Theory and Applications of Categories, Vol. 8, No. 23, 2001, pp. 555–575.

𝒱 -CAT IS LOCALLY PRESENTABLE OR LOCALLY BOUNDED IF 𝒱 IS SO

G. M. KELLY AND STEPHEN LACK

ABSTRACT. We show, for a monoidal closed category 𝒱 = (𝒱_{0}, ⊗, I), that the category 𝒱 -Cat of small 𝒱 -categories is locally λ-presentable if 𝒱_{0} is so, and that it is locally λ-bounded if the closed category 𝒱 is so, meaning that 𝒱_{0} is locally λ-bounded and that a side condition involving the monoidal structure is satisfied.

Many important properties of a monoidal category 𝒱 are inherited by the category 𝒱 -Cat of small 𝒱 -categories. For instance, if 𝒱 is symmetric monoidal, 𝒱 -Cat has a canonical symmetric monoidal structure, as was observed already in [4]. Much later [7, Remark 5.2], it was realized that if 𝒱 is only braided monoidal then 𝒱 -Cat still has a canonical monoidal structure, although it need not have a braiding unless the braiding on 𝒱 is in fact a symmetry. Similarly, it is straightforward to show that 𝒱 -Cat is monoidal closed when 𝒱 is closed and complete, and that 𝒱 -Cat is complete when 𝒱 is so. All of these results are essentially routine; the less trivial fact that 𝒱 -Cat is cocomplete when 𝒱 is so was first proved in [11].

The properties of 𝒱 or 𝒱 -Cat that we consider here are of a less basic nature, being conditions on 𝒱 which allow proofs by transfinite induction of the existence of various important adjoints. The best known of these conditions is local presentability [5], but there is also the notion of local boundedness [8], which is more general than local presentability, but also much more common, and sufficient for the central existence results of [8, Chapter 6], from which follow the basic results of the theory of enriched projective sketches. Recall that to be locally presentable is to be locally λ-presentable for some regular cardinal λ, and similarly that to be locally bounded is to be locally λ-bounded for some λ. It would be one thing to prove that 𝒱 -Cat is locally presentable if 𝒱 is so (in the sense that its underlying ordinary category 𝒱_{0} is so); here we prove the stronger result that 𝒱 -Cat is locally λ-presentable if 𝒱_{0} is so, so that the passage from 𝒱 to 𝒱 -Cat does not require the regular cardinal λ to be changed. When it comes to local boundedness, we prove that 𝒱 -Cat is locally λ-bounded when 𝒱 is so “as a closed category”, meaning that 𝒱_{0} is locally λ-bounded and satisfies a side condition involving the monoidal structure. We recall the precise definitions of local λ-presentability and local λ-boundedness in Section 2, but the common aspect is that 𝒱_{0} is cocomplete and has a small set 𝒢 of objects forming in some sense a generator of 𝒱_{0}, with the representables 𝒱_{0}(G, −) : 𝒱_{0} → Set preserving certain colimits: λ-filtered colimits in the locally λ-presentable case, and λ-filtered unions (with respect to a given factorization system on 𝒱_{0}) in the locally λ-bounded case.

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

Received by the editors 2001 November 12.

Transmitted by R. J. Wood. Published on 2001 December 19.

2000 Mathematics Subject Classification: 18C35, 18D20, 18A32.

Key words and phrases: enriched category, locally presentable category, locally bounded category.

\begin{matrix}c \\ ◯ \end{matrix} G. M. Kelly and Stephen Lack, 2001. Permission to copy for private use granted.

All categories are assumed to have small hom-sets.

1. Unions

For this section we consider a cocomplete category 𝒦 with a proper factorization system (ℰ , ℳ); recall that (ℰ , ℳ) is proper when each ℰ is an epimorphism and each ℳ a monomorphism — equivalently, when each ℳ is a monomorphism and each coretraction is in ℳ. Note that a map f : A → B lies in ℰ precisely when each factorization f = mg with m ∈ ℳ has m invertible.

A small family (m_{j} : A_{j} → B)_{j∈J} of maps with a common codomain is said to be jointly in ℰ if the induced map m : ∑_{j} A_{j} → B is in ℰ; this is equivalent to saying that there is no proper ℳ -subobject of B through which each m_{j} factorizes. When moreover each m_{j} is in ℳ, we say that the family constitutes an ℳ -union, or that B is the ℳ -union of the m_{j}.

More generally, the ℳ -union of a small family (m_{j} : A_{j} → B)_{j∈J} of ℳ -subobjects of B is defined to be the unique ℳ -subobject n : A → B containing the m_{j} for which the corresponding n_{j} : A_{j} → A constitute an ℳ -union. We may calculate this ℳ -union by taking the (ℰ , ℳ)-factorization of m : ∑_{j} A_{j} → B.

We shall need to speak of preservation of ℳ -unions only in the case of representable functors. We say that the representable functor 𝒦 (X, −) : 𝒦 → Set preserves the ℳ - union (m_{j} : A_{j} → B)_{j∈J} if the functions 𝒦 (X, m_{j}) : 𝒦 (X, A_{j}) → 𝒦 (X, B) are jointly surjective; in more concrete terms this says that any map f : X → B factorizes through some m_{j}.

Given a small family (m_{j} : A_{j} → B)_{j∈J} we can preorder the set J by setting j ≤ k whenever A_{j} ≤ A_{k} as ℳ -subobjects of B. Then the A_{j} are the object values of a functor A : J → 𝒦 , and we may form colimA and the induced h : colimA → B. It is easy to see that the m_{j} are an ℳ -union if and only if h ∈ ℰ.

Finally for a regular cardinal λ, the preorder J is said to be λ-filtered if it is so as a category: that is, if for each subset K of J with cardinality less than λ, the A_{k} with k ∈ K are all contained in some A_{j}. By a λ-filtered ℳ -union (m_{j} : A_{j} → B)_{j∈J} we mean one for which J is λ-filtered.

2. Locally presentable and locally bounded categories

In this section we continue to consider a cocomplete category 𝒦 ; from time to time we shall further suppose it to be equipped with a proper factorization system (ℰ , ℳ).

Let λ be a regular cardinal. An object X of 𝒦 is said to be λ-presentable [5] if the representable functor 𝒦 (X, −) : 𝒦 → Set preserves λ-filtered colimits, and λ-bounded [6] if 𝒦 (X, −) preserves λ-filtered ℳ -unions.

A small set 𝒢 of objects of 𝒦 is said to be a strong generator if an arrow f : A → B is invertible whenever 𝒦 (G, f) : 𝒦 (G, A) → 𝒦 (G, B) is bijective for each G ∈ 𝒢; while 𝒢 is an (ℰ , ℳ)-generator if this is true for arrows f : A → B in ℳ. Clearly 𝒢 is an (ℰ , ℳ)-generator precisely when, for each A ∈ 𝒦 , the family of all maps G → A with G ∈ 𝒢 is jointly in ℰ; that is, when the evident map ϵ_{A} : ∑_{G∈𝒢} 𝒦 (G, A) • G → A lies in ℰ; here we are writing X • A for the coproduct of X copies of A. When (ℰ , ℳ) is the proper factorization system (strong epimorphisms, monomorphisms), we have the well-known result (see for instance [6, Proposition 2.5.3]) that an (ℰ , ℳ)-generator is the same thing as a strong generator. In our cocomplete category 𝒦 , the pair (strong epimorphisms, monomorphisms) is certainly a proper factorization system if 𝒦 admits arbitrary cointersections of strong epimorphisms.

The cocomplete 𝒦 is said to be locally λ-presentable if it has a strong generator all of whose objects are λ-presentable; it is a consequence that 𝒦 is then complete. This and many other facts about locally presentable categories can be found in the books [1, 5, 10].

The cocomplete 𝒦 is said to be locally λ-bounded with respect to a proper factorization system (ℰ , ℳ) if it has an (ℰ , ℳ)-generator all of whose objects are λ-bounded, and if moreover 𝒦 admits arbitrary cointersections (even large ones, if need be) of maps in ℰ. The definition of locally λ-bounded category given in [8] included the further assumption of completeness, but once again this is a consequence of the other axioms, as we show in Corollary 2.2 below.

As well as being complete, every locally λ-presentable category is well-powered; it follows that it has a proper factorization system (ℰ , ℳ) in which ℳ consists of the monomorphisms and ℰ the strong epimorphisms. For this factorization system, an (ℰ , ℳ)-generator is, as we observed above, the same thing as a strong generator. Locally presentable categories are also well-copowered, and so arbitrary ℰ -cointersections exist. Finally, it turns out (see [6, Lemma 2.3.1]) that in a locally λ-presentable category every λ-presentable object is λ-bounded; we deduce that every locally λ-presentable category is locally λ-bounded. The converse, however, is false: see [5, p.104] or [6, p.190] for examples of locally λ-bounded categories that are not locally μ-presentable for any μ.

A cocomplete monoidal closed category is said to be locally λ-bounded as a closed category if its underlying ordinary category is locally λ-bounded and, in addition, the functors A ⊗ − and − ⊗ A map ℰ into ℰ for all objects A. The latter condition is clearly equivalent to the condition that e ⊗ e′ ∈ ℰ whenever e, e′ ∈ ℰ, and it turns out to be vacuous if ℳ consists of all the monomorphisms.

