Theory and Applications of Categories, Vol. 6, No. 7, 1999, pp. 83–93.

ENRICHED LAWVERE THEORIES

Dedicated to Jim Lambek

JOHN POWER

ABSTRACT. We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Moreover, the V -category of models of a Lawvere V -theory is equivalent to the V -category of algebras for the corresponding V -monad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where V is Cat, and explain how the correspondence extends to pseudo maps of algebras.

1. Introduction

In seeking a general account of what have been called notions of computation [13], one may consider a finitary 2-monad T on Cat, and a T -algebra (A, a), then make a construction of a category B and an identity on objects functor j : A ⟶ B. In making that construction, one is allowed to use the structure on A determined by the 2-monad, and that is all. For instance, given the 2-monad for which an algebra is a small category with a monad on it, then one possible construction would be the Kleisli construction. For another example, given the 2-monad for which an algebra is a small monoidal category A together with a specified object S, then a possible construction is that for which B(a, b) is defined to be A(S ⊗ a, S ⊗ b), with the functor j : A ⟶ B sending a map h to S ⊗ h.

We need a precise statement of what we mean by saying that these constructions only use structure determined by the 2-monad. In general, one may obtain one such definition by asserting that B(a, b) must be of the form A(fa, gb) for endofunctors f and g on A generated by the 2-monad; similarly for defining composition in B. Thus we need to know what exactly we mean by endofunctors on A and natural transformations between them that are generated by the 2-monad. This paper is motivated by a desire to make such notions precise.

In order to make precise the notion of functor generated by a finitary 2-monad T on a T -algebra (A, a), we first generalise from consideration of a finitary 2-monad on Cat to a finitary V -monad on V for any monoidal biclosed category that is locally finitely presentable as a closed category: we shall make that definition precise in the next section. We move to this generality primarily for simplicity but also because the computing application will need it later, for instance considering not mere categories but categories enriched in P oset or in the category of ω-cpo’s.

This work has been done with the support of EPSRC grants GR/J84205: Frameworks for programming language semantics and logic and R34723: The Structure of Programming Languages: syntax and semantics.

Received by the editors 1998 December 14 and, in revised form, 1999 August 27.

Published on 1999 November 30.

1991 Mathematics Subject Classification: 18C10, 18C15, 18D05.

Key words and phrases: Lawvere theory, monad.

\begin{matrix}c \\ ◯ \end{matrix} John Power 1999. Permission to copy for private use granted.

To support our main definition, we prove a theorem: we define the notion of a Lawvere V -theory, and we prove that to give a Lawvere V -theory is equivalent to giving a finitary V -monad on V. So, for a finitary V -monad T on V, there is a Lawvere V -theory L(T) for which the V -category of models of L(T), which we define, is equivalent to the V -category T -Alg of algebras for the monad. Conversely, given any Lawvere V -theory T , there is a corresponding finitary V -monad M(T ); and these constructions yield an equivalence between the category of finitary V -monads on V and that of Lawvere V -theories.

Once we have such a correspondence, we have the definition we seek: given a 2-monad T and a T -algebra (A, a), an endofunctor f on A is generated by T whenever it is in the image of L(T) in the model of L(T) determined by the algebra (A, a). Similarly, a natural transformation α : f ⇒ g is is said to be generated by T whenever it is the image of a 2-cell of L(T) in the model of L(T) determined by the algebra (A, a).

For cognoscenti of enriched categories, the definitions and results here, at least assuming our enrichment is over a symmetric monoidal category, should come as no great surprise. However, they are a little subtle. One’s first guess for a definition of Lawvere V -theory may be a small V -category with finite products, subject to a condition asserting single-sortedness. But that is too crude: not only does it not yield our theorem relating Lawvere theories and finitary monads, as in the case for Set (see [1]), but it does not agree with the known and very useful equivalences between finitary monads and universal algebra as in [9], [6], and in application to computation in [11] and [10], with an account for computer scientists in [14]. So we consider something a little more subtle: if V is symmetric, we define a Lawvere V -theory to be a small V -category with finite cotensors, subject to a single-sortedness condition. A model is therefore a finite cotensor preserving V -functor into V. In order to extend to nonsymmetric V, as for instance if V is the category of small locally ordered categories with the lax Gray tensor product [11], we need to introduce a twist: so a Lawvere V -theory is a small V_{t}-category with finite cotensors, subject to a single-sortedness condition, and a model is a finite cotensor preserving V_{t}-functor into V_{t}: that allows us to speak of the V -category of models, and prove our equivalence with finitary V -monads on V and their V -categories of algebras. There has been considerable development of categories enriched in monoidal biclosed categories [4, 5, 6, 11] recently: the main problem with extending the usual theory is that functor V -categories need not exist in general; but functor V -categories with base V_{t} do, and that suffices for our purposes here. If one is only interested in the case of symmetric V here, one may simply ignore all subscripts t, and one will have correct statements; of course, exponentials, i.e., terms of the form [X, −]_{l} and [X, −]_{r}, may also be abbreviated to [X, −].

