Theory and Applications of Categories, Vol. 5, No. 12, 1999, pp. 281–291.

LOCALIZATIONS OF MALTSEV VARIETIES

MARINO GRAN AND ENRICO MARIA VITALE

Transmitted by Aurelio Carboni

ABSTRACT. We give an abstract characterization of categories which are localizations of Maltsev varieties. These results can be applied to characterize localizations of naturally Maltsev varieties.

Introduction

In this paper we study localizations of Maltsev varieties.

A finitary (one sorted) variety V is called a Maltsev variety if its theory is equipped with a ternary operation p(x, y, z) satisfying the axioms p(x, y, y) = x, p(x, x, y) = y. The varieties of groups, abelian groups, modules over a fixed ring, crossed modules, rings, commutative rings, associative algebras, Lie algebras, quasi-groups and Heyting algebras are all examples of Maltsev varieties. The classical Maltsev theorem [11] asserts that the existence of such a ternary operation is equivalent to the fact that any reflexive relation in V is an equivalence relation. This semantical formulation of the Maltsev property has been the starting point for a purely categorical approach to Maltsev varieties developed in recent years in [3], [4], [8], [12] and [13].

The second aspect of our work concerns localizations. In 1964 P. Gabriel and N. Popescu proved that Grothendieck categories are exactly localizations of module categories [6]. This theorem has been extended to a non-abelian setting by the second author [16], who has characterized localizations of varieties as exact categories (in the sense of Barr [1]) with a regular generator and exact filtered colimits.

In this work we show that the categorical approach to Maltsev varieties and the exact completion technique developed in [15] and [16] fit together well, giving an abstract characterization of localizations of Maltsev varieties. In the first section we characterize localizations of Maltsev monadic categories, and we then restrict our attention to the finitary case (recall that varieties are exactly monadic categories over Set for a filtered colimit preserving monad). In the second section we specialize our results to localizations of naturally Maltsev monadic categories and naturally Maltsev varieties. These varieties are very closed to module categories, the only difference consisting in the existence of a zero object. In fact, the Gabriel-Popescu theorem is an obvious corollary of our result on naturally Maltsev varieties.

Received by the editors 1999 October 8.

Published on 1999 December 23.

1991 Mathematics Subject Classification: 18C15, 18E35.

\begin{matrix}c \\ ◯ \end{matrix} Marino Gran and Enrico Maria Vitale 1999. Permission to copy for private use granted.

1. Maltsev monadic categories

Let 𝕋 be a monad in Set: we denote by KL(𝕋) the Kleisli category of 𝕋 and by EM(𝕋) the Eilenberg-Moore category of 𝕋. The category KL(𝕋) is a projective cover of EM(𝕋), this meaning that each object of KL(𝕋) is regular projective in EM(𝕋) and for each object X in EM(𝕋) there is an object P in KL(𝕋) with a regular epimorphism P → X. The category EM(𝕋) is the exact completion of KL(𝕋) [14]: this fact has a central role in the characterization of the localizations of monadic categories over Set (we use the term “monadic category” for a category equivalent to EM(𝕋) for a monad 𝕋 in Set). In the following all categories we consider will be assumed to be locally small.

1.1. Theorem. [15] For a category B the following conditions are equivalent:

1. B is equivalent to a localization of a monadic category over Set

2. B is exact and has a regular generator G which admits all copowers

Let us recall the construction involved in the proof of the previous theorem and fix some notations. Let B be a category as in condition 2. of the theorem: we denote by C the full subcategory of B spanned by copowers of G. If S is a set we write S • G for the S-indexed copower of G. If C_{ex} is the exact completion of C, we denote by Γ and U the canonical full inclusions as in the diagram

\xymatrix{ & C_{ex} \ar[d]_{r} \\ C \ar[ur]^{Γ} \ar[r]_{U} & B. \ar[u]_{i}^{⊣}}  (1.)