In fact all the examples of closed categories considered in [8] have some factorization system for which they are locally bounded. Algebraic examples, such as the categories Set, Cat, and Ab of sets, categories, and abelian groups are all locally finitely presentable, as is the combinatorial example SSet, the category of simplicial sets. The reason for using the weaker notion of local boundedness rather than local presentability is the desire to include such topological examples as the categories CGTop, QTop, and Ban of compactly generated topological spaces, quasi-topological spaces, and Banach spaces, which are not locally presentable, but are locally bounded. The example QTop is not ℰ -wellcopowered, which explains why we must explicitly require arbitrary cointersections of maps in ℰ. For the details, and for many further examples, including Lawvere’s closed category given by the interval [0, ∞] of the reals, see [8, Chapter 6].

For our promised proof that every locally bounded category is complete we use an (apparently unpublished) (ℰ , ℳ)-variant of Freyd’s Special Adjoint Functor Theorem, namely:

2.1. Proposition. Let the cocomplete category 𝒦 have the factorization system (ℰ , ℳ) for which ℰ is contained in the epimorphisms; suppose that 𝒦 admits arbitrary cointersections of maps in ℰ, and that 𝒦 has an (ℰ , ℳ)-generator 𝒢. Then every cocontinuous functor S : 𝒦 → ℒ has a right adjoint.

Proof. To provide a right adjoint to S is equally to provide, for each D ∈ ℒ, a terminal object of the comma category S/D, whose objects are pairs (C, f : SC → D) and whose maps (C, f) → (C′, f′) are maps x : C → C′ with Sx.f = f′. The forgetful functor U : S/D → 𝒦 creates colimits (and hence reflects epimorphisms). We get an induced factorization system, still called (ℰ , ℳ), on S/D by taking x : (C, f) → (C′, f′) to be in ℰ or in ℳ when Ux is so; once again every ℰ is an epimorphism. Finally, the small set consisting of the (C, f) with C ∈ 𝒢 forms an (ℰ , ℳ)-generator for S/D. Thus (S/D, ℰ , ℳ) has just the properties required in the proposition of (𝒦 , ℰ , ℳ). So it suffices to prove that the 𝒦 of the proposition has a terminal object.

Form in 𝒦 the coproduct H = ∑_{G∈𝒢} G, and let ζ : H → K be the cointersection of all the maps in ℰ having domain H; of course ζ ∈ ℰ and is an epimorphism. Any two maps f, g : A → K must coincide: for their coequalizer h : K → L is in ℰ, so that hζ is in ℰ, whence khζ = ζ for some k by the definition of ζ as the smallest ℰ -quotient, so that in fact kh = 1 and h is invertible.

To exhibit K as the desired terminal object it remains only to show that, for each A ∈ 𝒦 , there is a map A → K. For each G ∈ 𝒢 and A ∈ 𝒦 we have the trivial function 𝒦 (G, A) → 1 into the singleton set, so that we have an induced map t : ∑_{G∈𝒢} 𝒦 (G, A)• G → ∑_{G∈𝒢} G. Form in 𝒞 the pushout

\xymatrix{∑_{G∈𝒢} 𝒦 (G, A) • G \ar[r]^{ϵ_{A}} \ar[d]_{t} & A \ar[d]^{r} \\ ∑_{G∈𝒢} G \ar[r]_{s} & L ;}

here ϵ_{A} lies in ℰ since 𝒢 is an (ℰ , ℳ)-generator, so that its pushout s also lies in ℰ. By the definition of K, therefore, there is a map v : L → K, and thus a map vr : A → K.

2.2. Corollary. Let the cocomplete category 𝒦 have a factorization system (ℰ , ℳ) for which every ℰ is an epimorphism, and suppose that 𝒦 admits arbitrary cointersections of maps in ℰ and has an (ℰ , ℳ)-generator 𝒢. Then 𝒦 is complete.

Proof. For each small category 𝒞 we seek a right adjoint to the diagonal Δ : 𝒦 → [𝒞 , 𝒦 ]; and this adjoint exists by the proposition, since [𝒞 , 𝒦 ] has colimits formed pointwise and Δ is cocontinuous.

2.3. Remark. Given a cocomplete category 𝒦 , to give a factorization system (ℰ , ℳ) having each ℰ epimorphic and admitting arbitrary cointersections of maps in ℰ, it suffices by [3, Lemma 3.1] to give a class ℰ of epimorphisms in 𝒦 , closed under composition and stable under pushout, for which arbitrary cointersections of maps in ℰ exist and lie in ℰ.

Before leaving this section, we make a final observation of rather lesser importance. We have discussed what it means for a monoidal closed category to be locally bounded as a closed category, but we have not considered local presentability for closed categories. In [9], a monoidal closed category 𝒱 was defined to be locally λ-presentable as a closed category if its underlying category 𝒱_{0} was locally λ-presentable and the λ-presentable objects of 𝒱_{0} were closed under the monoidal structure: that is, the unit I was λ-presentable and X ⊗ Y was λ-presentable whenever X and Y were so. The observation we wish to make here is the following:

2.4. Proposition. If 𝒱 is a monoidal closed category and 𝒱_{0} is locally λ-presentable, then there exists a regular cardinal μ for which 𝒱 is locally μ-presentable as a closed category.

Proof. Observe that the set of λ-presentable objects is (essentially) small, so the set of objects of the form G ⊗ H where G and H are λ-presentable is (essentially) small. Thus there exists a regular cardinal μ with the property that I is μ-presentable and G ⊗ H is μ-presentable whenever G and H are λ-presentable. But now if A and B are μ-presentable objects, then we may write A = colim_{i}G_{i} and B = colim_{j}H_{j} where the colimits in question are μ-small, and where each G_{i} and each H_{j} is λ-presentable. Then

A ⊗ B = colim_{i}G_{i} ⊗ colim_{j}H_{j}
= colim_{i,j}(G_{i} ⊗ H_{j})

and each G_{i}⊗H_{j} is μ-presentable; thus A⊗B is a μ-small colimit of μ-presentable objects, and thus is itself μ-presentable. This proves that 𝒱 is locally μ-presentable as a closed category.

3. 𝒱 -Cat is finitarily monadic over 𝒱 -Gph

For this section we suppose that 𝒱 is a monoidal category which is cocomplete, and that the functors A ⊗ − and − ⊗ A preserve colimits for all objects A of 𝒱 , as is certainly the case if the monoidal 𝒱 is closed.

As a preliminary to our investigation of 𝒱 -Cat, we consider the category 𝒱 -Gph of 𝒱 -graphs and their morphisms. Recall that a 𝒱 -graph is a pair (X, A), where X is a (small) set, and A is a family (A(x, y))_{x,y∈X} of objects of 𝒱 . A 𝒱 -graph morphism from (X, A) to (Y, B) is a pair (f, φ) where f : X → Y is a function from X to Y, and φ is a family (φ_{x,y} : A(x, y) → B(fx, fy))_{x,y∈X} of morphisms in 𝒱 . We write P : 𝒱 -Gph → Set for the functor sending a 𝒱 -graph (X, A) to its set X of objects, and sending (f, φ) to f.

There is an evident forgetful functor U : 𝒱 -Cat → 𝒱 -Gph which is monadic, as was proved in [2] under the hypotheses above, and more generally when 𝒱 is a suitable bicategory; and much earlier in [11] when 𝒱 is symmetric monoidal closed. In this section we shall show that the monad in question is finitary — meaning that it preserves filtered colimits; in the next, we show that 𝒱 -Gph is locally λ-presentable if 𝒱 is so; it will then follow that 𝒱 -Cat is locally λ-presentable if 𝒱 is so, by [5, Satz 10.3]. Accordingly we begin by studying colimits in 𝒱 -Gph.

Following [2], we shall analyze 𝒱 -graphs in terms of the more general 𝒱 -matrices. If X and Y are sets, a 𝒱 -matrix S from X to Y is a family (S(y, x))_{(x,y)∈X×Y} of objects of 𝒱 ; thus a 𝒱 -graph is just a set X equipped with a 𝒱 -matrix A : X → X. The value of 𝒱 -matrices is that they can be composed: if S : X → Y and T : Y → Z are 𝒱 -matrices, then their composite T S : X → Z is defined by

(T S)(z, x) = ∑_{y∈Y} T (z, y) ⊗ S(y, x).

There is now a bicategory 𝒱 -Mat in which the objects are the (small) sets, the 1-cells are the 𝒱 -matrices, and a 2-cell between 𝒱 -matrices S, S′ : X → Y is a family (σ_{y,x} : S(y, x) → S′(y, x))_{(x,y)∈X×Y} of morphisms of 𝒱 .

For objects X and Y of 𝒱 -Mat, the hom-category 𝒱 -Mat(X, Y) is just 𝒱 ^{Y ×X}, which is cocomplete since 𝒱 is so, with colimits formed pointwise from those in 𝒱 . Furthermore, if S : Y → Y′ and R : X′ → X are arbitrary 𝒱 -matrices, the functors 𝒱 -Mat(X, S) : 𝒱 -Mat(X, Y) → 𝒱 -Mat(X, Y′) and 𝒱 -Mat(R, Y) : 𝒱 -Mat(X, Y) → 𝒱 -Mat(X′, Y) are cocontinuous; we express this fact by saying that “composition commutes with colimits”.

