} : 𝔸_{s} → ℂ for UJ; of course at this level of generality there is no reason why U_{s} should have a left biadjoint, let alone a left adjoint, but we shall provide conditions under which U_{s} does indeed have a left adjoint.

A special case is that where 𝔸_{s} is the 2-category Φ-Colim for a small class Φ = {ϕ : D_{ϕ}^{op} → V} of weights, 𝔸 is Φ-Cocts_{c}, J : Φ-Colim → Φ-Cocts_{c} is the inclusion, and U : Φ-Cocts_{c} → V-Cat is the forgetful 2-functor; of course the left biadjoint to U takes A to ΦA. Henceforth this special case will be called the MAIN EXAMPLE.

We further suppose given a class M of arrows in ℂ, containing the equivalences and the components y_{C} of y, and satisfying the property that if mf ≅ 1 and m ∈ M then fm ≅ 1. Since mf ≅ 1 implies mfm ≅ m1, the latter condition will be satisfied if mx ≅ my implies that x ≅ y; this in turn is clearly the case if each m is representably fully faithful in ℂ, in the sense that the functor ℂ(C, m) is fully faithful for each C in ℂ. In the MAIN EXAMPLE, M will be the class of fully faithful V-functors, which are indeed representably fully faithful.

We recall the notation ℂ ↓ UΦC defined in the Introduction, and we shall also consider 𝔸_{s} ↓ ΦC, and the 2-functor U_{ΦC} : 𝔸_{s} ↓ ΦC → ℂ ↓ UΦC induced by U_{s}; here the chosen arrows in 𝔸_{s} are those whose image under U_{s} lie in M, and so we shall often write f ∈ M to mean U_{s}f ∈ M.

4.1. Lemma. [Main Lemma] Let 𝔸_{s}, 𝔸, ℂ, and M be as above, and suppose that the following conditions are satisfied:

(Ax1) Any arrow f in 𝔸 for which Uf is an equivalence is itself an equivalence;

(Ax2) 𝔸 has, and U preserves, pseudolimits of arrows; furthermore if u : L → A and v : L → B are the projections for the pseudolimit of an arrow f : A → B in 𝔸, then u and v lie in 𝔸_{s}, and an arrow x : C → L in 𝔸 lies in 𝔸_{s} if and only ux and vx do so;

(Ax3) U_{A} : 𝔸_{s} ↓ A → ℂ ↓ A has a left adjoint F_{A} for every object A in 𝔸_{s}.

Then U_{s} has a left adjoint F whose value at C is the object F C appearing in F_{ΦC}(y : C → ΦC) = (w : F C → ΦC); furthermore Jw is an equivalence.

Proof. We shall often identify objects, arrows, and 2-cells of 𝔸_{s} with their images under J. The left adjoint F_{ΦC} sends y : C → U_{s}ΦC to w : F C → ΦC; writing z for the y-component of the unit for this adjunction, we have a commutative triangle

\xymatrix{U_{s}F C \ar[rr]^{U_{s}w} &  & U_{s}ΦC \\  & C \ar[ul]^{z} \ar[ur]_{y}}

in which w and y lie in M. By the universal property of ΦC, there is an (essentially unique) arrow w′ : ΦC → F C in 𝔸 with an isomorphism θ : Uw′.y ≅ z in ℂ. Now U(Jw.w′).y ≅ U_{s}w.z = y, and so by the universal property of ΦC once again, there is an isomorphism Jw.w′ ≅ 1. Thus UJw.Uw′ ≅ 1 and UJw ∈ M, and so Uw′.UJw ≅ 1 by the hypotheses on M; whence UJw is an equivalence in ℂ. Thus Jw is an equivalence in 𝔸 by (Ax1), which implies in particular that (JF C, z) “has the same universal property as (ΦC, y)”.

We shall now show that z : C → U_{s}F C exhibits F C as the free object on C with respect to U_{s}. Suppose then that an arrow f : C → U_{s}B is given. By the universal property of ΦC we can find an arrow g : ΦC → B in 𝔸 and an isomorphism ζ : f ≅ Ug.y. Now form the pseudolimit

[figure: ΦC; g; L; u; v; ⇑; λ; B]

in 𝔸. Since U preserves this pseudolimit by (Ax2), the isomorphism ζ : f → Ug.y induces a unique arrow h : C → UL in ℂ satisfying Uu.h = y, Uv.h = f, and Uλ.h = ζ.

Now u is an equivalence in 𝔸 by Lemma 3.1; thus Uu is an equivalence and so Uu lies in M. Since y : C → UΦC also lies in M, we can see h as an arrow h : (y : C → UΦC) → (Uu : UL → UΦC) in ℂ ↓ ΦC. The adjunction F_{ΦC} ⊣ U_{ΦC} therefore gives a unique arrow k : (w : F C → ΦC) → (u : L → ΦC) in 𝔸_{s} ↓ ΦC satisfying Uk.z = h. Now vk : F C → B lies in 𝔸_{s} since v and k do so, and U(vk).z = Uv.Uk.z = Uv.h = f, giving the existence part of the one-dimensional aspect of the universal property making F left adjoint to U_{s}.

As for the uniqueness, suppose that \overline{f} : F C → B in 𝔸_{s} satisfies U\overline{f}.z = f. Since F C shares the universal property of ΦC, the isomorphism ζ : U\overline{f}.z = f ≅ Ug.y = Ug.Uw.z = U(gw).z is of the form U\overline{ζ}.z for a unique isomorphism \overline{ζ} : \overline{f} → gw in 𝔸. Now by the definition of pseudolimit, there is a unique arrow k′ : F C → L in 𝔸 satisfying uk′ = w, vk′ = \overline{f}, and λk′ = \overline{ζ}; furthermore uk′ and vk′ lie in 𝔸_{s}, hence so too by (Ax2) does k′. Finally Uλ.Uk′.z = U\overline{ζ}.z = ζ = Uλ.h and so Uk′.z = h, giving k = k′; whence \overline{f} = vk′ = vk, which is the desired uniqueness.

This completes the proof of the one-dimensional aspect of the universal property of the left adjoint; and the two-dimensional aspect is immediate, since F C is already known to share the universal property of ΦC.