The universal property of C_{ex} gives a unique exact (left exact and regular epimorphism preserving) functor r: C_{ex} → B with r ∘ Γ = U. Moreover, there is a fully faithful functor i: B → C_{ex} (right adjoint to r) defined as follows: if A ∈ B, one considers its canonical cover a: B(G, A) • G → A, then the kernel pair of a

\xymatrix{a_{1}, a_{2} : N(a) \ar[r] \ar[r] & B(G, A) • G}

and finally its canonical cover n: B(G, N(a)) → N(a). The object i(A) in C_{ex} is given by the pseudo equivalence relation

\xymatrix{a_{1} ∘ n, a_{2} ∘ n: B(G, N(a)) • G \ar[r] \ar[r] & B(G, A) • G} ,

with a similar definition on arrows.

The next simple result is stated separately for future references; it can be proved by using the explicit description of the exact completion of a category with finite weak limits 1.2. Lemma. Let B be an exact category with a regular generator G which admits all copowers. Then, with the notations as above, one has that Γ = i ∘ U (up to natural isomorphisms).

We are interested in localizations of Maltsev monadic categories over Set. In a category with finite limits there are several equivalent definitions of the Maltsev property [3]. We adopt the simplest one: a category C is Maltsev if any reflexive relation in C is symmetric.

In [13] Pedicchio proved that, for a monadic category, the Maltsev property is completely determined by the behaviour of the regular projective regular generator P = F (1) (the free algebra on one generator). In order to state her result we recall that an object X in a category with finite sums is an internal Maltsev coalgebra if there exists an arrow t: X → X + X + X such that the following diagram commutes

\xymatrix{ & X \ar[dl]^{i_{2}} \ar[dr]_{i_{1}} \ar[d]_{t} \\ X + X & X + X + X \ar[l]^{∇+1} \ar[r]_{1+∇} & X + X,}

where i_{1} and i_{2} are the canonical injections in the sum and ∇ is the codiagonal.

1.3. Theorem. [13] The following conditions are equivalent for a category B:

1. B is a Maltsev monadic category over Set

2. B is exact with a regular projective regular generator P which admits all copowers and P is an internal Maltsev coalgebra

It is then natural to ask whether the Maltsev property can be determined by a condition on the regular generator in the case of localizations of Maltsev monadic categories as well. A positive answer to this question is given in the Proposition 1.6 below; for this, the following lemma, which is a special case of Lemma 7. in [12], will be needed.

1.4. Lemma. Let A and B be finitely complete categories and let \xymatrix{R \ar[r]^{d} \ar[r]_{c} & A} be a relation

in A. If F: A → B is a left exact and conservative functor, then \xymatrix{R \ar[r]^{d} \ar[r]_{c} & A} is symmetric

precisely when \xymatrix{F (R) \ar[r]^{F (d)} \ar[r]_{F (c)} & F (A)} is symmetric.

Let us introduce a point of terminology:

1.5. Definition. Let A be a regular category. A graph \xymatrix{X_{1} \ar[r]^{x_{0}} \ar[r]_{x_{1}} & X_{0}} in A is pseudo-

symmetric if its image factorisation \xymatrix{I \ar[r]^{^{\overline{x}^{0}}} \ar[r]_{_{\overline{x}_{1}}} & X_{0}} in A is a symmetric relation.

We can now prove our characterization:

1.6. Proposition. The following conditions are equivalent:

1. B is equivalent to a localization of a Maltsev monadic category over Set

2. B is exact with a regular generator G which admits all copowers and G is an internal Maltsev coalgebra

3. B is exact with a regular generator G which admits all copowers and the functor B(G, −) sends reflexive graphs to pseudo-symmetric graphs

Proof. By Theorem 1.1 the category B is equivalent to a localization of a monadic category over Set if and only if it is exact with a regular generator G which admits all copowers.

1. ⇒ 2. By assumption B is a localization of a monadic Maltsev category A as in the diagram

\xymatrix{B \ar[r]_{i} & A \ar[l]^{⊥}_{r}} ,

where r preserves finite limits. Let P be the regular projective regular generator which is an internal Maltsev coalgebra in A by Theorem 1.3. The regular generator G = r(P) is then an internal Maltsev coalgebra in B, since r preserves sums.

2. ⇒ 3. Let C denote the full subcategory of B spanned by copowers of the regular generator G. The category B is equivalent to a localization of the exact completion C_{ex} of the category C (see Proposition 2.1 in [15] for a proof).

\xymatrix{ & C_{ex} \ar[d]_{r} \\ C \ar[ur]^{Γ} \ar[r]_{U} & B \ar[u]_{i}^{⊣}}  (1.)