A function f : X → Y determines 𝒱 -matrices f_{∗} : X → Y and f^{∗} : Y → X with

f_{∗}(y, x) = f^{∗}(x, y) = \begin{cases}I if fx = y \\ 0 otherwise \end{cases}

where I denotes the unit object and 0 the initial object of 𝒱 . The reader will easily construct a natural bijection between 2-cells f_{∗}A → B and 2-cells A → f^{∗}B, and so deduce that f_{∗} is left adjoint to f^{∗} in the bicategory 𝒱 -Mat. In fact it is also easy to describe explicitly the unit 1_{X} → f^{∗}f_{∗} and the counit f_{∗}f^{∗} → 1_{Y}.

We have already observed that a 𝒱 -graph is an object X of 𝒱 -Mat equipped with a 1-cell A : X → X; a morphism of 𝒱 -graphs from (X, A) to (Y, B) can be seen as a function f : X → Y equipped with a 2-cell φ : A → f^{∗}Bf_{∗}, as the following calculation shows:

(f^{∗}Bf_{∗})(z, x) = ∑_{y∈Y} f^{∗}(z, y) ⊗ (Bf_{∗})(y, x)

= (Bf_{∗})(fz, x)

= ∑_{y∈Y} B(fz, y) ⊗ f_{∗}(y, x)

= B(fz, fx).

In fact, because of the adjunction f_{∗} ⊣ f^{∗} in the bicategory 𝒱 -Mat, there is a bijection (of “mates”) between 2-cells φ : A → f^{∗}Bf_{∗} and 2-cells φ̂ : f_{∗}Af^{∗} → B; explicitly, we find that

(f_{∗}Af^{∗})(u, v) = ∑_{{fx=u \atop fy=v}} A(x, y),

and now for x ∈ f^{−1}(u) and y ∈ f^{−1}(v) the (x, y)-component of φ̂_{u,v} : (f_{∗}Af^{∗})(u, v) → B(u, v) is φ_{x,y}.

As shown in [2], colimits in 𝒱 -Gph can be described as follows. Let 𝒥 be a small category, and (X, A) : 𝒥 → 𝒱 -Gph a functor; we denote the image of an object j under (X, A) by (X_{j}, A_{j}) and the image of a morphism θ : j → k by (X_{θ}, A_{θ}). Consider the functor X = P (X, A) : 𝒥 → Set, and form its colimit X̄ with colimit cone (q_{j} : X_{j} → X̄)_{j∈𝒥} . There is a functor Ã: 𝒥 → 𝒱 -Mat(X̄, X̄) sending j to (q_{j})_{∗}A_{j}(q_{j})^{∗} and sending a morphism θ : j → k to (q_{k})_{∗}Â_{θ}(q_{k})^{∗} : (q_{k})_{∗}(X_{θ})_{∗}A_{j}(X_{θ})^{∗}(q_{k})^{∗} → (q_{k})_{∗}A_{k}(q_{k})^{∗}, where Â_{θ} : (X_{θ})_{∗}A_{j}(X_{θ})^{∗} → A_{k} is the mate, as above, of A_{θ} : A_{j} → (X_{θ})^{∗}A_{k}(X_{θ})_{∗}. As we saw above, the colimit of Ã is formed pointwise from colimits in 𝒱 : write Ā : X̄ → X̄ for this colimit, with colimit cone α′_{j} : (q_{j})_{∗}A_{j}(q_{j})^{∗} → Ā. Now we have in 𝒱 -Mat a cone (q_{j}, α_{j}) : (X_{j}, A_{j}) → (X̄, Ā), where α_{j} : A_{j} → (q_{j})^{∗}Ā(q_{j})_{∗} is the 2-cell for which α̂_{j} : (q_{j})_{∗}A_{j}(q_{j})^{∗} → Ā is α′_{j}; and it is shown in [2] that this is a colimit cone for (X, A) : 𝒥 → 𝒱 -Gph. (Of course we henceforth drop the name α′_{j} in favour of α̂_{j}.)

We need below to consider functors (X, A) : 𝒥 → 𝒱 -Gph and (X, B) : 𝒥 → 𝒱 -Gph with the same X : 𝒥 → Set; accordingly we introduce the category 𝒱 -Gph^{(2)} defined by the pullback

\xymatrix{𝒱 -Gph^{(2)} \ar[r]^{Q} \ar[d]_{R} & 𝒱 -Gph \ar[d]^{P} \\ 𝒱 -Gph \ar[r]_{P} & Set}

in Cat; observe that, since 𝒱 -Gph and Set are cocomplete and P is cocontinuous, 𝒱 -Gph^{(2)} is cocomplete and the functors Q and R jointly create colimits. An object of 𝒱 -Gph^{(2)} is a pair ((X, A), (X, B)) of 𝒱 -graphs with the same underlying set X, which we henceforth write as (X, A, B); and a morphism has the form (f, α, β) : (X, A, B) → (X′, A′, B′) where (f, α) : (X, A) → (X′, A′) and (f, β) : (X, B) → (X′, B′) are morphisms in 𝒱 -Gph. To give a pair of functors as in the first sentence of this paragraph is of course to give a single functor from 𝒥 to 𝒱 -Gph^{(2)}. In the same way we can define 𝒱 -Gph^{(n)} with objects (X, A_{1}, …, A_{n}) by taking the fibred product in Cat of n copies of P : 𝒱 -Gph → Set, and 𝒱 -Gph^{(ℕ)} by taking the fibred product of copies indexed by the set ℕ of natural numbers; and we have the corresponding results about colimits in 𝒱 -Gph^{(n)} and 𝒱 -Gph^{(ℕ)}.

Consider the functor S : 𝒱 -Gph^{(2)} → 𝒱 -Gph sending (X, A, B) to (X, A + B), where the sum A + B of matrices is of course the coproduct in 𝒱 ^{X×X}; the value of S on morphisms is given by the evident sum of 2-cells using the distributive law for matrices.

This functor S preserves colimits, for if (X, A, B) : 𝒥 → 𝒱 -Gph^{(2)}, it is clear from the description above of colimits in 𝒱 -Gph that the colimit of (X, A + B) is (X̄, Ā + B̄), where (X̄, Ā) and (X̄, B̄) are the colimits of (X, A) and (X, B). Similarly of course for sums of any size: the form we need below is:

3.1. Lemma. The functor S : 𝒱 -Gph^{(ℕ)} → 𝒱 -Gph sending (X, (A_{n})_{n∈ℕ}) to (X, ∑_{n∈ℕ} A_{n}) preserves colimits.

We also need to consider the functor M : 𝒱 -Gph^{(2)} → 𝒱 -Gph which sends (X, A, B) to (X, AB), where AB denotes as before the matrix product. We must of course define M on morphisms too. Recall that the α of a morphism (f, α) : (X, A) → (X′, A′) can be seen as a matrix α : A → f^{∗}A′f_{∗}, but can equally be described by its mate α̂ : f_{∗}Af^{∗} → A′ under the adjunction f_{∗} ⊣ f^{∗}. But there is of course yet another equivalent form, namely ᾱ : f_{∗}A → A′f_{∗}. In fact we find that (f_{∗}A)(x′, x) = ∑_{fy=x′} A(y, x), that (A′f_{∗})(x′, x) = A′(x′, fx), and that ᾱ_{x′,x} has α_{y,x} as its y-component. Now the value of M on (f, α, β) : (X, A, B) → (X′, A′, B′) is (f, γ) : (X, AB) → (X′, A′B′) where γ is determined in terms of its mate γ̄ by the pasting composite

[figure: X; f_{∗}; AB; γ̄; X′; A′B′; X; f_{∗}; B; β̄; X′; B′; =; X; f_{∗}; A; ᾱ; X′; A′; X; f_{∗}; X′; X; f_{∗}; X′.]

This comes, as the reader will easily see, to taking for γ_{z,x} : (AB)(z, x) → (A′B′)(fz, fx) the composite

\xymatrix{∑_{y∈X} A(z, y)B(y, x) \ar[r]^{∑ α_{z,y}β_{y,x}} & ∑_{y∈X} A′(fz, fy)B′(fy, fx) \ar[r]|{κ} & ∑_{y′∈X′} A′(fz, y′)B′(y′, fx) ,}

where the y-component of κ is the fy-injection into the final sum; we included the less elementary description of γ given above since it makes clearer the functoriality of M. The result we need is:

3.2. Lemma. The functor M : 𝒱 -Gph^{(2)} → 𝒱 -Gph preserves filtered colimits.

Proof. Consider a functor (X, A, B) : 𝒥 → 𝒱 -Gph^{(2)} with 𝒥 filtered. Using the notation above, we recall that the colimit of (X, A) : 𝒥 → 𝒱 -Gph is (X̄, Ā) with colimit cone (q_{j}, α_{j}) : (X_{j}, A_{j}) → (X̄, Ā), where q_{j} : X_{j} → X̄ is the colimit cone for X : 𝒥 → Set and α̂_{j} : (q_{j})_{∗}A_{j}(q_{j})^{∗} → Ā is the colimit cone for the functor Ã : 𝒥 → 𝒱 -Mat(X̄, X̄) sending j to Ãj = (q_{j})_{∗}A_{j}(q_{j})^{∗} and sending θ : j → k to Ãθ = (q_{k})_{∗}Â_{θ}(q_{k})^{∗}. Similarly the colimit of (X, B) is (X̄, B̄) with colimit cone (q_{j}, β_{j}), where β̂_{j} : (q_{j})_{∗}B_{j}(q_{j})^{∗} → B̂ is the colimit cone for B̃ : 𝒥 → 𝒱 -Mat(X̄, X̄).