5. Verification of the axioms in the MAIN EXAMPLE

In the MAIN EXAMPLE, (Ax1) is obviously satisfied, and it is also not hard to see that (Ax2) is satisfied: for given the pseudolimit L of a V-functor f : A → B in the notation of Section 3, and given a V-functor s : D_{ϕ} → L, we may choose the colimit ϕ ∗ s in L to be the object (ϕ ∗ us, ϕ ∗ vs, β : ϕ ∗ vs → f(ϕ ∗ us)) of L, where β is the composite of ϕ ∗ λs : ϕ ∗ vs → ϕ ∗ fus and the canonical isomorphism ϕ ∗ fus ≅ f(ϕ ∗ us). The straightforward verifications are left to the reader.

The key step therefore involves (Ax3). We begin by observing that the 2-categories Φ-Colim ↓ A and V-Cat ↓ A are locally chaotic, by which is meant that there is a unique 2-cell between any parallel pair of arrows. It will therefore suffice to prove that the ordinary functor (U_{A})_{0} : (Φ-Colim ↓ A)_{0} → (V-Cat ↓ A)_{0} has a left adjoint, for each V-category A with chosen Φ-colimits; here (Φ-Colim ↓ A)_{0} and (V-Cat ↓ A)_{0} denote the ordinary categories underlying Φ-Colim ↓ A and V-Cat ↓ A. To do this, we shall construct an endofunctor E of (V-Cat ↓ A)_{0} for which (Φ-Colim ↓ A)_{0} is the category of algebras, and then prove that free E-algebras exist.

Given an object (B, m : B → A) of (V-Cat ↓ A)_{0}, we write (EB)_{0} for the set \{(ϕ, s)|ϕ ∈ Φ, s : D_{ϕ} → B\}, seen as a discrete V-category, and \overline{m}_{0} : (EB)_{0} → A for the V-functor taking (ϕ, s) to ϕ ∗ ms. We now factorize \overline{m}_{0} as

\xymatrix{(EB)_{0} \ar@{|->}[r]^{e} & EB \ar@{^{(}->}[r]^{\overline{m}} & A}

where e is bijective on objects and \overline{m} is fully faithful; recall that the bijective-on-objects V-functors and the fully faithful ones constitute a factorization system on V-Cat_{0}, the arrows of which we decorate as in the preceding diagram.

Given an arrow f : (B, m) → (B′, m′) in (V-Cat ↓ A)_{0} we now write (Ef)_{0} : (EB)_{0} → (EB′)_{0} for the V-functor taking (ϕ, s) to (ϕ, fs); since (\overline{m′})_{0}(Ef)_{0}(ϕ, s) = (\overline{m′})_{0}(ϕ, fs) = ϕ ∗ m′fs = ϕ ∗ ms = (\overline{m})_{0}(ϕ, s), we have (\overline{m′})_{0}(Ef)_{0} = \overline{m}_{0}, so that there is a unique V-functor Ef rendering commutative

\xymatrix{(EB)_{0} \ar@{|->}[r]^{e} \ar[dd]_{(Ef)_{0}} & EB \ar@{^{(}->}[dr]^{\overline{m}} \ar[dd]_{Ef} \\  &  & A \\ (EB′)_{0} \ar@{|->}[r]_{e′} & EB′ \ar@{^{(}->}[ur]_{\overline{m′}} & .}

We now define E to be the endofunctor of (V-Cat ↓ A)_{0} taking (B, m) to (EB, \overline{m}) and f : (B, m) → (B′, m′) to Ef.

To give to an object (B, m) of (V-Cat ↓ A)_{0} the structure of an E-algebra is to give a V-functor b : EB → B satisfying mb = \overline{m}. This determines a V-functor b_{0} = be satisfying mb_{0} = mbe = \overline{m}e; but since e is bijective on objects and m is fully faithful, such a b_{0} equally determines b, so that to give to (B, m) the structure of an E-algebra is just to give a V-functor b_{0} : (EB)_{0} → B satisfying mb_{0} = (\overline{m})_{0}. This, however, is just to give, for each ϕ ∈ Φ and each s : D_{ϕ} → B, an object ϕ ⋄ s of B satisfying m(ϕ ⋄ s) = ϕ ∗ ms. Finally m is fully faithful and so reflects colimits, whence ϕ ⋄ s must be a ϕ-weighted colimit of s; thus we see that an E-algebra structure on (B, m) is precisely a choice in B of Φ-colimits, strictly preserved by m.

Given two such E-algebras (B, m) and (B′, m′) with structure maps b : EB → B and b′ : EB′ → B′, an arrow f : (B, m) → (B′, m′) in (V-Cat ↓ A)_{0} is a morphism of E-algebras just when fb = b′.Ef, which happens if and only if fb_{0} = b′_{0}.(Ef)_{0}; that is, if f(ϕ ∗ s) = ϕ ∗ fs for each ϕ and each s. Thus an arrow in (V-Cat ↓ A)_{0} between E-algebras is a morphism of E-algebras if and only if it (strictly) preserves the chosen Φ-colimits. This now proves that (Φ-Colim ↓ A)_{0} is precisely the category of algebras for E, and (U_{A})_{0} : (Φ-Colim ↓ A)_{0} → (V-Cat ↓ A)_{0} is the forgetful functor.

Thus we have reduced the problem of finding a left adjoint to U_{A} : Φ-Colim ↓ A → V-Cat ↓ A to the problem of showing that free E-algebras exist. This in turn will be the case — see for example [8, Proposition 3.1] and the references contained there — if we can show that (V-Cat ↓ A)_{0} is cocomplete and E preserves α-filtered colimits for some regular cardinal α. Recall that an object c of a cocomplete category K is said to be α-presentable if the representable functor K(c, −) : K → Set preserves α-filtered colimits.

The functor ob : V-Cat_{0} → Set taking a V-category to its set of objects induces a functor ob_{A} : (V-Cat ↓ A)_{0} → Set/obA, and this latter functor is easily seen to be an equivalence. Furthermore the functor ∂_{0} : Set/obA → Set taking a function with codomain obA to its domain creates colimits; so too, therefore, does the composite ob : (V-Cat ↓ A)_{0} → Set of ob_{A} and ∂_{0}. Thus in particular (V-Cat ↓ A)_{0} is cocomplete, and an object (B, m) is α-presentable if and only if B has fewer than α objects.