The special case that V = Cat also contains a mild surprise. Given a 2-monad T on Cat, one’s primary interest lies in pseudo maps of algebras, i.e., maps of algebras in which the structure need only be preserved up to coherent isomorphism rather than strictly. Such maps are analysed in [2], with an explanation for computer scientists in [12]. One might imagine that, extending the above correspondence in this case, pseudo maps of algebras would correspond exactly to pseudo natural transformations between the corresponding models of the corresponding Lawvere 2-theory. But the position is a little more delicate than that: pseudo natural transformations allow too much flexibility to yield a bijection. However, given a pair of algebras, one still has an equivalence between the category of pseudo maps of algebras between the two algebras and that of pseudo natural transformations between the corresponding models of the corresponding Lawvere 2-theory. Hence one has a biequivalence between T -Alg_{p} and the 2-category of models of the Lawvere theory, with pseudo natural transformations. We explain that in Section 5.

I am surprised I have not been able to find our main results, in the case of symmetric V, in the literature on enriched categories. There has been work on enriched monads, for instance [9] and [6], and there has been work on enriched finite limit theories, primarily [8] and [5]. But the closest work to this of which I am aware is by Dubuc [3], and that does not account for the finitary nature of Lawvere theories, which is central for us. Of course, what we say here is not explicitly restricted to finitariness: one could easily extend everything to cardinality < κ for any regular κ.

The paper is organized as follows. In Section 2, we recall the basic facts about enriched categories we shall use to define our terms. In Section 3, we define Lawvere V -theories and their models, and prove that every finitary V -monad on V gives rise to a Lawvere V -theory with the same V -category of models. And in Section 4, we give the converse, i.e., to each Lawvere V -theory, we discover a finitary V -monad with the same V -category of models. We also prove the equivalence between the categories of Lawvere V -theories and finitary V -monads on V. Finally, in Section 5, we explain the more fundamental pseudo maps of algebras when V = Cat, and show how they relate to maps of Lawvere 2-theories.

For symmetric V, the standard reference for all basic structures other than monads is Kelly’s book [7]. For nonsymmetric V, a reasonable reference is [11], but the central results were proved in the more general setting of categories enriched in bicategories as in [4, 5, 6]. For 2-categories, the best relevant reference is [2], and for an explanation directed towards computer scientists, see [12].

2. Background on enriched categories

A monoidal category V is called biclosed if for every object X of V, both − ⊗ X: V_{∘} ⟶ V_{∘} and X ⊗ −: V_{∘} ⟶ V_{∘} have right adjoints, denoted [X, −]_{r} and [X, −]_{l} respectively. For monoidal biclosed locally small V, it is evident how to define V -categories, V -functors and V -natural transformations, yielding the 2-category V -Cat of small V -categories. The category V_{∘} enriches to a V -category with hom given by [X, Y]_{r}. Note that [X, Y]_{r} cannot be replaced by [X, Y]_{l} here, using the usual conventions of Kelly’s book [7]. One can speak of representable V -functors and there is an elegant theory of V -adjunctions, see for instance [4]. If V_{∘} is complete, then it is shown in [7] that V − Cat is complete. So we can speak of the Eilenberg-Moore V -category for a V -monad. If V is also cocomplete, there is an elegant theory of limits and colimits in V -categories generalising the situation for symmetric monoidal closed V, see for instance [5]. Note that for a V -category C, there is a construction of what should clearly be called C^{op}, but C^{op} is a V_{t}-category, not a priori a V -category.