The composite of M with the functor (X, A, B) is a functor (X, C) : 𝒥 → 𝒱 -Gph where C_{j} = A_{j}B_{j} and where C_{θ} for θ : j → k is such that C̄_{θ} is a pasting composite of Ā_{θ} and B̄_{θ}: see the definition of M on morphisms above. This functor, of course, has the colimit cone (q_{j}, γ_{j}) : (X_{j}, C_{j}) → (X̄, C̄) where γ̂_{j} : (q_{j})_{∗}A_{j}B_{j}(q_{j})^{∗} = (q_{j})_{∗}C_{j}(q_{j})^{∗} → C̄ is the colimit cone for the functor C̃ : 𝒥 → 𝒱 -Mat(X̄, X̄) sending j to C̃_{j} = (q_{j})_{∗}A_{j}B_{j}(q_{j})^{∗}.

The functor M, however, sends the colimit (X̄, Ā, B̄) of (X, A, B) to (X̄, ĀB̄), and sends the colimit cone (q_{j}, α_{j}, β_{j}) of (X, A, B) to the cone (q_{j}, δ_{j}) : (X_{j}, A_{j}, B_{j}) → (X̄, ĀB̄) where δ_{j} is determined through the pasting equation

[figure: X_{j}; (q_{j})_{∗}; A_{j}B_{j}; δ̄_{j}; X̄; _{ĀB̄}; X_{j}; (q_{j})_{∗}; B_{j}; β̄_{j}; X̄; B̄; =; X_{j}; (q_{j})_{∗}; A_{j}; ᾱ_{j}; X̄; Ā; X_{j}; (q_{j})_{∗}; X̄; X_{j}; (q_{j})_{∗}; X̄ .]

To say that M preserves the colimit of (X, A, B) is to say that the cone (q_{j}, δ_{j}) is a colimit cone, and hence, by the above, to say that the cone

\xymatrix{δ̂_{j} : (q_{j})_{∗}A_{j}B_{j}(q_{j})^{∗} \ar[r] & ĀB̄}

is a colimit cone in 𝒱 -Mat(X̄, X̄) over the functor C̃.

On the other hand, since composition of matrices commutes with colimits, the colimit cones α̂_{j} : Ã_{j} → Ā and β̂_{j} : B̃_{j} → B̄ give by composition a colimit cone α̂_{j}β̂_{k} : Ã_{j}B̃_{k} → ĀB̄ over the functor 𝒥 × 𝒥 → 𝒱 -Mat(X̄, X̄) sending (j, k) to Ã_{j}B̃_{k} and similarly defined on morphisms. Because 𝒥 is filtered, however, the diagonal 𝒥 → 𝒥 × 𝒥 is final; so that α̂_{j}β̂_{j} : Ã_{j}B̃_{j} → ĀB̄ is a colimit cone for the functor ÃB̃ : 𝒥 → 𝒱 -Mat(X̄, X̄) sending j to Ã_{j}B̃_{j} = (q_{j})_{∗}A_{j}(q_{j})^{∗}(q_{j})_{∗}B_{j}(q_{j})_{∗}.

We have the unit η_{j} : 1_{X_{j}} → (q_{j})^{∗}(q_{j})_{∗} of the adjunction (q_{j})_{∗} ⊣ (q_{j})^{∗}, and thus for each j a 2-cell

(q_{j})_{∗}A_{j}η_{j}B_{j}(q_{j})^{∗} : (q_{j})_{∗}A_{j}B_{j}(q_{j})^{∗} → (q_{j})_{∗}A_{j}(q_{j})^{∗}(q_{j})_{∗}B_{j}(q_{j})^{∗} ,

which we may write as ζ_{j} : C̃_{j} → Ã_{j}B̃_{j}; a straightforward calculation verifies that these are the components of a natural transformation ζ : C̃ → ÃB̃ : 𝒥 → 𝒱 -Mat(X̄, X̄). Using the adjunction (q_{j})_{∗} ⊣ (q_{j})^{∗} to express the δ̂_{j} in terms of their mates δ̄_{j} and hence in terms of α and β, we find that the cone δ̂_{j} : C̃_{j} → ĀB̄ is just the composite of ζ_{j} with the colimit cone α̂_{j}β̂_{j} : Ã_{j}B̃_{j} → ĀB̄. So the δ̂_{j} constitute a colimit cone if and only if the ζ̄: C̄ → ĀB̄ induced by ζ : C̃ → ÃB̃ is invertible.

Recall our earlier calculation of a matrix composite f_{∗}Af^{∗}. This gives us, for x, y ∈ X̄,

C̃_{j}(x, y) = ((q_{j})_{∗}A_{j}B_{j}(q_{j})^{∗})(x, y)

= ∑_{ρ,σ,τ∈X_{j} ; {q_{j}ρ=x \atop q_{j}σ=y}} A_{j}(ρ, τ)B_{j}(τ, σ)

and

(Ã_{j}B̃_{j})(x, y) = ((q_{j})_{∗}A_{j}(q_{j})^{∗}(q_{j})_{∗}B_{j}(q_{j})^{∗})(x, y)

= ∑_{z∈X̄} ∑_{r,t∈X_{j} ; {q_{j}r=x \atop q_{j}t=z}} ∑_{p,s∈X_{j} ; {q_{j}p=z \atop q_{j}s=y}} A_{j}(r, t)B_{j}(p, s) ;

and it follows easily from the explicit description of the unit 1_{X_{j}} → (q_{j})^{∗}(q_{j})_{∗} that (ζ_{j})_{x,y} : C̃_{j}(x, y) → (Ã_{j}B̃_{j})(x, y) is the map whose (ρ, σ, τ)-component is the (z, r, t, p, s)- coprojection where r = ρ, s = σ, t = p = τ, and z = q_{j}τ.

We complete the proof by constructing an inverse ξ̄: ĀB̄ → C̄ of ζ̄, or equally inverses ξ̄_{x,y} : (ĀB̄)(x, y) → C̄(x, y) of ζ̄_{x,y}; here ξ̄_{x,y} is to be the map induced on the colimit by a cone (ξ_{j})_{x,y} : (Ã_{j}B̃_{j})(x, y) → C̄(x, y). By the formula above for (Ã_{j}B̃_{j})(x, y), it suffices to give for each (z, r, t, p, s) the appropriate component (ξ_{j})_{x,y,z;r,t,p,s} : A_{j}(r, t)B_{j}(p, s) → C̄(x, y). Now since q_{j}t = q_{j}p, there is by the filteredness of 𝒥 some θ : j → k with X_{θ}t = X_{θ}p = t′ ∈ X_{k}, say. Write r′ for X_{θ}r and s′ for X_{θ}s. We take for (ξ_{j})_{x,y,z;r,t,p,s} the composite

\xymatrix{A_{j}(r, t)B_{j}(p, s) \ar[r]^{(A_{θ})_{r,t}(B_{θ})_{p,s}} & A_{k}(r′, t′)B_{k}(t′, s′) \ar[r]^{λ} & C̃_{k}(x, y) \ar[r]^{(γ̂_{k})_{x,y}} & C̄(x, y) ,}

where λ is the appropriate coprojection in the expression above for C̃_{j}(x, y), but now with k in place of j. It is easy to verify, first, that (ξ_{j})_{x,y,z;r,t,p,s} is independent of our choice of a θ : j → k with X_{θ}t = X_{θ}p, so that (ξ_{j})_{x,y} is well-defined; and second that the (ξ_{j})_{x,y} : (Ã_{j}B̃_{j})(x, y) → C̄(x, y) constitute a cone, thus inducing a map ξ̄_{x,y} : ĀB̄(x, y) → C̄(x, y) determined by ξ̄_{x,y}(α̂_{j}β̂_{j})_{x,y} = (ξ_{j})_{x,y}.