Each V-category D_{ϕ} is small, as is the class Φ, and so we may choose a regular cardinal α in such a way that for every ϕ ∈ Φ, the V-category D_{ϕ} has fewer than α objects. We shall now show that E preserves α-filtered colimits for this α.

Suppose then that J is an α-filtered category, and H : J → (V-Cat ↓ A)_{0} is a diagram with colimit

\xymatrix{ & B \ar[dr]^{m} \\ B_{j} \ar@{-->}[ur]^{k_{j}} \ar[rr]_{m_{j}} &  & A,}

where we have written m_{j} for the value of H at the object j of J; observe that k_{j} is necessarily fully faithful. We shall show that

\xymatrix{ & EB \ar[dr]^{\overline{m}} \\ EB_{j} \ar@{-->}[ur]^{Ek_{j}} \ar[rr]_{\overline{m_{j}}} &  & A}

exhibits (EB, \overline{m}) as the colimit in (V-Cat ↓ A)_{0} of EH, using the fact that ob : (V-Cat ↓ A)_{0} → Set creates colimits.

If (ϕ, s) is an object of EB then we can factorize s as

\xymatrix{D_{ϕ} \ar@{|->}[r]^{e} & E_{ϕ} \ar@{^{(}->}[r]^{n} & B}

with e bijective on objects and n fully faithful. Now E_{ϕ} has fewer than α objects since D_{ϕ} does so, thus (E_{ϕ}, mn : E_{ϕ} → A) is α-presentable in (V-Cat ↓ A)_{0}. Since J is αfiltered, we may factorize n : (E_{ϕ}, mn) → (B, m) as n = k_{j}n_{j} for some j ∈ J. Thus s = ne = k_{j}n_{j}e, and so (ϕ, s) = (Ek_{j})(ϕ, n_{j}e).

On the other hand, if (ϕ_{j}, s_{j}) ∈ EB_{j} and (ϕ_{i}, s_{i}) ∈ EB_{i} satisfy (Ek_{j})(ϕ_{j}, s_{j}) = (Ek_{i})(ϕ_{i}, s_{i}), then (ϕ_{j}, k_{j}s_{j}) = (ϕ_{i}, k_{i}s_{i}), and so ϕ_{j} = ϕ_{i} and k_{j}s_{j} = k_{i}s_{i}. Since k_{j} and k_{i} are fully faithful, we may factorize s_{j} and s_{i} as s_{j} = n_{j}e and s_{i} = n_{i}e with e bijective on objects and with n_{i} and n_{j} fully faithful. Writing n for mk_{j}n_{j} (=mk_{i}n_{i}) and E_{ϕ} for the domain of n, we now have an α-presentable object (E_{ϕ}, n : E_{ϕ} → A) of (V-Cat ↓ A)_{0} with arrows n_{j} : (E_{ϕ}, n) → (B_{j}, mk_{j}) and n_{i} : (E_{ϕ}, n) → (B_{i}, mk_{i}) satisfying k_{j}n_{j} = k_{i}n_{i}. It follows that there exist arrows ξ : j → h and ζ : i → h in J with (Hξ)n_{j} = (Hζ)n_{i}, whence finally (EHξ)(ϕ_{j}, s_{j}) = (ϕ_{j}, Hξ.s_{j}) = (ϕ_{j}, Hξ.n_{j}e) = (ϕ_{i}, Hζ.n_{i}e) = (ϕ_{i}, Hζ.s_{i}) = (EHζ)(ϕ_{i}, s_{i}), and so E preserves α-filtered colimits as claimed.

This completes the verification of the hypotheses of the main lemma, and we now apply it to obtain:

5.1. Theorem. For a small class Φ of weights, the forgetful 2-functor U_{s} : Φ-Colim → V-Cat has a left adjoint F. Furthermore, if z_{C} : C → U_{s}F C is the unit at C of the adjunction, then z_{C} exhibits F C as the free Φ-cocompletion of C.

6. The monadicity of V-categories with chosen colimits

Now that the 2-functor U_{s} : Φ-Colim → V-Cat is known to have a left adjoint, we shall prove it to be monadic by using Beck’s criterion, in the “strict” form of Mac Lane’s account [16, Theorem VI.7.1]; recall from Dubuc’s thesis [6, Theorem II.2.1] that this applies unchanged to enriched categories, provided that we then understand “coequalizer” in the enriched sense; so that it applies in particular to our 2-categorical case.

We therefore consider in V-Cat a diagram

\xymatrix{A \ar[r]^{f} \ar[r]^{g} & B \ar[r]^{q} \ar[l]^{j} & C \ar[l]^{i}}

satisfying the “split fork” conditions qf = qg, qi = 1, fj = 1, iq = gj; wherein A and B have chosen Φ-colimits strictly preserved by f and g. We are to prove that C admits a unique choice of Φ-colimits for which q strictly preserves Φ-colimits, and that q is the coequalizer of f and g not only in V-Cat but also in Φ-Colim.

There is no difficulty about the uniqueness of the Φ-colimits in C for which q is strictly Φ-cocontinuous: if ϕ : D_{ϕ}^{op} → V is in Φ and s : D_{ϕ} → C, we are obliged to define the colimit ϕ ∗ s in C by

ϕ ∗ s = q(ϕ ∗ is)

with the unit

\xymatrix{ϕ \ar[r]^{η} & B(is, ϕ ∗ is) \ar[r]^{q_{∗}} & C(s, q(ϕ ∗ is)),}

where η is the unit for the colimit ϕ ∗ is and q_{∗} is here short for q_{is,ϕ∗is}, the effect of q on hom-objects; here and elsewhere, we make use without comment of the fact that qis = s.