In general, there is no definition of a functor V -category. However, if V is complete, for a small V -category C, one does have a functor V -category of the form [C^{op}, V_{t}], whose objects are V_{t}-functors from C^{op} into V_{t}, and with homs given by the usual construction for symmetric V. Details appear in [5], where it is denoted P C. This also agrees with Street’s construction [16], but the latter is formulated in terms of modules.

A V -monad on a V -category C consists of a V -functor T: C ⟶ C and V -natural transformations η: 1 ⇒ T and μ: T^{2} ⇒ T satisfying the usual three axioms. The Eilenberg-Moore V -category T -Alg has as objects the T_{∘}-algebras, where T_{∘} is the ordinary monad underlying T, on the ordinary category C_{∘} underlying C. Given algebras (A, a) and (B, b), the hom object T -Alg((A, a), (B, b)) is the equaliser in V of the diagram

[figure: C(A, B); C(a, B); C(T (A), B); T; C(T (A), b); C(T (A), T (B))]  (1)

Composition in T -Alg is determined by that in C. Moreover, there is a forgetful V - functor U: T -Alg ⟶ C, which has a left adjoint. If C is complete, then so is T -Alg, and U preserves limits.

Given monads T and S on C, a map of monads from T to S is a V -natural transformation α : T ⇒ S that commutes with the unit and multiplication data of the monads. With the usual composition of V -natural transformations, this yields the ordinary category Mnd(C) of monads on C.

A monoidal biclosed category V is called locally finitely presentable as a closed category if V_{∘} is locally finitely presentable, if the unit I of V is finitely presentable, and if x ⊗ y is finitely presentable whenever x and y are. Henceforth, all monoidal biclosed categories to which we refer will be assumed to be locally finitely presentable as closed categories.

A V -category C is said to have finite tensors if for every finitely presentable object x of V, and for every object A of C, [x, C(A, −)]_{r}: C ⟶ V is representable, i.e., if there exists an object x ⊗ A of C together with a natural isomorphism [x, C(A, −)]_{r} ≅ C(x ⊗ A, −). The V -category C has finite cotensors if it satisfies the dual condition that for every finitely presentable x in V and every A in C, the V_{t}-functor [x, C(−, A)]_{l}: C^{op} ⟶ V_{t} is representable, where V_{t} is the dual of V; i.e., there exists an object A^{x} of C together with a natural isomorphism [ x, C(−, A) ]_{l} ≅ C(−, A^{x}). A V -category C is cocomplete whenever C_{∘} is cocomplete, C has finite tensors, and x ⊗ −: C_{∘} ⟶ C_{∘} preserves colimits for all finitely presentable x. A cocomplete V -category C is locally finitely presentable if C_{∘} is locally finitely presentable, C has finite cotensors, and (−)^{x}: C_{∘} ⟶ C_{∘} preserves filtered colimits for all finitely presentable x. In general, a V -functor is called finitary if the underlying ordinary functor is so, i.e., if it preserves filtered colimits. We therefore denote the full subcategory of Mnd(C) determined by the finitary monads on C by Mnd_{f}(C). In [6], finitary monads on an lfp V -category are characterized in terms of algebraic structure.

For any V that is locally finitely presentable as a closed category, V is necessarily a locally finitely presentable V -category. The cotensor A^{X} is given by [X, A]_{l}. Given a small finitely cotensored V_{t}-category C, we write F C(C, V_{t}) for the full sub-V -category of [C, V_{t}] determined by those V -functors that preserve finite cotensors. It follows from Freyd’s adjoint functor theorem, and from the fact that the inclusion preserves cotensors, that F C(C, V_{t}) is a full reflective sub-V-category of [C, V_{t}], i.e., the inclusion has a left adjoint. It follows immediately from the definitions that the inclusion is also finitary.

Note that, in all that follows, if V is symmetric, one may simply disregard all subscripts t, and one will have correct statements, with correct proofs: if V is symmetric, then the symmetric monoidal category V_{t} is isomorphic to V, and of course [X, −]_{l} and [X, −]_{r} agree.

3. Enriched Lawvere theories

We seek a definition of a Lawvere V -theory. In order to validate that definition, we shall prove that the V -category of models of a Lawvere V -theory is finitarily monadic over V; and for any finitary V -monad T on V, the V -category T -Alg is the V -category of models of a Lawvere theory. More elegantly, we shall establish an equivalence between the ordinary category of finitary V -monads on V and the category of Lawvere V -theories.