The Maltsev coalgebra structure t: G → G + G + G on the generator G induces a Maltsev coalgebra structure on Γ(G), simply because Γ preserves all sums that exist in C [5]. By Theorem 1.3 the category C_{ex} is accordingly Maltsev: indeed, C is equivalent to KL(𝕋) for a monad 𝕋 over Set and C_{ex} is then equivalent to EM(𝕋) (by the way, this proves the implication 2. ⇒ 1.). Let us then consider a reflexive graph

\xymatrix{X_{1} \ar[r]^{c} \ar[r]|{d} & X_{0} \ar[l]_{e}}

in B. The reflexive graph \xymatrix{i(X_{1}) \ar[r] \ar[r] & i(X_{0})} can be factorised in C_{ex} as in the diagram

\xymatrix{i(X_{1}) \ar[r]^{i(c)} \ar[r]_{i(d)} \ar[dr]_{p} & i(X_{0}) \\  & I, \ar[u] \ar[u]}

where p is a regular epi and \xymatrix{I \ar[r] \ar[r] & i(X_{0})} is a reflexive relation in C_{ex}. C_{ex} is Maltsev so

that \xymatrix{I \ar[r] \ar[r] & i(X_{0})} is also symmetric: since Γ(G) is regular projective, it follows that the

image factorisation in Set of the reflexive graph \xymatrix{C_{ex}(Γ(G), i(X_{1})) \ar[r] \ar[r] & C_{ex}(Γ(G), i(X_{0}))} is a symmetric relation. By Lemma 1.2 one has that the graph

\xymatrix{C_{ex}(Γ(G), i(X_{1})) \ar[r] \ar[r] & C_{ex}(Γ(G), i(X_{0}))}

is isomorphic to the graph

\xymatrix{C_{ex}(i(G), i(X_{1})) \ar[r] \ar[r] & C_{ex}(i(G), i(X_{0}))} .

The functor i is fully faithful: it follows that the graph \xymatrix{B(G, X_{1}) \ar[r] \ar[r] & B(G, X_{0})} is a pseudo-symmetric graph in Set.

3. ⇒ 1. Let \xymatrix{X_{1} \ar[r]^{c} \ar[r]_{d} & X_{0}} be a reflexive relation in C_{ex} (with the same notations as above)

and we are going to prove that it is symmetric. Recall that there is a commutative diagram of functors

\xymatrix{Set^{𝕋} = C_{ex} \ar[r]^{V} & Set \ar[dr]_{−•G} \ar[rr]^{F} &  & C_{ex} = Set^{𝕋} \\  &  & C \ar[ur]_{Γ}}

where V is the forgetful functor and F is the “free algebra” functor. The reflexive relation above can be covered by a regular epi between reflexive graphs, as in the diagram

\xymatrix{Γ(X′_{1}) \ar[r] \ar[r] \ar[d]_{ϵ_{X_{1}}} & Γ(X′_{0}) \ar[l] \ar[d]^{ϵ_{X_{0}}} \\ X_{1} \ar[r] \ar[r] & X_{0}, \ar[l]}

where the functor ( )′ is the composite (− • G) ∘ V and the vertical arrows are the counit components of the adjunction F ⊣ V. By applying the functor C_{ex}(Γ(G), −) to the commutative diagram above and by taking the regular image factorisation J of the reflexive graph on the top one gets the diagram

\xymatrix{C_{ex}(Γ(G), Γ(X_{1}′)) \ar[rr] \ar[rr] \ar[dd] \ar[dr]^{p} &  & C_{ex}(Γ(G), Γ(X_{0}′)) \ar[dd] \\  & J \ar[ur] \ar[ur] \ar@{.>}[dl]^{α} \\ C_{ex}(Γ(G), X_{1}) \ar[rr] \ar[rr] &  & C_{ex}(Γ(G), X_{0}).}

The factorization α making commutative the diagram above comes from the fact that

\xymatrix{C_{ex}(Γ(G), X_{1}) \ar[r] \ar[r] & C_{ex}(Γ(G), X_{0})}

is a relation; the arrow α is surjective because ϵ_{X_{1}} is a regular epi and Γ(G) is regular projective. Since Γ and U are full and faithful, the reflexive graph

\xymatrix{C_{ex}(Γ(G), Γ(X_{1}′)) \ar[r] \ar[r] & C_{ex}(Γ(G), Γ(X_{0}′))}

is isomorphic to the reflexive graph

\xymatrix{B(G, X_{1}′) \ar[r] \ar[r] & B(G, X_{0}′)} .