We must now show that q(ϕ ∗ is), with the unit above, is indeed a colimit ϕ ∗ s in C. That is, we are to prove the invertibility of the V-natural transformation α whose composite α_{c} for c ∈ C is the composite appearing below:

\xymatrix{C(q(ϕ ∗ is), c) \ar[r]^{C(s,−)} & [D_{ϕ}^{op}, V](C(s, q(ϕ ∗ is)), C(s, c)) \ar[d]^{[D_{ϕ}^{op},V](q_{∗},C(s,c))} \\ [D_{ϕ}^{op}, V](ϕ, C(s, c)) & [D_{ϕ}^{op}, V](B(is, ϕ ∗ is), C(s, c)) \ar[l]^{[D_{ϕ}^{op},V](η,C(s,c))}}

We assert that α has the inverse β, whose component β_{c} is the following composite

\xymatrix{[D_{ϕ}^{op}, V](ϕ, C(s, c)) \ar[r]^{[D_{ϕ}^{op},V](ϕ,i_{∗})} & [D_{ϕ}^{op}, V](ϕ, B(is, ic)) \ar[d]^{π} \\ C(q(ϕ ∗ is), c) & B(ϕ ∗ is, ic) \ar[l]^{q_{∗}}}

wherein π denotes the natural isomorphism expressing the universal property of the colimit ϕ ∗ is in B. To ease the burden of writing these long expressions, let us simplify somewhat by writing [X, Y] for [D_{ϕ}^{op}, V](X, Y), and, for instance, writing [q_{∗}, 1] for [D^{op}_{ϕ}, V](q_{∗}, C(s, c)).

We first show that βα = 1. Since the domain of βα is the representable V-functor C(q(ϕ∗is), −), it suffices by the Yoneda lemma to put c = q(ϕ∗is) and to show that β_{c}α_{c} sends the identity 1 ∈ C_{0}(q(ϕ ∗ is), q(ϕ ∗ is)) to itself, where C_{0} is the ordinary category underlying the V-category C. The composite [1, i_{∗}][η, 1][q_{∗}, 1]C(s, −) sends 1_{q(ϕ∗is)} to the top leg of the diagram

\xymatrix{ϕ \ar[r]^{η} \ar[d]_{η} & B(is, ϕ ∗ is) \ar[r]^{q_{∗}} & C(s, q(ϕ ∗ is)) \ar[d]^{i_{∗}} \\ B(is, ϕ ∗ is) \ar[rr]_{B(is,τ)} &  & B(is, iq(ϕ ∗ is)),}

which, since η is the unit for the colimit ϕ ∗ is, is of the form B(is, τ)η for a unique τ : ϕ ∗ is → iq(ϕ ∗ is), as shown in the diagram. In fact this τ is just what we obtain by applying π to the top leg of the diagram, so that finally we have

β_{c}α_{c}(1) = qτ : q(ϕ ∗ is) → qiq(ϕ ∗ is) = q(ϕ ∗ is);

and it remains to show that qτ = 1.

In fact τ is precisely the canonical comparison morphism

\xymatrix{ϕ ∗ is = ϕ ∗ iqis \ar[r]^{τ} & iq(ϕ ∗ is),}

and since iq = gj, this is equally the canonical comparison morphism ϕ∗gjis → gj(ϕ∗is), which in turn is the composite

\xymatrix{ϕ ∗ gjis \ar[r]^{\overline{g}} & g(ϕ ∗ jis) \ar[r]^{g\overline{j}} & gj(ϕ ∗ is) ,}

where \overline{g} is the canonical comparison morphism associated to g, and \overline{j} that associated to j. But \overline{g} = 1, since g preserves Φ-colimits strictly; so that finally τ = g\overline{j}, and qτ = qg\overline{j}; which is equally qf\overline{j}. Now if \overline{f} is the canonical comparison morphism associated to f, that associated to fj is the composite

\xymatrix{ϕ ∗ fjis \ar[r]^{\overline{f}} & f(ϕ ∗ jis) \ar[r]^{f\overline{j}} & fj(ϕ ∗ is),}

which is the identity since fj = 1; while \overline{f} = 1 since f preserves ϕ-colimits strictly. It follows that f\overline{j} = 1, whence qf\overline{j} = 1; giving qτ = 1, as desired, and so βα = 1.

The proof that αβ = 1 must be more direct, since now the domain is no longer representable. A first simplification arises as follows: that the V-functor q respects composition is expressed by the commutativity of

\xymatrix{B(ϕ ∗ is, ic) ⊗ B(is, ϕ ∗ is) \ar[r] \ar[d]_{q_{∗}⊗q_{∗}} & B(is, ic) \ar[d]^{q_{∗}} \\ C(q(ϕ ∗ is), c) ⊗ C(s, q(ϕ ∗ is)) \ar[r] & C(s, c),}

whose transpose under the tensor-hom adjunction is the commutative diagram

\xymatrix{B(ϕ ∗ is, ic) \ar[r]^{q_{∗}} \ar[d]_{B(is,−)} & C(q(ϕ ∗ is), c) \ar[r]^{C(s,−)} & [C(s, q(ϕ ∗ is)), C(s, c)] \ar[d]^{[q_{∗},1]} \\ [B(is, ϕ ∗ is), B(is, ic)] \ar[rr]_{[1,q_{∗}]} &  & [B(is, ϕ ∗ is), C(s, c)].}

Accordingly, in the composite

α_{c}β_{c} = [η, 1][q_{∗}, 1]C(s, −)q_{∗}π[1, i_{∗}],

we can replace [q_{∗}, 1]C(s, −)q_{∗} by [1, q_{∗}]B(is, −); and since we can then trivially replace [η, 1][1, q_{∗}] by [1, q_{∗}][η, 1], each being [η, q_{∗}], we get

α_{c}β_{c} = [1, q_{∗}][η, 1]B(is, −)π[1, i_{∗}].

Now the composite [η, 1]B(is, −)π here is just the identity of [ϕ, B(is, ic)], since η is by definition the unit of the representation π; moreover [1, q_{∗}][1, i_{∗}] = 1 since qi = 1; so that we do indeed have α_{c}β_{c} = 1, or αβ = 1.

So q(ϕ ∗ is) does provide a ϕ-colimit of s in C; let us write ϕ ∗ s = q(ϕ ∗ is), it being understood here and below that such an equation asserts the equality not only of the objects but also of the respective units. We are next to show that q does indeed strictly preserve Φ-colimits; but for r : D_{ϕ} → B we have

q(ϕ ∗ r) = q(ϕ ∗ fjr)
= qf(ϕ ∗ jr)
= qg(ϕ ∗ jr)
= q(ϕ ∗ gjr)
= q(ϕ ∗ iqr)
= ϕ ∗ qr.