That ξ̄_{x,y}ζ̄_{x,y} = 1 follows easily because, in applying ξ̄_{x,y} on the image of ζ̄_{x,y} we may, since here t = p = τ, take θ : j → k to be 1_{j}. To say that ζ̄_{x,y}ξ̄_{x,y} = 1 is to say that ζ̄_{x,y}ξ̄_{x,y}(α̂_{j}β̂_{j})_{x,y} = (α̂_{j}β̂_{j})_{x,y} for each j. However ζ̄_{x,y}ξ̄_{x,y}(α̂_{j}β̂_{j})_{x,y} = ζ̄_{x,y}(ξ_{j})_{x,y}, whose (z; r, t, p, s)-component by the above is

ζ̄_{x,y}(γ̂_{k})_{x,y}λ ((A_{θ})_{r,t}(B_{θ})_{p,s}) = (α̂_{k}β̂_{k})_{x,y}(ζ_{k})_{x,y}λ ((A_{θ})_{r,t}(B_{θ})_{p,s}) ,

and it follows from the explicit description above of (ζ_{k})_{x,y} that (ζ_{k})_{x,y}λ is just the coprojection A_{k}(r′, t′)B_{k}(t′, s′) → (Ã_{k}B̃_{k})(x, y), which we shall write as κ_{k}. If we similarly write κ_{j} for the coprojection A_{j}(r, t)B_{j}(p, s) → (Ã_{j}B̃_{j})(x, y), we have κ_{k}((A_{θ})_{r,t}(B_{θ})_{p,s}) = (Ã_{θ}B̃_{θ})_{x,y}κ_{j}, so that (α̂_{k}β̂_{k})_{x,y}κ_{k}((A_{θ})_{r,t}(B_{θ})_{p,s}) = (α̂_{k}β̂_{k})_{x,y}(Ã_{θ}B̃_{θ})_{x,y}κ_{j} = (α̂_{j}β̂_{j})_{x,y}κ_{j}, which is the (z; r, t, p, s)-component of (α̂_{j}β̂_{j})_{x,y}, as desired. So the ζ̄_{x,y} are indeed invertible, which completes the proof.

We shall now describe the endofunctor T of 𝒱 -Gph underlying the “free 𝒱 -category” monad. Recall from [2] that T sends a 𝒱 -graph (X, A) to (X, A′) where A′ = ∑_{n∈ℕ} A^{n} is the free monoid on A in the monoidal category given by 𝒱 -Mat(X, X) with matrix multiplication as its tensor product; and that the unit (X, A) → (X, A′) of the adjunction is (1, ρ_{A}) where ρ_{A} : A → A′ is the injection of the summand A = A^{1} into ∑ A^{n}. From this we can calculate the value of T on morphisms, which leads to the following description of T. For each n ∈ ℕ there is an endofunctor T_{n} of 𝒱 -Gph sending (X, A) to (X, A^{n}); and because P T_{n} = P, these T_{n} are the components of a functor T_{ℕ} : 𝒱 -Gph → 𝒱 -Gph^{(ℕ)}; whereupon T is the composite ST_{ℕ}, where S : 𝒱 -Gph^{(ℕ)} → 𝒱 -Gph is the functor so denoted in Lemma 3.1. Since S preserves all colimits by Lemma 3.1, T will be finitary (that is, will preserve filtered colimits) if T_{ℕ} is so. Since the projections 𝒱 -Gph^{(ℕ)} → 𝒱 -Gph jointly create colimits, T_{ℕ} will be finitary if each T_{n} is so. However T_{1} is the identity endofunctor 1 of 𝒱 -Gph, while T_{2} is the composite M(1, 1), where (1, 1) : 𝒱 -Gph → 𝒱 -Gph^{(2)} is the functor each of whose components is 1; and T_{n+1} for n ≥ 1 is (isomorphic to) the composite M(T_{n}, 1). Since the projections 𝒱 -Gph^{(2)} → 𝒱 -Gph jointly create colimits, it follows inductively from Lemma 3.2 that T_{n} is finitary for n ≥ 1.

It remains to consider the endofunctor T_{0} of 𝒱 -Gph sending (X, A) to (X, 1_{X}), where 1_{X} is the identity matrix with (1_{X})_{x,y} being I for x = y and 0 otherwise. This is the composite of the forgetful functor P : 𝒱 -Gph → Set and the evident functor H : Set → 𝒱 -Gph sending X to (X, 1_{X}). Since P preserves all colimits, it will suffice to show that H preserves filtered colimits. Suppose then that X : 𝒥 → Set with 𝒥 filtered has as before the colimit cone (q_{j} : X_{j} → X̄), and consider the colimit of HX; our claim is that the colimit of the (X_{j}, 1_{X_{j}}) is (X̄, 1X̄). By our description of colimits in 𝒱 -Gph, we have to show that 1X̄ is the colimit in 𝒱 -Mat(X̄, X̄) of the (q_{j})_{∗}1_{X_{j}}(q_{j})^{∗}. Since

((q_{j})_{∗}1_{X_{j}}(q_{j})^{∗}) (x, y) = ∑_{{q_{j}r=x \atop q_{j}s=y}}(1_{X_{j}})(r, s) ,

there is nothing to prove for x ≠ y, the cone being constant at 0. For x = y the above gives

((q_{j})_{∗}1_{X_{j}}(q_{j})^{∗}) (x, x) = q_{j}^{−1}(x) • I ,

the coproduct of q_{j}^{−1}(x) copies of I; and we are claiming that the colimit in 𝒱 of the q_{j}^{−1}(x) • I is I. However ( ) • I : Set → 𝒱 preserves colimits, so that it suffices to observe that in Set we have colim(q_{j}^{−1}(x)) = 1. But filtered colimits in Set commute with finite limits; and the above is precisely what we get on pulling back the colimit q_{j} : X_{j} → X̄ along x : 1 → X̄. This completes the proof of:

3.3. Theorem. The monad on 𝒱 -Gph whose algebras are 𝒱 -categories is finitary.

An equivalent formulation is:

3.4. Corollary. The forgetful functor U : 𝒱 -Cat → 𝒱 -Gph is finitary.

4. 𝒱 -Cat is locally presentable if 𝒱 is so

As in Section 3, we continue to suppose that the monoidal category 𝒱 is cocomplete and that the functors A ⊗ − and − ⊗ A preserve colimits, as they surely do when 𝒱 is closed. To avoid pathologies in our use of the “strong generator” notion, we further suppose that 𝒱_{0} admits arbitrary cointersections of strong epimorphisms, which ensures that (strong epimorphisms, monomorphisms) is a factorization system on 𝒱_{0}. This presents no problem, since our main goal is the study of the case where 𝒱_{0} is locally presentable.

It is convenient to introduce, for each object G of 𝒱 , the 𝒱 -graph (2, Ḡ) having 2 = {0, 1} for its set of objects and having

Ḡ(0, 1) = G, Ḡ(0, 0) = Ḡ(1, 1) = Ḡ(1, 0) = 0,

where this last 0 is the initial object of 𝒱 ; that is to say, Ḡ is the 2-by-2 matrix (\begin{matrix}0 & G \\ 0 & 0 \end{matrix}).

To give a morphism (2, Ḡ) → (X, A) of 𝒱 -graphs is just to give a pair x, y ∈ X and a morphism u : G → A(x, y) in 𝒱 .

The forgetful functor P : 𝒱 -Gph → Set sending the 𝒱 -graph (X, A) to X clearly has a left adjoint D sending the set X to the 𝒱 -graph (X, 0_{X}), where 0_{X} is the initial object of 𝒱 -Mat(X, X) given by 0_{X}(x, x′) = 0.

4.1. Lemma. A morphism (f, α) : (X, A) → (Y, B) in 𝒱 -Gph is monomorphic if and only if f : X → Y is an injective function and each α_{x,x′} : A(x, x′) → B(fx, fx′) is a monomorphism in 𝒱 (that is, in 𝒱_{0}).

Proof. The “if” part being clear from the definition of composition in 𝒱 -Gph, it suffices to prove the “only if” part; so suppose that (f, α) is monomorphic in 𝒱 -Gph. Then f is injective because P : 𝒱 -Gph → Set, having a left adjoint, preserves monomorphisms. Suppose that, for some x, x′ ∈ X, maps β, γ : G → A(x, x′) in 𝒱 satisfy α_{x,x′}β = α_{x,x′}γ, and define g : 2 → X by setting g0 = x and g1 = x′; now the morphisms (g, β), (g, γ) : (2, Ḡ) → (X, A) have the same composite with (f, α) : (X, A) → (Y, B), whence β = γ. Thus α_{x,x′} is indeed monomorphic.

4.2. Lemma. If a set 𝒢 of objects constitutes a strong generator of 𝒱_{0}, then the set {(2, Ḡ) | G ∈ 𝒢 or G = 0} constitutes a strong generator of 𝒱 -Gph.

Proof. We prove the assertion in the equivalent form — see Section 2 above — that the totality of maps in 𝒱 -Gph into the object (Y, B) having domain one of the (2, Ḡ) with G ∈ 𝒢 ∪ {0} factorizes through no proper subobject of (Y, B) and is therefore jointly a strong epimorphism. Suppose then that (f, α) : (X, A) → (Y, B) is a monomorphism in 𝒱 -Gph through which every (g, β) : (2, Ḡ) → (Y, B) with G ∈ 𝒢 ∪{0} factorizes. To give a map from (2, ¯0) = (2, 0_{2}) into (Y, B) is just to give two elements of Y; and since every such map factorizes through (f, α), the injection f is in fact a bijection. Since every (g, β) : (2, Ḡ) → (Y, B) with G ∈ 𝒢 factorizes through (f, α), every map G → B(fx, fx′) in 𝒱 factorizes through the monomorphism α_{x,x′} : A(x, x′) → B(fx, fx′), which is therefore invertible, because 𝒢 is a strong generator for 𝒱_{0}. Thus the monomorphism (f, α) is indeed invertible.