First we shall give our definition of Lawvere V -theory. Observe that the full subcategory V_{f} of V determined by (the isomorphism classes of) the finitary objects of V has finite tensors given as in V and is small.

3.1. Definition. A Lawvere V -theory consists of a small V_{t}-category T with finite cotensors, together with an identity on objects strictly finite cotensor preserving V_{t}-functor ι : V_{f}^{op} ⟶ T .

It is immediate that if V = Set, this definition agrees with the classical one. We extend the usual convention for Set by informally referring to T as a Lawvere V -theory, leaving the identity on objects V_{t}-functor implicit. A map of Lawvere V -theories from T to S is a strict finite cotensor preserving V_{t}-functor from T to S that commutes with the V_{t}-functors from V_{f}^{op}. Together with the usual composition of V_{t}-functors, this yields a category we denote by Law_{V}.

We now extend the usual definition of a model of a Lawvere theory.

3.2. Definition. A model of a Lawvere V -theory T is a finite cotensor preserving V_{t}- functor from T to V_{t}. The V-category of models of T is F C(T , V_{t}), the full sub-V -category of [T , V_{t}] determined by the finite cotensor preserving V_{t}-functors.

3.3. Example. Let T be a V -monad on V. Then, one can speak of the Kleisli V -category Kl(T) for T, and there is a V -functor j : V ⟶ Kl(T) that is the identity on objects, has a right adjoint, and satisfies the usual universal property of Kleisli constructions, as explained in Street’s article [15]. Since j has a right adjoint, it preserves tensors. So if we restrict j to the finitely presentable objects of V, we have an identity on objects (strictly) finite tensor preserving V -functor from V_{f} to the full sub-V -category Kl(T)_{f} of Kl(T) determined by the objects of V_{f}. Dualising, we have a Lawvere V -theory, which we denote by L(T). This construction extends to a functor L : Mnd(V) ⟶ Law_{V}. Since our primary interest is in finitary V -monads, we also use L to denote its restriction to Mnd_{f}(V).

3.4. Theorem. Let T be a finitary V -monad on V. Then F C(L(T), V_{t}) is equivalent to T -Alg.

Proof. There is a comparison V -functor from Kl(T) to T -Alg, and it preserves tensors since the canonical functors from V into each of Kl(T) and T -Alg do. By composition with the inclusion of Kl(T)_{f} into Kl(T), we have a V -functor c : Kl(T)_{f} ⟶ T -Alg, and this yields a V -functor c̃: T -Alg ⟶ [Kl(T)_{f}^{op}, V_{t}] sending a T -algebra (A, a) to the V_{t}-functor T -Alg(c(−), (A, a)). This V -functor factors through F C(Kl(T)_{f}^{op}, V_{t}) since c preserves finite tensors and since representables preserve finite cotensors. Moreover, it is fully faithful since every object of T -Alg is a canonical colimit of a diagram in Kl(T)_{f}. So it remains to show that c̃ is essentially surjective.

Suppose h : Kl(T)_{f}^{op} ⟶ V_{t} preserves finite cotensors. Let A = h(I), where I is the unit of V. The behaviour of h on all objects is fully determined by its behaviour on I since h preserves finite cotensors and every object of Kl(T)_{f}^{op}, i.e., every object of V_{f}^{op}, is a finite cotensor of I. The behaviour of h on homs gives, for each finitely presentable x, a map in V from Kl(T)(I, x) to [A^{x}, A]_{l}, or equivalently, since cotensors in V are given by a left exponential, and by the usual properties of maps in Kleisli categories and maps from units, a map in V from [x, A]_{l} ⊗ T x to A. This must all be natural in x, thus, by finitariness of T, we have a map a : T A ⟶ A. Functoriality of h, together with the V_{t}-functor from V_{f}^{op} into Kl(T)^{op}, force a to be an algebra map. It is routine to verify that c̃((A, a)) is isomorphic to h.

4. The converse

In this section, we start by proving a converse to Theorem 3.4. This involves constructing a finitary monad M(T ) from a Lawvere V -theory T . Having proved that result, we establish that L together with M form an equivalence of categories between the categories of finitary V -monads on V and Lawvere V -theories, for any monoidal biclosed V that is locally finitely presentable as a closed category.