Finally we must show that q is the coequalizer of f and g in Φ-Colim. If E is an object of Φ-Colim and r : C → E is such that rq preserves Φ-colimits strictly, then r too preserves them strictly; for if s : D_{ϕ} → C we have

r(ϕ ∗ s) = rq(ϕ ∗ is) = ϕ ∗ rqis = ϕ ∗ rs.

Thus q is certainly the coequalizer of f and g in the underlying ordinary category Φ-Colim_{0} of the 2-category Φ-Colim. To show that q is the coequalizer of f and g in Φ-Colim, we use the following argument, based on the existence in Φ-Colim of cotensors with the arrow-category 2 = {0 → 1}; see [9, Section 3.8] for the general principle behind it.

To abbreviate, we temporarily introduce the notation C for Φ-Colim and C_{0} for Φ-Colim_{0}. For an object E of C, consider the functor category [2, E]; this becomes an object of C when we give it the Φ-colimits formed pointwise from those in E, and then the evaluations ∂_{0}, ∂_{1} : [2, E] → E strictly preserve Φ-colimits. To give a functor h : X → [2, E] is to give two functors h_{0}, h_{1} : X → E and a natural transformation λ : h_{0} → h_{1}; and h is a morphism in C precisely when each of h_{0} and h_{1} is so. Accordingly we have a natural bijection

C_{0}(X, [2, E]) ≅ Cat_{0}(2, C(X, E)),  (∗)

where Cat_{0} is the ordinary category underlying the 2-category Cat.

Now since q is the coequalizer in C_{0}, we have in Set the equalizer

\xymatrix{C_{0}(C, [2, E]) \ar[r]^{C_{0}(q,1)} & C_{0}(B, [2, E]) \ar[r]^{C_{0}(f,1)} \ar[r]_{C_{0}(g,1)} & C_{0}(A, [2, E]),}

which by (∗) is also an equalizer

\xymatrix{Cat_{0}(2, C(C, E)) \ar[r]^{Cat_{0}(1,C(q,1))} & Cat_{0}(2, C(B, E)) \ar[r]^{Cat_{0}(1,C(f,1))} \ar[r]_{Cat_{0}(1,C(g,1))} & Cat_{0}(2, C(A, E)).}

But Cat_{0}(2, −) : Cat_{0} → Set preserves equalizers, and also reflects them, since it reflects isomorphisms; and so

\xymatrix{C(C, E) \ar[r]^{C(q,1)} & C(B, E) \ar[r]^{C(f,1)} \ar[r]_{C(g,1)} & C(A, E)}

is an equalizer in Cat_{0} for all C; which is what it means for q to be the coequalizer of f and g in C. This completes the proof of:

6.1. Theorem. If Φ is a small class of weights, then the 2-functor U_{s} : Φ-Colim → V-Cat is monadic.

We have used z : 1 → U_{s}F for the unit of the (2-)adjunction F ⊣ U_{s}; let us use e : F U_{s} → 1 for the counit. This adjunction determines on V-Cat the 2-monad T = (T, z, m), where T = U_{s}F, where z : 1 → T is the z : 1 → U_{s}F above, and where m : T^{2} → T is U_{s}eF : U_{s}F U_{s}F → U_{s}F. Recall from the study [3] of 2-monads that it is convenient to use the name T -Alg_{s} for the Eilenberg-Moore 2-category given by the T -algebras (A, a : T A → A), the strict T -algebra morphisms (or just strict T -morphisms) f : (A, a) → (B, b), which are those f : A → B with fa = b.T f, and the T -transformations α : f → g : (A, a) → (B, b), which are those for which αa = b.T α; this leaves available the name T -Alg for the related 2-category whose arrows from (A, a) to (B, b), called T -algebra morphisms or just T -morphisms, are pairs (f, \overline{f}) where f : A → B is a V-functor and \overline{f} is an invertible V-natural transformation \overline{f} : b.T f → fa satisfying the usual two coherence conditions as in [3].

Theorem 6.1 asserts that the canonical comparison 2-functor K : Φ-Colim → T -Alg_{s} is invertible. This K sends the object A of Φ-Colim to the T -algebra (U_{s}A, U_{s}eA : U_{s}F U_{s}A → U_{s}A). Here U_{s}A is A itself, seen as merely a V-category, and U_{s}eA is then an action a : T A → A. Since K is invertible, to give an action a : T A → A of T on A is equally to give a choice of Φ-colimits in A — which is of course possible only when A is Φ-cocomplete. Again, K sends a morphism f : A → B in Φ-Colim to this same f seen as a strict T -morphism f : (A, a) → (B, b); thus, since K is invertible, a V-functor f : A → B preserves the chosen Φ-colimits strictly precisely when it satisfies fa = b.T f. As a final consequence of the invertibility of K, every V-natural α : f → g between morphisms in Φ-Colim is automatically a T -transformation. It follows that we may actually, for convenience, identify Φ-Colim and T -Alg_{s} without harm; and to make the identification more useful, it is helpful to describe the action a : T A → A of the T -algebra KA = (A, a) without explicit mention of the 2-monad T. In fact a : T A → A, as we said above, is U_{s}eA : U_{s}F U_{s}A → U_{s}A, so that it may be seen as the morphism a = eA : T A → A of Φ-Colim; and it satisfies the unit axiom az_{A} = 1. Accordingly, since z_{A} : A → T A is the unit z_{A} : A → U_{s}F A of the adjunction F ⊣ U_{s} of Theorem 5.1, we can characterize a : T A → A as the unique V-functor strictly preserving the chosen Φ-colimits and satisfying az_{A} = 1.

Let us turn now to the 2-category Φ-Cocts_{c}, which has the same objects as Φ-Colim, and consider a morphism f : A → B therein — that is, a Φ-cocontinuous V-functor. Write (A, a) and (B, b) as above for the associated T -algebras, and consider the two V-functors b.T f and fa from T A to B. Here f preserves Φ-colimits, while a and b do so strictly, as we have just seen, and T f too does so strictly, since as U_{s}F f it underlies the morphism F f of Φ-Colim. Recalling from Theorem 5.1 that z_{A} : A → T A = U_{s}F A also exhibits F A (which is just T A seen as a Φ-cocomplete V-category) as the free Φ-cocompletion of A in the sense of Section 2, we see that the equality b.T f.z_{A} = bz_{B}f = f = faz_{A} implies the existence of a unique invertible 2-cell \overline{f} of the form

[figure: T A; a; T f; ⇒; \overline{f}; A; f; T B; b; B]

whose composite with z_{A} : A → T A is the identity. Of course, by the last paragraph f is an identity if and only if f strictly preserves the chosen Φ-colimits. Recall from [3] that the pair (f, \overline{f}) is said to be a T -morphism if \overline{f} also satisfies

[figure: T^{2}A; m_{A}; T^{2}f; T A; a; T f; \overline{f}; ⇒; A; f; =; T^{2}A; T a; T^{2}f; T \overline{f}; ⇒; T A; a; T f; \overline{f}; ⇒; A; f; T^{2}B; m_{B}; T B; b; B; T^{2}B; T b; T B; b; B.]