We now examine the “presentability” of such a strong generator for 𝒱 -Gph. Note that 𝒱_{0}(0, −) : 𝒱_{0} → Set is the functor constant at 1, which preserves all connected colimits; so that the object 0 of 𝒱_{0} is λ-presentable for any regular cardinal λ.

4.3. Lemma. If, for some regular cardinal λ, the object G is λ-presentable in 𝒱_{0}, then (2, Ḡ) is λ-presentable in 𝒱 -Gph.

Proof. Consider as in Section 3 above the colimit cone (q_{j}, α_{j}) : (X_{j}, A_{j}) → (X̄, Ā) of a functor (X, A) : 𝒥 → 𝒱 -Gph, where the category 𝒥 is λ-filtered; we are to show that the functor 𝒱 -Gph((2, Ḡ), −) : 𝒱 -Gph → Set preserves every such colimit; equivalently, we are to prove bijective the canonical comparison

κ : colim_{k∈𝒥} 𝒱 -Gph((2, Ḡ), (X_{k}, A_{k})) → 𝒱 -Gph((2, Ḡ), (X̄, Ā)

of sets. We begin by proving κ surjective; that is to say, that every map (g, τ) : (2, Ḡ) → (X̄, Ā) factorizes through some (q_{k}, α_{k}) : (X_{k}, A_{k}) → (X̄, Ā). To give (g, τ) is to give a function g : 2 → X̄ picking out elements x, y ∈ X̄ and to give a map τ : G → Ā(x, y) in 𝒱 . We recall from Section 3, however, that the (α̂_{j})_{x,y} : Ã_{j}(x, y) → Ā(x, y) constitute a colimit cone for the functor Ã(x, y) : 𝒥 → 𝒱 ; so, G being λ-presentable in 𝒱_{0}, the map τ : G → Ā(x, y) factorizes as

\xymatrix{G \ar[r]^{σ} & Ã_{j}(x, y) \ar[r]^{(α̂_{j})_{x,y}} & Ā(x, y)}

for some j ∈ 𝒥. Here Ã_{j} is the object (q_{j})_{∗}A_{j}(q_{j})^{∗} of 𝒱 -Mat(X̄, X̄), so that

Ã_{j}(x, y) = ∑_{{q_{j}t=x \atop q_{j}s=y}} A_{j}(t, s).

This coproduct, however, is the λ-filtered colimit of its sub-coproducts indexed by subsets of q_{j}^{−1}(x)×q_{j}^{−1}(y) of cardinality less than λ; so, G being λ-presentable, σ factorizes through such a sub-coproduct, say ∑_{ν∈N} A_{j}(t_{ν}, s_{ν}) where q_{j}t_{ν} = x and q_{j}s_{ν} = y for all ν ∈ N and where cardN < λ. Using yet again the λ-filteredness of 𝒥, there is some arrow θ : j → k in 𝒥 for which all the X_{θ}t_{ν} are equal and all the X_{θ}s_{ν} are equal: say

X_{θ}t_{ν} = t̄∈ X_{k} and X_{θ}s_{ν} = s̄ ∈ X_{k} for all ν ∈ N.

Using the factorization above of τ, we have

τ = (α̂_{j})_{x,y}σ = (α̂_{k})_{x,y}(Ã_{θ})_{x,y}σ = (α̂_{k})_{x,y}ρ,

where ρ = (Ã_{θ})_{x,y}σ, which by Section 3 is in fact ((q_{k})_{∗}Â_{θ}(q_{k})^{∗})_{x,y} σ. The point is that

this map

ρ : G → Ã_{k}(x, y) = ∑_{{q_{k}t′=x \atop q_{k}s′=y}} A_{k}(t′, s′)

factorizes through the coprojection of a single summand A_{k}(t̄, s̄), say as the composite of this coprojection with the map φ : G → A_{k}(t̄, s̄) of 𝒱 . Now the pair (t̄, s̄) determines a function h : 2 → X_{k}, so that we have in 𝒱 -Gph the map (h, φ) : (2, Ḡ) → (X_{k}, A_{k}); but the composite of this with (q_{k}, α_{k}) is (g, τ). So (g, τ) does indeed factorize through some (q_{k}, α_{k}), which completes the proof that the canonical comparison κ is surjective.

It remains to show that κ is injective. Suppose then that (h, φ) : (2, Ḡ) → (X_{k}, A_{k}) and (h′, φ′) : (2, Ḡ) → (X_{k′}, A_{k′}) are maps in 𝒱 -Gph with (q_{k}, α_{k})(h, φ) = (q_{k′}, α_{k′})(h′, φ′). We are to show that there exist θ : k → j and θ′ : k′ → j in 𝒥, with (X_{θ}, Ã_{θ})(h, φ) = (X_{θ′}, Ã_{θ′})(h′, φ′). Since 𝒥 is λ-filtered, there certainly do exist maps θ : k → j and θ′_{0} : k′ → j, and so without loss of generality we may suppose that k = k′.

Write (t, s) for (h0, h1) and (t′, s′) for (h′0, h′1), so that φ : G → A_{k}(t, s) and φ′ : G → A_{k}(t′, s′). Since q_{k}h = q_{k}h′, there is some θ : k → j with X_{θ}t = X_{θ}t′ and X_{θ}s = X_{θ}s′; in other words, we may suppose without loss of generality that t = t′ and s = s′. Now, therefore, h = h′ : 2 → X_{k} corresponds to (t, s) ∈ X_{k}, and φ, φ′ : G → A_{k}(t, s) have α_{k}φ = α_{k}φ′. Write t̄ for q_{k}t ∈ X̄, and s̄ for q_{k}s, and recall from Section 3 that we have in 𝒱 the colimit cone ((α̂_{j})_{t̄,s̄} : Ã_{j}(t̄, s̄) → Ā(t̄, s̄))_{j∈𝒥} , where

Ã_{j}(t̄, s̄) = ∑_{{q_{j}t̃=t̄ \atop q_{j} s̃=s̄}}A_{j}(t̃, s̃).

Composing φ and φ′ with the (t, s)-coprojection for Ã_{k} gives us two maps ψ, ψ′ : G → Ã_{k}(t̄, s̄) in 𝒱 with (α̂_{k})_{t̄,s̄}ψ = (α̂_{k})_{t̄,s̄}ψ′. Because 𝒥 is λ-filtered and G is λ-presentable in 𝒱_{0}, there is some θ : k → j in 𝒥 for which Ã_{θ}(t̄, s̄)ψ = Ã_{θ}(t̄, s̄)ψ′. When we recall the definition of the functor Ã, we see that this gives exactly the equality (X_{θ}, A_{θ})(h, φ) = (X_{θ}, A_{θ})(h, φ′) that we need for the injectivity of κ.

Since, as we have remarked, the object 0 of 𝒱_{0} is λ-presentable for any regular cardinal λ, it follows from Lemmas 4.2 and 4.3 that, under the standing hypotheses of this section:

4.4. Proposition. 𝒱 -Gph is locally λ-presentable when 𝒱_{0} is so.

Combining this with Theorem 3.3 and using [5, Satz 10.3], we conclude that :

4.5. Theorem. If 𝒱 is a monoidal closed category whose underlying ordinary category 𝒱_{0} is locally λ-presentable, then 𝒱 -Cat is also locally λ-presentable.

We can be more specific, in the sense of actually exhibiting a strong generator for 𝒱 -Cat consisting of λ-presentable objects. We make use of the following simple and well-known general observations:

4.6. Lemma. Let F ⊣ U : 𝒜 → ℬ where 𝒜 and ℬ are cocomplete categories. Then (i) for an object G of ℬ, the object F G of 𝒜 is λ-presentable if G is λ-presentable and U preserves λ-filtered colimits; and (ii) if a small set 𝒢 of objects of ℬ constitutes a strong generator of ℬ, and if U reflects isomorphisms (as it surely does whenever it is monadic), then the set {F G | G ∈ 𝒢 } constitutes a strong generator of 𝒜.

Let us use F now for the left adjoint of the forgetful U : 𝒱 -Cat → 𝒱 -Gph. When 𝒱_{0} is locally λ-presentable, we can take for 𝒢 the full subcategory 𝒱_{λ} of 𝒱_{0} given by the λ-presentable objects, noting that it contains the initial object 0. By Lemmas 4.2 and 4.3, the (2, Ḡ) for G ∈ 𝒱_{λ} constitute a strong generator of 𝒱 -Gph consisting of λ-presentable objects. By Lemma 4.6 and Corollary 3.4, therefore, the F (2, Ḡ) for G ∈ 𝒱_{λ} constitute a strong generator of 𝒱 -Cat consisting of λ-presentable objects. In future we shall write 2_{G} for the 𝒱 -category F (2, Ḡ); it is characterized by the observation that to give a 𝒱 - functor from 2_{G} to a 𝒱 -category B is to give objects x and y of B along with a map G → B(x, y) in 𝒱 . The reader will easy verify that 2_{G} has two objects 0 and 1, with 2_{G}(0, 0) = 2_{G}(1, 1) = I, 2_{G}(0, 1) = G, 2_{G}(1, 0) = 0, and with the evident composition.

A standard result from the theory of locally presentable categories now gives:

4.7. Proposition. When 𝒱_{0} is locally λ-presentable, the class of λ-presentable objects in 𝒱 -Cat is the closure in 𝒱 -Cat under λ-small colimits of the 𝒱 -categories 2_{G}, where G is a λ-presentable object of 𝒱 .