By assumption this latter is a pseudo-symmetric graph, so that the relation \xymatrix{J \ar[r] \ar[r] & C_{ex}(Γ(G), Γ(X′_{0}))} is symmetric. In the category of sets one can easily check that the symmetry of this relation induces a symmetry on the relation

\xymatrix{C_{ex}(Γ(G), X_{1}) \ar[r] \ar[r] & C_{ex}(Γ(G), X_{0}),}

by using the fact that the regular epimorphism α splits. The functor C_{ex}(Γ(G), −) is left exact and conservative (it is isomorphic to the forgetful functor V), so that by Lemma 1.4 the proof is complete.

We remark that the condition 3. in the Proposition 1.6 implies in particular that B is Maltsev. However, this property on reflexive graphs does not seem to be equivalent to the Maltsev property, since the functor B(G, −) does not preserve regular epimorphisms. In other words, we don’t know if an exact Maltsev category with a regular generator (with copowers) is necessarily a localization of a Maltsev monadic category.

1.7. Corollary. The following conditions are equivalent:

1. B is a Maltsev monadic category over Set

2. B is exact with a regular projective regular generator G which admits all copowers and the functor B(G, −) sends reflexive graphs to pseudo-symmetric graphs

Proof. With the same notations as above, this follows from the fact that when G ∈ B is regular projective, then the adjunction

\xymatrix{B \ar[r]_{i} & C_{ex} \ar[l]^{⊥}_{r}}

is an equivalence.

In the following we shall use the term variety for a finitary variety in the sense of universal algebra. An application of our previous result can be given to characterize localizations of Maltsev varieties.

1.8. Proposition. The following conditions are equivalent:

1. B is a localization of a Maltsev variety

2. B is exact with a regular generator G which admits all copowers, G is an internal Maltsev coalgebra and filtered colimits commute with finite limits in B

3. B is exact with a regular generator G which admits all copowers, the functor B(G, −) sends reflexive graphs to pseudo-symmetric graphs and filtered colimits commute with finite limits in B

Proof. The implications 1. ⇒ 2. and 1. ⇒ 3. follow from Proposition 1.6 above and Corollary 1.2 in [16]. To see that 2. ⇒ 1., we first fix some notations. Let C′ be the subcategory of C (as in Proposition 1.6) with same objects as C and arrows f: R•G → S•G the arrows in C such that, for each r ∈ R, f ∘ σ_{r}: G → R • G → S • G factorises through S′ • G for some finite subset S′ of S, where σ_{r}: G → R • G is the rth injection in the coproduct. We denote by C′_{ex} the exact completion of the category C′ and by Γ′: C′ → C′_{ex} the canonical embedding. We know that C′_{ex} is the category of algebras for the finitary part 𝕋′ of the monad 𝕋 which has C_{ex} as category of algebras.

\xymatrix{ & C′_{ex} \ar[r]_{q}|{⊤} & C_{ex} \ar[d]_{r} \ar[l]_{j} \\ C′ \ar[r]_{U′} \ar[ur]^{Γ′} & C \ar[r]_{U} \ar[ur]^{Γ} & B \ar[u]_{i}^{⊣}}  (2.)

In the diagram above the forgetful functor U′ gives rise to a morphism from 𝕋′ to 𝕋, inducing an adjunction q ⊣ j: C_{ex} → C′_{ex}. Observe that B is a localization of the Maltsev monadic category C_{ex} and it is also a localization of the variety C′_{ex} by Corollary 1.2 in [16]. The regular generator G has an internal Maltsev coalgebra operation t: G → G + G + G, and this t is an arrow that belongs to C′. It follows that Γ′(t): Γ′(G) → Γ′(G) + Γ′(G) + Γ′(G) is an internal Maltsev coalgebra operation in C′_{ex}, proving that C′_{ex} is Maltsev.