Since this equation expresses the equality of two 2-cells between Φ-cocontinuous Vfunctors with codomain T^{2}A, and since z_{T A} : T A → T^{2}A expresses T^{2}A as the free Φ-cocompletion of T A, it holds if the two composites with z_{T A} coincide; however an easy calculation using naturality and \overline{f}.z_{A} = id shows each composite with z_{T A} to be \overline{f} itself.

Again, if α : f → g : A → B is a V-natural transformation between Φ-cocontinuous V-functors, we have the equality

[figure: T A; a; T g; T f; T α; A; g; \overline{g}; =; T A; a; T f; \overline{f}; A; g; f; α; T B; b; B; T B; b; B,]

since each has the same composite with z_{A} : A → T A; that is to say, α is a T - transformation α : (f, \overline{f}) → (g, \overline{g}) : (A, a) → (B, b) in the sense of [3]. Writing as there T -Alg for the 2-category of T -algebras, T -morphisms, and T -transformations, we have exhibited a 2-functor K′ : Φ-Cocts_{c} → T -Alg which extends K : Φ-Colim → T -Alg_{s}. In fact:

6.2. Theorem. The above 2-functor K′ : Φ-Cocts_{c} → T -Alg extending the isomorphism K : Φ-Colim → T -Alg_{s} is itself an isomorphism of 2-categories.

Proof. It remains only to show that K′ is bijective on morphisms. In fact, for objects A and B of Φ-Cocts_{c}, giving rise to the T -algebras (A, a) and (B, b), any V-functor f : A → B having fa ≅ b.T f is Φ-cocontinuous by Theorem 2.5; so that any T -morphism f : (A, a) → (B, b) is the image under K′ of a unique morphism A → B in Φ-Cocts_{c}, namely f itself.

Recall from [11, Theorem 6.2] that a 2-monad T = (T, z, m) on a 2-category K is lax-idempotent when for each T -algebra (A, a : T A → A) there is an adjunction a ⊣ z_{A} with identity counit; this is also called [14] the Kock-Zöberlein property. That this is the case here follows from Theorem 2.4, since z_{A} : A → T A is equivalent to y : A → ΦA, and since a : T A → A is Φ-cocontinuous with az_{A} = 1. Thus:

6.3. Theorem. The 2-monad T whose Eilenberg-Moore objects is U_{s} : Φ-Colim → V-Cat is lax-idempotent.

Before ending this section we make some observations on changing the class Φ, along with a kind of warning. Given two classes Φ and Ψ of weights with Φ ⊆ Ψ, we have an evident forgetful 2-functor P′′ : Ψ-Cocts → Φ-Cocts over V-Cat (the words “over V-Cat” meaning that P′′ commutes with the forgetful 2-functors Φ-Cocts → V-Cat and Ψ-Cocts → V-Cat), which underlies a forgetful P′ : Ψ-Cocts_{c} → Φ-Cocts_{c}, which in turn restricts to a forgetful P : Ψ-Colim → Φ-Colim, all over V-Cat. It may be the case that P′′ is an equality of 2-categories: that is, every Φ-cocomplete V-category is Ψ-cocomplete and every Φ-cocontinuous V-functor is Ψ-cocontinuous. For a given Φ there is clearly a greatest Ψ with this property, namely the class Φ^{∗} consisting of those weights ψ : D_{ψ}^{op} → V such that every Φ-cocomplete V-category is ψ-cocomplete and every Φ-cocontinuous V-functor is ψ-cocontinuous; and an explicit description of Φ^{∗} was given by Albert and Kelly [2], who showed that ψ ∈ Φ^{∗} if and only if the object ψ of [D_{ψ}^{op}, V] lies in the closure Φ(D_{ψ}^{op}) of the representables under Φ-colimits.

We want to consider now the case where Φ ⊆ Ψ ⊆ Φ^{∗}, so that Ψ-Cocts = Φ-Cocts and the forgetful Ψ-Cocts_{c} → Φ-Cocts_{c} is an equivalence over V-Cat. For an example when V = Set, we may take Φ to consist of the weights for the initial object, binary coproducts, and coequalizers, while Ψ consists of the weights Δ1 : J^{op} → Set for all finite categories J; so that Φ-Colim consists of categories with chosen initial object, binary coproducts, and coequalizers, while Ψ-Colim consists of categories with a choice of all finite colimits in the usual sense. (Here Φ^{∗} consists of all weights ψ : D_{ψ}^{op} → Set which are finitely presentable in [D_{ψ}^{op}, Set], which is much bigger than Ψ, and bigger even than the class of weights Δ1 : J^{op} → Set with J finitely presentable.) The point we want to emphasize is that the forgetful P : Ψ-Colim → Φ-Colim is not in general an equivalence, even though the forgetful P′ : Ψ-Cocts_{c} → Φ-Cocts_{c} is clearly an equivalence, since it overlies the equality P′′ : Ψ-Cocts = Φ-Cocts. In fact, P here is not even fully faithful: there is no reason why a functor strictly preserving the chosen initial object, the chosen binary coproducts, and the chosen coequalizers should strictly preserve, say, the chosen ternary coproducts.