Recall from Section 2 above that a monoidal closed category 𝒱 is locally λ-presentable as a closed category when its underlying ordinary category 𝒱_{0} is locally λ-presentable, and the λ-presentable objects of 𝒱_{0} are closed under the monoidal structure. Although our interest in local presentability for closed categories is rather secondary, we nonetheless record:

4.8. Proposition. If the symmetric monoidal closed category 𝒱 is locally λ-presentable as a closed category, then so is 𝒱 -Cat.

Proof. We must show that the λ-presentable 𝒱 -categories are closed under tensor product. By [9, (5.2)] it suffices to show that 2_{G} ⊗ 2_{H} is λ-presentable for all G, H ∈ 𝒱_{λ}.

Write ℐ for the 𝒱 -category with a single object ∗ and ℐ (∗, ∗) = I; to give a 𝒱 - functor ℐ → 𝒜 is just to give an object of 𝒜. Thus 𝒱 -Cat(ℐ , −) : 𝒱 -Cat → Set is the functor sending a 𝒱 -category to its set of objects. This has a right adjoint, and so preserves all colimits, whence ℐ is certainly λ-presentable.

The 𝒱 -category 𝒞 = 2_{G} ⊗ 2_{H} has four objects: (0, 0), (0, 1), (1, 0), and (1, 1), and hom-objects

𝒞 ((i, j), (i′, j′)) = \begin{cases}G if i = 0, i′ = 1, j = j′ \\ H if i = i′, j = 0, j′ = 1 \\ G ⊗ H if i = i′ = 0, j = j′ = 1 \\ I if i = i′, j = j′ \\ 0 otherwise \end{cases}

with the obvious composition maps. To give a 𝒱 -functor T : 2_{G} ⊗ 2_{H} → 𝒜, therefore, is to give four objects A = S(0, 0), B = S(0, 1), C = S(1, 0), D = S(1, 1) of 𝒜, along with maps α : G → 𝒜 (A, C), β : G → 𝒜 (B, D), γ : H → 𝒜 (A, B), and δ : H → 𝒜 (C, D) in 𝒱 rendering commutative the diagram

\xymatrix{G ⊗ H \ar[d]_{τ} \ar[rr]^{β⊗γ} &  & 𝒜 (B, D) ⊗ 𝒜 (A, B) \ar[d]^{M} \\ H ⊗ G \ar[r]_{δ⊗α} & 𝒜 (C, D) ⊗ 𝒜 (A, C) \ar[r]_{M} & 𝒜 (A, D)}  (∗)

wherein τ denotes the symmetry isomorphism and M the composition maps in 𝒜.

To give A, C, D along with α and δ is to give a 𝒱 -functor R : 3_{G,H} → 𝒜, where 3_{G,H} is the pushout

\xymatrix{ℐ \ar[r]^{0} \ar[d]_{1} & 2_{H} \ar[d] \\ _{2_{G}} \ar[r] & _{3_{G,H}}}

in 𝒱 -Cat. Similarly to give A, B, D along with β and γ is to give a 𝒱 -functor S : 3_{H,G} → 𝒜. To give T : 2_{G} ⊗ 2_{H} → 𝒜, therefore, is to give R and S with the same A and D and satisfying (∗); which is to say that 2_{G} ⊗ 2_{H} is the pushout

\xymatrix{2_{G⊗H} \ar[r]^{N} \ar[d]_{M} & 3_{H,G} \ar[d] \\ _{3_{G,H}} \ar[r] & 2_{G} ⊗ 2_{H}}

in 𝒱 -Cat, where M and N are the evident 𝒱 -functors. Since ℐ, 2_{G}, and 2_{H} are λpresentable, and since the λ-presentables are closed under finite colimits, it follows that 2_{G} ⊗ 2_{H} is λ-presentable, as desired.

5. 𝒱 -Cat is locally bounded if 𝒱 is so

For the first part of this section we suppose only that 𝒱 is cocomplete and monoidal closed. The functor ob : 𝒱 -Cat → Set sending a 𝒱 -category to its set of objects then has both adjoints: the left adjoint D sends a set X to the “discrete” 𝒱 -category with object-set X and DX(x, y) equal to 0 unless x = y in which case it is the unit I, and the right adjoint C sends X to the “chaotic” 𝒱 -category with object-set X and CX(x, y) = 1.

For a 𝒱 -functor F : 𝒜 → ℬ, we consider the set K of pairs (A, B) of objects of 𝒜 with F A = F B, and form the coequalizer Q : 𝒜 → 𝒞 in 𝒱 -Cat of the two “projections” P_{1}, P_{2} : DK → 𝒜. Since F P_{1} = F P_{2}, there is a unique 𝒱 -functor I : 𝒞 → ℬ satisfying IQ = F. Clearly Q is invertible if and only if F is injective on objects; if I is invertible then we say that F is a quotient on objects. Since the “congruence” K arising from F is the same as that arising from Q, we see that Q is a quotient on objects, while I is injective on objects by construction. The factorization is clearly functorial, and so we obtain a factorization system (𝒬, ℐ) on 𝒱 -Cat in which 𝒬 consists of the quotients on objects, and ℐ consists of those 𝒱 -functors that are injective on objects. (Although each Q ∈ 𝒬 is an epimorphism in 𝒱 -Cat—in fact a regular one — there can be 𝒱 -functors which are injective on objects but not monomorphic, and so (𝒬, ℐ) is not proper.)

Before turning to the main results of the section, recall that if ℋ is a class of arrows in 𝒱 , a 𝒱 -functor F : 𝒜 → ℬ is said to be locally in ℋ if each F : 𝒜 (A, B) → ℬ(F A, F B) is in ℋ .

We now suppose that 𝒱 is locally λ-bounded as a closed category, with respect to the proper factorization system (ℰ , ℳ).

Write ℳ′ for the class of 𝒱 -functors which are injective on objects and locally in ℳ; clearly every such 𝒱 -functor is a monomorphism. Because of the (ℰ , ℳ)-generator 𝒢, an object of 𝒱 has only a small set of ℳ -subobjects, from which it follows that an object of 𝒱 -Cat has only a small set of ℳ′-subobjects, and therefore admits arbitrary intersections of ℳ′-subobjects. Since ℳ′ is clearly closed under composition and intersections, and stable under pullback, it forms, by [3, Lemma 3.1], part of a factorization system (ℰ′, ℳ′ ) on 𝒱 -Cat. Since (ℰ , ℳ) is proper, every coretraction in 𝒱 lies in ℳ, and one now easily shows that every coretraction in 𝒱 -Cat lies in ℳ′, and so that (ℰ′, ℳ′ ) is proper.

It takes a little work to compute ℰ′, although it is easy to see that a 𝒱 -functor F : 𝒜 → ℬ in ℰ′ must be surjective on objects, since otherwise it would factorize through some non-invertible J : 𝒞 → ℬ which is injective on objects and fully faithful, and therefore lies in ℳ′. Now consider, for an F : 𝒜 → ℬ that is surjective on objects, its (𝒬, ℐ)-factorization F = IQ. Since here I, like F, is surjective on objects, it is in fact bijective on objects. The quotient-on-objects 𝒱 -functor Q, being a regular epimorphism in 𝒱 -Cat, certainly lies in the ℰ′ of the proper factorization system (ℰ′, ℳ′ ); whence it follows by [6, Proposition 2.1.1] that F lies in ℰ′ if and only if I lies in ℰ′. The case of a bijective-on-objects 𝒱 -functor, however, is dealt with in the following:

5.1. Lemma. A 𝒱 -functor F : 𝒜 → ℬ which is bijective on objects lies in ℰ′ if and only if it is locally in ℰ.

Proof. Suppose that F : 𝒜 → ℬ is bijective on objects and in ℰ′; without loss of generality we may suppose F to be the identity on objects. Let the (ℰ , ℳ)-factorization of F : 𝒜 (A, B) → ℬ(A, B) be

\xymatrix{𝒜 (A, B) \ar[r]^{E_{A,B}} & 𝒟(A, B) \ar[r]^{M_{A,B}} & ℬ(A, B).}

For objects A, B, C of 𝒜, we have E_{B,C} ⊗ E_{A,B} in ℰ, since the class ℰ is by assumption closed under tensor products, and we also have M_{A,C} in ℳ; thus there is a unique map M′ making commutative the diagram

\xymatrix{𝒜 (B, C) ⊗ 𝒜 (A, B) \ar[r]^{E_{B,C}⊗E_{A,B}} \ar[d]_{M} & 𝒟(B, C) ⊗ 𝒟(A, B) \ar[r]^{M_{B,C}⊗B_{A,B}} \ar[d]_{M′} & ℬ(B, C) ⊗ ℬ(A, B) \ar[d]^{M′′} \\ 𝒜 (A, C) \ar[r]_{E_{A,C}} & 𝒟(A, C) \ar[r]_{M_{A,C}} & ℬ(A, C),}

in which M and M′′ are the composition maps for 𝒜 and ℬ. The M′ give to the 𝒟(A, B) the structure of a 𝒱 -category 𝒟, for which the E_{A,B} constitute a 𝒱 -functor M : 𝒟 → ℬ which is the identity on objects: the point is that the 𝒱 -category axioms for 𝒟 follow from those for ℬ, since the M_{A,B} are monomorphic. Now the E_{A,B} constitute a 𝒱 -functor E : 𝒜 → 𝒟 which is the identity on objects, and F = ME provides a factorization of F with M ∈ ℳ′ . Since F ∈ ℰ′, this implies that M is invertible, and in particular that each M_{A,B} is so, so that F_{A,B} = M_{A,B}E_{A,B} lies in ℰ as required.