First we recall Beck’s monadicity theorem. Given a functor U : C ⟶ D, a pair of arrows h_{1}, h_{2} : X ⟶ Y in C is called a U-split coequaliser pair if there exist arrows h : UY ⟶ Z, i : Z ⟶ UY, and j : UY ⟶ UX in D such that h · Uh_{1} = h · Uh_{2}, i splits h, j splits h_{1}, and Uh_{2} · j = i · h. It follows that h is a coequaliser of Uh_{1} and Uh_{2}, and that coequaliser is preserved by all functors. Beck’s theorem says

4.1. Theorem. A functor U : C ⟶ D is monadic if and only if

• U has a left adjoint

• U reflects isomorphisms, and

• C has and U preserves coequalisers of U-split coequaliser pairs.

For a detailed account of Beck’s theorem, see Barr and Wells’ book [1]. Now we are ready to prove our main result.

4.2. Theorem. Let T be an arbitrary Lawvere V -theory. Then there is a finitary monad M(T ) on V such that M(T ) − Alg is equivalent to F C(T , V_{t}).

Proof. Recall from the definition of Lawvere V -theory, we have a finite cotensor preserving V_{t}-functor from V_{f}^{op} to T . By composition, this yields a V -functor U : F C(T , V_{t}) ⟶ F C(V_{f}^{op}, V_{t}), but the latter is equivalent to V: one proof of this is by applying Theorem 3.4 to the identity monad. The V -functor U is given by evaluation at I. Moreover, it has a left adjoint since the inclusion of F C(T , V_{t}) into [T , V_{t}] has a left adjoint, as mentioned in Section 2, and since evaluation functors from V -presheaf categories have left adjoints given by tensors. Moreover, it is finitary since (−)^{x} preserves filtered colimits for finitely presentable x. It remains to show that U is monadic.

We shall apply Beck’s monadicity theorem. It is routine to verify that U reflects isomorphisms, and one can readily calculate that U preserves the coequalisers of U-split coequaliser pairs: since F C(T , V_{t}) is a full reflective sub-V -category of [T , V_{t}], it is cocomplete; and the required coequaliser lifts to all objects as they are all finite cotensors of the unit I, so one need merely apply the associated cotensor to the given map and its splitting. That proves monadicity of the underlying ordinary functor of U; extending that to the enriched functor follows immediately from the fact that U respects cotensors.

Our construction M extends routinely to a functor M : Law_{V} ⟶ Mnd_{f}(V). Recall from the previous section that we also have a functor L : Mnd_{f}(V) ⟶ Law_{V}. In fact we have

4.3. Theorem. The functors M : Law_{V} ⟶ Mnd_{f}(V) and L : Mnd_{f}(V) ⟶ Law_{V} form an equivalence of categories.

Proof. Given a finitary V -monad T on V, it follows from Theorem 3.4 and the construction of M that ML(T) is isomorphic to T. For the converse, given a Lawvere V -theory T , we need to study the construction of M(T ). It is given by taking the left adjoint to the functor from F C(T , V_{t}) to V given by evaluation at the unit I. The left adjoint, applied to a finitely presentable object x of V, yields the representable functor T (x, −) : T ⟶ V_{t}: this follows since x is a finite tensor of I, and by preservation of finite cotensors. Now applying Yoneda yields the result that LM(T ) is isomorphic to T . Thus we are done.

5. 2-Monads on Cat

We now restrict our attention to the case that V = Cat. So we consider finitary 2-monads on Cat. An example is that there is a finitary 2-monad for which the algebras are small monoidal categories with a specified object S, as mentioned in the introduction. Another finitary 2-monad on Cat has an algebra given by a small category with a monad on it, again as in the introduction. There are many variants of these examples: see [2] or [12] for many more examples and more detail.

When one does have a 2-monad T, the maps of primary interest are the pseudo maps of algebras. These correspond to the usual notion of structure preserving functor. They are defined as follows.

5.1. Definition. Given T -algebras (A, a) and (B, b), a pseudo map of algebras from (A, a) to (B, b) consists of a functor f : A ⟶ B and a natural isomorphism

T A \overline{T f ^{}}T B
⇓f̄
[figure: A; a; f; B; b]

such that

[figure: T^{2}A; T^{2}f; T B; T^{2}A; T^{2}f; T B]
⇓^{T f̄}
[figure: T A; T a; T f; T B; T b; =; T A; μ_{A}; T f; T B; μ_{B}]
⇓f̄  ⇓f̄
[figure: A; a; f; B; b; A; a; f; B; b]

and

[figure: A; f; B; A; f; B]

[figure: =; T A; η_{A}; T f; T B; η_{B}]
⇓f̄
[figure: A; id_{A}; f; B; id_{B}; A; a; f; B; b]

A 2-cell between pseudo maps (f, f̄) and (g, ḡ) is a natural transformation between f and g that respects f̄ and ḡ.

By using the composition of Cat, it follows that T -algebras, pseudo maps of T -algebras, and 2-cells form a 2-category, which we denote by T -Alg_{p}. This agrees with the notation in some of the relevant literature, and this situation has undergone extensive study, in particular in [2]; and for an account directed towards computer scientists, see [12].

Similarly, given a small 2-category C, one has the notion of pseudo natural transformation between 2-functors h, k : C ⟶ Cat: one has isomorphisms where the definition of natural transformation has commuting squares, and those isomorphisms are subject to two coherence conditions, expressing coherence with respect to composition and identities in C. Thus, for any Lawvere 2-theory T , we have the 2-category F C_{p}(T , Cat), given by finite cotensor preserving 2-functors, pseudo natural transformations, and the evident 2-cells. If a finitary 2-monad T corresponds to the Lawvere 2-theory T , one might guess that the 2-equivalence between T -Alg and F C(T , Cat) would extend to a 2-equivalence between T -Alg_{p} and F C_{p}(T , Cat), but it does not!

5.2. Example. Let T be the identity 2-monad on Cat. Then T -Alg_{p} is 2-equivalent to Cat. But F C_{p}(T , Cat) is not, as it contains more maps. First, all functors lie in F C_{p}(T , Cat) via the inclusion of Cat, which is 2-equivalent to F C(T , Cat), in F C_{p}(T , Cat). But also, one may vary any component of a pseudo natural transformation α, say α_{2}, by an isomorphism, leaving every other component fixed, and vary the structural isomorphisms of α by conjugation, and one still has a pseudo natural transformation.

So the correspondence we seek is a little more subtle. The notion of 2-equivalence between 2-categories is quite rare. More commonly, one has a biequivalence. This amounts to relaxing fully faithfulness to an equivalence on homcategories, and similarly for essential surjectivity. In our case, we already have essential surjectivity. So we could ask whether our 2-functor c̃ : T -Alg ⟶ F C(T , Cat) extends to giving, for each pair of algebras ((A, a), (B, b)), an equivalence between T -Alg_{p}((A, a), (B, b)) and the category of pseudo natural transformations between c̃((A, a)) and c̃((B, b)), hence a biequivalence of 2-categories. In fact, we have

5.3. Theorem. The 2-functor c̃ : T -Alg ⟶ F C(T , Cat) extends to a biequivalence between T -Alg_{p} and F C_{p}(T , Cat).

Proof. If one routinely follows the argument for essential surjectivity of c̃ as in the proof of Theorem 3.4, and ones uses the 2-dimensional property of the colimit defining T A, one obtains a bijection between pseudo maps of T -algebras from (A, a) to (B, b), and those pseudo natural transformations between c̃((A, a)) and c̃((B, b)) that respect finite cotensors strictly, i.e., up to equality, not just isomorphism. Now, since cotensors are pseudo limits, they are automatically bilimits, and so every pseudo natural transformation is isomorphic to one that respects finite cotensors strictly. Local fully faithfulness is straightforward.

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THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is archived electronically and in printed paper format.

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Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at URL http://www.tac.mta.ca/tac/ or by anonymous ftp from ftp.tac.mta.ca in the directory pub/tac/info. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org Lawrence Breen, Université de 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@pmms.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.ruu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mscs.dal.ca Andrew Pitts, University of Cambridge: ap@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@charlie.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 Sydney: walters b@maths.usyd.edu.au R. J. Wood, Dalhousie University: rjwood@mscs.dal.ca