There is, however, a positive result. Using an argument by transfinite induction, Adámek and Kelly prove in the Appendix of the forthcoming article [1] that, when Φ ⊆ Ψ ⊆ Φ^{∗} as above, there is a 2-functor Q : Φ-Colim → Ψ-Colim over V-Cat satisfying P Q = 1; we may use their result as follows. If T and S are the 2-monads on V-Cat corresponding respectively to Φ-Colim and Ψ-Colim, it follows from [13, Proposition 3.4] that P and Q, being 2-functors P : S-Alg_{s} → T -Alg_{s} and Q : T -Alg_{s} → S-Alg_{s} over V-Cat, are of the form ρ^{∗} and σ^{∗} for unique (strict) 2-monad morphisms ρ : T → S and σ : S → T, which moreover satisfy σρ = 1 since P Q = 1. But by [13, (3.2)], ρ and σ also induce ρ^{†} : S-Alg → T -Alg and σ^{†} : T -Alg → S-Alg, with of course ρ^{†}σ^{†} = 1. Here, however, ρ^{†} is the equivalence P′, so that there is an isomorphism α : σ^{†}ρ^{†} ≅ 1. Now it follows from [13, Proposition 3.5] that there is a unique invertible modification θ : ρσ → 1 of 2-monad morphisms having θ^{†} = α, and we conclude that ρ and σ constitute an equivalence T ≃ S of 2-monads, inducing the equivalence between T -Alg and S-Alg, while T -Alg_{s} and S-Alg_{s} remain inequivalent. Of course we expect Φ-Colim to be a stranger creature than Φ-Cocts, but we need it to construct the 2-monad T for which T -Alg = Φ-Cocts_{c}.

7. The case of a large class Φ

We now turn to the case where the class Φ of weights is not small. Our approach was sketched in the Introduction: we have been supposing the existence of an inaccessible cardinal ∞ and saying that a set is small if its cardinality is less than ∞, and we now suppose the existence of a further inaccessible cardinal ∞′ such that the collection of isomorphism classes of small weights, seen as objects of V-Cat/V, has cardinality less than ∞′. One easily sees that it suffices to suppose that the set of isomorphism classes of objects of Set and the set of isomorphism classes of objects of V both have cardinality less than ∞′, and in the typical practical cases the latter cardinality is, like the former, equal to ∞, so that in these cases any inaccessible cardinal greater than ∞ will suffice. We now define a set to be big if its cardinality is less than ∞′, a V-category to be big if its set of objects is big, a V-category to be big-complete if it has all big-limits, and big-cocomplete if it has all big-colimits.

Suppose now that Φ is a big class of small weights. Thus Φ might for example be the class of all small weights; but then, as mentioned in the Introduction, the only cocomplete small categories are preorders. There is little point then in restricting our attention to small V-categories; rather, we seek a monadicity result involving big V-categories, in the sense of the Introduction. Our previous results would carry over unchanged, replacing V-Cat by the 2-category V-CAT of big V-categories, were it not for the fact that V was assumed only to be small-complete and small-cocomplete, not big-complete and bigcocomplete. To deal with this problem we adapt to our present needs the analysis of [9, Section 3.12] for dealing with enrichment in monoidal categories which fail to be complete or cocomplete.

Accordingly we write SET for the category of big sets; then the presheaf category [V_{0}^{op}, SET] has a symmetric monoidal closed structure given by convolution [5], and is big-complete and big-cocomplete, with the limits and colimits computed pointwise. The Yoneda embedding Y : V_{0} → [V_{0}^{op}, SET] preserves the symmetric monoidal closed structure, as well as any limits which exist in V_{0}, but does not preserve colimits. In order to overcome this, we write V′_{0} for the full subcategory of [V_{0}^{op}, SET] consisting of those functors which preserve small limits. This subcategory is reflective, by the results of [7], and so big-complete and big-cocomplete, and the restricted Yoneda embedding Y : V_{0} → V′_{0} preserves small colimits and all existing limits. Furthermore V′_{0} has a symmetric monoidal closed structure, with the tensor product of two objects being the reflection into V′_{0} of their tensor product in [V_{0}^{op}, SET]; we write V′ for V′_{0} with this symmetric monoidal closed structure. The restricted Yoneda embedding Y : V_{0} → V′_{0} also preserves the symmetric monoidal closed structure, and so induces a fully faithful 2-functor

Y_{∗} : V-CAT → V′-CAT

(which we henceforth treat as an inclusion) from V-CAT to the 2-category V′-CAT of big V′-categories; and moreover this inclusion preserves small limits and small colimits.

A small weight, consisting of a V-functor ϕ : D_{ϕ}^{op} → V, which can also be seen as a V′-functor, gives on composition with the V′-functor Y : V → V′ a “V′-weight” Y ϕ : D_{ϕ}^{op} → V′. Thus, for a V′-functor s : D_{ϕ}^{op} → A, we can speak of the colimit (Y ϕ) ∗ s. When, however, the V′-category A is in fact a V-category, the colimit (Y ϕ) ∗ s coincides with the colimit ϕ ∗ s (if either exists). The point is that [D_{ϕ}^{op}, V′](Y ϕ, Y A(s, a)), which is the end in V′ of [ϕd, A(sd, a)]′ (where [−, −]′ is the internal hom in V′), coincides with [D_{ϕ}^{op}, V](ϕ, A(s, a)), since [Y a, Y b]′ ≅ Y [a, b] and since Y : V → V′ preserves limits. Thus it does no harm to speak of (Y ϕ) ∗ s as a “ϕ-colimit”.

It follows, if Φ is a big class of small weights, that we have in the category 2-Cat of 2-categories and 2-functors a pullback diagram

\xymatrix{Φ-COLIM \ar[d]_{U_{s}} \ar[r] & Φ-COLIM′ \ar[d]^{U′_{s}} \\ V-CAT \ar[r]_{Y_{∗}} & V′-CAT,}

where Φ-COLIM′ is the 2-category of big V′-categories with chosen Φ-colimits, V′- functors strictly preserving these, and V′-natural transformations, while Φ-COLIM, as defined in the Introduction, is the 2-category of big V-categories with chosen Φ-colimits, V-functors strictly preserving these, and V-natural transformations.

Now U′_{s} is monadic by Theorem 6.1, so that its pullback U_{s} along the fully faithful Y_{∗} is monadic provided that it has a left adjoint. Write F′ for the left adjoint to U′_{s}, and let A be a big V-category. Although [A^{op}, V] does not exist as a V-category unless A is small, there is no problem forming [A^{op}, V] as a V′-category; it is small-cocomplete since V is so, with colimits being formed pointwise. Now PA is closed in [A^{op}, V] under small colimits by [9, Proposition 5.34], and the V′-functor [A^{op}, V] → [A^{op}, V′] induced by Y : V → V′ preserves small colimits since Y does so; hence PA is closed in [A^{op}, V′] under small colimits and so in particular under Φ-colimits.