3. ⇒ 2. By Proposition 1.6 C_{ex} is a Maltsev monadic category. Accordingly, the object r(P) = r(Γ(G)) = r(i(G)) = G (using Lemma 1.2) is an internal Maltsev coalgebra.

2. Naturally Maltsev varieties

We first recall the notion of naturally Maltsev category introduced by Johnstone in [9]:

2.1. Definition. A finitely complete category C is naturally Maltsev if there exists a natural transformation μ from the functor A ↦ A × A × A to the identity functor on C, such that μ_{A}: A × A × A → A is a Maltsev operation on A, for any A ∈ C.

Let us denote by Grpd(C) and by RG(C) the categories of internal groupoids and internal reflexive graphs in a finitely complete category C. We shall need the following results:

2.2. Theorem. [9] For a finitely complete category C the following conditions are equivalent:

1. C is naturally Maltsev

2. the forgetful functor U: Grpd(C) → RG(C) is an isomorphism

2.3. Theorem. [7] If C is an exact Maltsev category, then the category Grpd(C) is closed in RG(C) under quotients.

We can now prove our result on localizations of naturally Maltsev monadic categories. Observe that, unlike the Maltsev case, the condition is now given on the subcategory B and not on the regular generator G.

2.4. Proposition. The following conditions are equivalent:

1. B is equivalent to a localization of a naturally Maltsev monadic category over Set

2. B is exact naturally Maltsev with a regular generator G which admits all copowers

Proof. 1. ⇒ 2. Trivial.

2. ⇒ 1. In the proof we shall refer to the diagram (1.). First remark that any reflexive graph in B is symmetric by Theorem 2.2. Accordingly, the functor B(G, −) sends reflexive graphs to pseudo-symmetric graphs in Set so that the category C_{ex} is Maltsev by Proposition 1.6. Let us then prove that any reflexive graph

\xymatrix{X_{1} \ar[r]^{c} \ar[r]|{d} & X_{0} \ar[l]_{e}}

in C_{ex} is an internal groupoid. As in the proof of Proposition 1.6, one can apply the functor ( )′ = (− • G) ∘ V to this reflexive graph and the counit components of the adjunction F ⊣ V give a regular epimorphism of reflexive graphs

\xymatrix{Γ(X′_{1}) \ar[r] \ar[r] \ar[d]_{ϵ_{X_{1}}} & Γ(X′_{0}) \ar[l] \ar[d]^{ϵ_{X_{0}}} \\ X_{1} \ar[r] \ar[r] & X_{0}. \ar[l]}

By Theorem 2.3 to complete the proof it suffices to show that

\xymatrix{Γ(X′_{1}) \ar[r] \ar[r] & Γ(X′_{0})}

is an internal groupoid in C_{ex}. The reflexive graph

\xymatrix{U(X′_{1}) \ar[r] \ar[r] & U(X′_{0})}

is an internal groupoid in B, the category B being naturally Maltsev by assumption. It follows that

\xymatrix{iU(X′_{1}) \ar[r] \ar[r] & iU(X′_{0})}

is an internal groupoid in C_{ex}, and this latter is isomorphic to the graph

\xymatrix{Γ(X′_{1}) \ar[r] \ar[r] & Γ(X′_{0})}

by Lemma 1.2.

2.5. Proposition. The following conditions are equivalent:

1. B is equivalent to a localization of a naturally Maltsev variety

2. B is exact naturally Maltsev with a regular generator G which admits all copowers and filtered colimits commute with finite limits

Proof. The non trivial implication is 2. ⇒ 1. We use the same notations as in the proof of Proposition 1.8 and we first consider the diagram (2.): the category C_{ex} is naturally Maltsev by Proposition 2.4 and C′_{ex} is Maltsev by Proposition 1.8. The rest of the proof runs as in Proposition 2.4, once one has proved that there exists a natural transformation from the functor j ∘ i ∘ U ∘ U′ to the functor Γ′ such that any component α_{A}: jiUU′(A) → Γ′(A) is a split epimorphism. We leave this verification to the reader (hint: use the explicit

description of the composite functor \xymatrix{B \ar[r]^{i} & C_{ex} \ar[r]^{j} & C′_{ex}} given in [16]). Any reflexive graph

\xymatrix{X_{1} \ar[r]^{c} \ar[r]|{d} & X_{0} \ar[l]_{e}}

in C′_{ex} is accordingly covered by a regular epi between reflexive graphs as the vertical composite in the diagram

\xymatrix{jiUU′(X′_{1}) \ar[r] \ar[r] \ar[d]_{α_{X′_{1}}} & jiUU′(X′_{0}) \ar[l] \ar[d]^{α_{X′_{0}}} \\ Γ′(X′_{1}) \ar[r] \ar[r] \ar[d]_{ϵ_{X_{1}}} & Γ′(X′_{0}) \ar[l] \ar[d]^{ϵ_{X_{0}}} \\ X_{1} \ar[r] \ar[r] & X_{0}. \ar[l]}

The reflexive graph

\xymatrix{jiUU′(X_{1}′) \ar[r] \ar[r] & jiUU′(X′_{0}) \ar[l]}

is an internal groupoid in the exact Maltsev category C′_{ex} and by Theorem 2.3 the proof is complete.

As a corollary we get the classical Gabriel-Popescu characterization of Grothendieck categories [6]. Recall that by the Tierney theorem [2] a category is abelian if and only if it is exact and additive, while Johnstone proved that a category with products is additive if and only if it is naturally Maltsev and has a zero object [9]. With the same notations as above, the result follows by Proposition 2.5 and the fact that B has a zero object precisely when C′_{ex} has one. Indeed, Proposition 2.5 states that C′_{ex} is an abelian variety and then, by Lawvere theorem in [10], it is equivalent to the category of modules over a unital ring. We have then proved the following result:

2.6. Corollary. [6] The following conditions are equivalent:

1. B is equivalent to a localization of a module category

2. B is abelian with a regular generator G which admits all copowers and filtered colimits commute with finite limits

References

[1] M. Barr, Exact Categories, LNM, 236, Springer-Verlag, Berlin, 1971, 1-120.

[2] F. Borceux, Handbook of Categorical Algebra 2, Encyclopedia of Mathematics, vol. 51, Cambridge University Press, Cambridge, 1994.

[3] A. Carboni, G.M. Kelly, M.C. Pedicchio, Some remarks on Maltsev and Goursat categories, Appl. Categ. Structures, 1, 1993, 385-421.

[4] A. Carboni, J. Lambek, M.C. Pedicchio, Diagram chasing in Mal’cev categories, J. Pure Appl. Algebra, 69, 1990, 271-284.

[5] A. Carboni, E.M. Vitale, Regular and exact completions, J. Pure Appl. Algebra, 125, 1998, 79-116.

[6] P. Gabriel, N. Popescu, Caractérisation des catégories abéliennes avec générateurs et limites inductives, Comp. Rend. Acad. Sc. Paris , 258, 4188-4190, 1964.

[7] M. Gran, Internal categories in Mal’cev categories, J. Pure Appl. Algebra, 143, 1999, 221-229.

[8] M. Gran, Central extensions of internal groupoids in Maltsev categories, Ph.D. Thesis, Université catholique de Louvain, Belgium, 1999.

[9] P.T. Johnstone, Affine categories and naturally Mal’cev categories, J. Pure Appl. Algebra, 61, 1989, 251-256.

[10] F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci., 50, 1963, 869-872.

[11] A.I. Mal’cev, On the general theory of algebraic systems, Mat. Sbornik N. S., 35, 1954, 3-20.

[12] M.C. Pedicchio, Maltsev categories and Maltsev operations, J. Pure Appl. Algebra, 98, 1995, 67-71.

[13] M.C. Pedicchio, On k-permutability for categories of T -algebras, Logic and algebra, Lecture Notes in Pure and Applied Mathematics, 180, 1996, 637-646.

[14] E.M. Vitale, On the characterization of monadic categories over Set, Cahiers de Top. et Géom. Diff. Catégoriques, XXXV, 1994, 351-358.

[15] E.M. Vitale, Localizations of algebraic categories, J. Pure Appl. Algebra, 108, 1996, 315-320.

[16] E.M. Vitale, Localizations of algebraic categories II, J. Pure Appl. Algebra, 133, 1998, 317-326.

Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium.

Email: gran@agel.ucl.ac.be and vitale@agel.ucl.ac.be

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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