Conversely, a 𝒱 -functor which is bijective on objects and locally in ℰ factorizes through no proper ℳ′-subobject, and so must be in ℰ′.

This now gives:

5.2. Proposition. A 𝒱 -functor F : 𝒜 → ℬ is in ℰ′ if and only if it can be written as F = IQ where Q is a quotient on objects and I is bijective on objects and locally in ℰ.

As a first step to proving that 𝒱 -Cat is locally λ-bounded with respect to (ℰ′, ℳ′ ), we prove:

5.3. Lemma. 𝒱 -Cat admits arbitrary cointersections of maps in ℰ′.

Proof. Let (E_{i} : 𝒜 → ℬ_{i})_{i∈I} be a family of 𝒱 -functors, each lying in ℰ′. By wellordering the indexing set I, we can write these instead in the form (E_{α} : 𝒜 → ℬ_{α})_{α<δ} for some initial ordinal δ. We set out to define by transfinite induction a “descending” family F_{α} : 𝒜 → 𝒞_{α} of 𝒱 -functors in ℰ′. We set F_{0} : 𝒜 → 𝒞_{0} to be equal to 1 : 𝒜 → 𝒜. We take for F_{α+1} : 𝒜 → 𝒞_{α+1} the cointersection of F_{α} : 𝒜 → 𝒞_{α} and E_{α} : 𝒜 → ℬ_{α}; it lies in ℰ′, because ℰ′ is closed under any cointersections that exist. Finally, for a limit ordinal α, we take for F_{α} : 𝒜 → 𝒞_{α} the cointersection of all the F_{β} with β < α, provided that this exists; then F_{δ} : 𝒜 → 𝒞_{δ} is clearly the required cointersection of the E_{α} : 𝒜 → ℬ_{α}, if it exists.

Suppose it does not. Let γ be the first ordinal for which F_{γ} fails to exist. Then γ cannot be of the form α + 1, since binary cointersections certainly exist; thus γ is a limit ordinal. It cannot be small, since small cointersections exist. Since ob𝒜 has only a small set of epimorphic images in Set, the surjections obF_{α} : ob𝒜 → ob𝒞_{α} have become constant at some ordinal β < γ; so that the comparison functor F_{ρ}^{σ} : 𝒞_{ρ} → 𝒞_{σ} is bijective on objects whenever β ≤ ρ < σ < γ. Since F_{ρ}^{σ} is in ℰ′ by [6, Proposition 2.1.1], it is locally in ℰ by Lemma 5.1. But now the non-existence of the cointersection F_{γ} : 𝒜 → 𝒞_{γ} contradicts the hypothesis that 𝒱_{0} admits arbitrary cointersections of maps in ℰ.

We have seen that 𝒱 -Cat is a cocomplete category with a proper factorization system (ℰ′, ℳ′ ) for which 𝒱 -Cat admits arbitrary ℰ′-cointersections. It will therefore be locally λ-bounded if it has an (ℰ′, ℳ′ )-generator consisting of λ-bounded objects. Let 𝒢 be an (ℰ , ℳ)-generator for 𝒱_{0} consisting of λ-bounded objects; without loss of generality we may suppose that 𝒢 contains the initial object 0. Write 𝒢′ for the set of those 𝒱 - categories of the form 2_{G} for some G ∈ 𝒢. The reader will easily verify that 𝒢′ is an (ℰ′, ℳ′ )-generator for 𝒱 -Cat: the argument is essentially that used to prove Lemma 4.2. A little more work is required in showing that 2_{G} is λ-bounded in 𝒱 -Cat when G is so in 𝒱_{0}, since we first need the following lemma:

5.4. Lemma. Consider a small filtered family (F_{j} : 𝒜_{j} → ℬ)_{j∈J} in ℳ′; without loss of generality we take the functions obF_{j} : ob𝒜_{j} → obℬ to be set-inclusions. Then (F_{j} : 𝒜_{j} → ℬ)_{j∈J} is an ℳ′-union in 𝒱 -Cat precisely when obℬ is the union in Set of the ob𝒜_{j} and, for each pair X, Y of objects of ℬ, the family (F_{j} : 𝒜_{j}(X, Y) → ℬ(X, Y))_{j∈J_{X,Y}} is an ℳ -union in 𝒱_{0}, where J_{X,Y} is {j ∈ J|X and Y lie in ob𝒜_{j} }.

Proof. The F_{j} are an ℳ′-union if and only if the induced 𝒱 -functor F : colim_{j}𝒜_{j} → ℬ lies in ℰ′. This means in particular that it is surjective on objects; but F is in any case injective on objects, since the F_{j} are so, and 𝒥 is filtered. Thus F would need to be bijective on objects, and we saw in Lemma 5.1 that such a 𝒱 -functor lies in ℰ′ if and only if it is locally in ℰ. Thus the F_{j} are an ℳ′-union if and only if obℬ is the union of the ob𝒜_{j} and F is locally in ℰ.

Since 𝒥 is filtered, the colimit of the 𝒜_{j} is preserved by U : 𝒱 -Cat → 𝒱 -Gph. Thus

(colim_{j}𝒜_{j}) (X, Y) = colim_{j} ∑_{{F_{j}A_{j}=X \atop F_{j}B_{j}=Y}} 𝒜_{j}(A_{j}, B_{j}),

but then to say that F : colim_{j}𝒜_{j}(X, Y) → ℬ(X, Y) is in ℰ for all X and Y is just to say that (F_{j} : 𝒜_{j}(X, Y) → ℬ(X, Y))_{j∈J_{X,Y}} is an ℳ -union in 𝒱_{0}.

It now follows easily that 𝒱 -Cat(2_{G}, −) : 𝒱 -Cat → Set preserves λ-filtered ℳ′- unions if 𝒱_{0}(G, −) : 𝒱_{0} → Set preserves λ-filtered ℳ -unions, and so we have:

5.5. Proposition. 𝒱 -Cat is locally λ-bounded with respect to (ℰ′, ℳ′ ).

Finally, we look at the closed structure of 𝒱 -Cat in this context:

5.6. Theorem. If 𝒱 is a symmetric monoidal closed category which is locally λ-bounded as a closed category with respect to the proper factorization system (ℰ , ℳ), then 𝒱 -Cat is locally λ-bounded as a closed category with respect to the proper factorization system (ℰ′, ℳ′ ).

Proof. We must prove for each 𝒱 -category 𝒳 that 𝒳 ⊗ E is in ℰ′ if E is so; or, equivalently, that [𝒳 , M] is in ℳ′ if M is so. Suppose then that 𝒳 is a 𝒱 -category and that M : 𝒜 → ℬ lies in ℳ′. An object of [𝒳 , 𝒜] is a 𝒱 -functor from 𝒳 to 𝒜; since M is a monomorphism, [𝒳 , M] : [𝒳 , 𝒜] → [𝒳 , ℬ] is injective on objects. To see that [𝒳 , M] is locally in ℳ, let F, G : 𝒳 → 𝒜 be 𝒱 -functors. Then the hom-object [𝒳 , 𝒜](F, G) is given by the end

∫_{X∈𝒳} 𝒜 (F A, GA).

Each M : 𝒜 (F X, GX) → ℬ(MF X, MGX) lies in ℳ, and ℳ is closed under limits; it follows that

∫_{X∈𝒳} M : ∫_{X∈𝒳} 𝒜 (F X, GX) → ∫_{X∈𝒳} ℬ(MF X, MGX)

lies in ℳ; that is, that M : 𝒜 (F, G) → ℬ(MF, MG) does so.

References

[1] J. Adámek and J. Rosický, Locally presentable categories and accessible categories, LMS Lecture Notes Series 189, Cambridge University Press, Cambridge, 1994.

[2] R. Betti, A. Carboni, R. Street, and R. Walters, Variation through enrichment, J. Pure Appl. Alg. 29(1983), 109–127.

[3] C. Cassidy, M. Hébert, and G.M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc.(A) 38(1985), 287–329.

[4] S. Eilenberg and G.M. Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra (La Jolla, 1965), 421–562, Springer-Verlag, 1966.

[5] P. Gabriel and F. Ulmer, Lokal-Präsentierbare Kategorien, Lecture Notes in Mathematics 221, Springer-Verlag, 1971.

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

[7] André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102(1993), 20-78.

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

[9] G.M. Kelly, Structures defined by finite limits in the enriched context I, Cah. de Top. et Géom. Diff., XXIII(1982), 3–42.

[10] Michael Makkai and Robert Paré, Accessible categories: The foundations of categorical model theory, Contemporary Mathematics 104, Amer. Math. Soc., Providence, 1989.

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School of Mathematics and Statistics University of Sydney, NSW 2006 AUSTRALIA

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