Now F′A is equivalent to the closure Φ′A of the representables in [A^{op}, V′] under Φcolimits; and PA is closed in [A^{op}, V′] under Φ-colimits, and contains the representables, and so contains Φ′A. It follows that PA contains Φ′A, and so that Φ′A is a V-category since PA is one. Thus F′A is equivalent to a V-category; but it then follows easily that F′A is isomorphic to a V-category, and this last V-category now gives the value at A of a left adjoint to U_{s}.

This completes the proof of:

7.1. Theorem. The 2-functor U_{s} : Φ-COLIM → V-CAT is monadic, for any big class Φ of weights.

7.2. Remark. Once again, it follows as in Theorem 6.2 and Theorem 6.3 that the 2monad T on V-CAT given by Theorem 7.1 is lax-idempotent, and that the isomorphism K : Φ-COLIM → T -Alg_{s} extends to an isomorphism K′ : Φ-COCTS_{c} → T -Alg.

8. Further applications

In this final section we briefly sketch another application of our results. Write Reg_{c} for the 2-category of small regular categories with chosen finite limits and chosen factorizations, regular functors, and natural transformations; and U : Reg_{c} → Cat for the forgetful 2-functor. Write Reg_{s} for the sub-2-category of Reg_{c} consisting of all the objects, those regular functors which strictly preserve the finite limits and the factorizations, and all the 2-cells between them; and write J : Reg_{s} → Reg_{c} for the inclusion and U_{s} for UJ. We seek a left adjoint to U_{s}.

U is well known to have a left biadjoint, whose value at a category A may be computed by a two-step process: first one forms the free finite-limit-completion of A, and then the free regular category [4] on that; moreover the unit y of the biadjunction is fully faithful. If we take M to be the class of fully faithful functors, then the standing hypotheses of Section 4 are satisfied. (Ax1) and (Ax2) are easily verified, and so once again the crucial step is in proving (Ax3): that U_{R} : Reg_{s} ↓ R → Cat ↓ R has a left adjoint for each object R of Reg_{s}.

The approach is similar to that taken in the case of the MAIN EXAMPLE. Once again Reg_{s} ↓ R and Cat ↓ R are locally chaotic, and so it suffices to show that (U_{R})_{0} : (Reg_{s} ↓ R)_{0} → (Cat ↓ R)_{0} has a left adjoint, and we do this by showing that it is the forgetful functor from the category of algebras for an endofunctor E of (Cat ↓ R)_{0}, and that free E-algebras exist. The earlier argument proves once again the cocompleteness of (Cat ↓ R)_{0}, and so free E-algebras will exist if E preserves α-filtered colimits for some regular cardinal α.

Let |Cat_{f}| be a set of representatives of the isomorphism classes of finite categories. Given an object (B, m : B → R) of (Cat ↓ R)_{0}, let E′B be the category

B^{2} + Σ_{C∈|Cat_{f} |}Cat(C, B)

and let m′ : E′B → R be the evident functor taking k : a → b in B^{2} to the object appearing in the chosen factorization in R of mk, and taking x : C → B to the chosen limit in R of mx : C → R. Factorizing m′ as

\xymatrix{E′B \ar@{|->}[r]^{e} & EB \ar@{^{(}->}[r]^{\overline{m}} & R}

where e is bijective on objects and \overline{m} is fully faithful, one can make (EB, \overline{m}) into the value at (B, m) of an endofunctor E of (Cat ↓ R)_{0}, much as in the case of the MAIN EXAMPLE.

The straightforward verifications that (Reg_{s} ↓ R)_{0} is the category of algebras for E, and that E preserves filtered colimits, are left to the reader, giving:

8.1. Theorem. The 2-functor U_{s} : Reg_{s} → Cat admits a left adjoint.

References

[1] J. Adámek and G.M. Kelly, M-completeness is seldom monadic over graphs, Theory Appl. Categ., 7:171–205, 2000.

[2] M.H. Albert and G.M. Kelly, The closure of a class of colimits, J. Pure Appl. Alg., 51:1–17, 1988.

[3] R. Blackwell, G.M. Kelly, and A. J. Power, Two-dimensional monad theory, J. Pure Appl. Alg., 59:1–41, 1989.

[4] A. Carboni and R. Celia Magno, The free exact category on a left exact one, J. Austral. Math. Soc. (Ser. A) 33:295–301, 1982.

[5] Brian Day, On closed categories of functors, in Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 137, pages 1–38. Springer-Verlag, 1970.

[6] E.J. Dubuc, Kan extensions in enriched category theory, Lecture Notes in Mathematics 145, Springer-Verlag, 1970.

[7] P.J. Freyd and G.M. Kelly, Categories of continuous functors I, J. Pure Appl. Alg., 2:169–191, 1972.

[8] G. Janelidze and G.M. Kelly, The reflectiveness of covering morphisms in algebra and geometry, Theory Appl. Categ. 3:132–159, 1997.

[9] G.M. Kelly, Basic Concepts of Enriched Category Theory, LMS Lecture Notes Series 64, Cambridge University Press, Cambridge, 1982.

[10] G.M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc. 36:137–152, 1987.

[11] G.M. Kelly and Stephen Lack, On property-like structures, Theory Appl. Categ. 3:213–250, 1997.

[12] G.M. Kelly and I. Le Creurer, On the monadicity over graphs of categories with limits, Cahiers de Topologie et Géom. Diff. Catégoriques 38:179–191, 1997.

[13] G.M. Kelly and R.H. Street, Review of the elements of 2-categories, in Sydney Category Seminar, Lecture Notes in Mathematics 420, pp. 75–103, Springer-Verlag, 1974.

[14] A. Kock, Monads for which structures are adjoint to units, J. Pure Appl. Alg. 104:41–59, 1993.

[15] H. Lindner, Morita equivalences of enriched categories, Cahiers de Topologie et Géom. Diff. 15:377–397, 1974.

[16] Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, New York, 1971.

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: carboni@fis.unico.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: walters@fis.unico.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca