Theory and Applications of Categories, Vol. 7, No. 11, pp. 227–235.

ON THE OBJECT-WISE TENSOR PRODUCT OF FUNCTORS TO MODULES

MAREK GOLASIŃSKI

Transmitted by Michael Barr

ABSTRACT. We investigate preserving of projectivity and injectivity by the object-wise tensor product of Rℂ-modules, where ℂ is a small category. In particular, let O(G, X) be the category of canonical orbits of a discrete group G, over a G-set X. We show that projectivity of RO(G, X)-modules is preserved by this tensor product. Moreover, if G is a finite group, X a finite G-set and R is an integral domain then such a tensor product of two injective RO(G, X)-modules is again injective.

1. Introduction

It is well-known that the tensor product of two projective R-modules is projective. One can easily check that the tensor product of two injective R-modules is injective for R being an integral domain or more generally, the product of a finite number of integral domains. But (even for a commutative ring R) this is not the case, in general. By [3], for a commutative noetherian ring R, the tensor product of any two injective R-modules is injective if and only if the local ring R_{𝔭} is quasi-Frobenius for any prime ideal 𝔭 in R.

Let G be a discrete group and O(G) the associated category of canonical orbits. The injectivity of the object-wise tensor product of injective functors from some categories associated with O(G) to vector spaces has been first extensively used but not proved in [4, 11] to study the equivariant rational homotopy theory and then applied in [5, 10] for further generalized investigations. The balance of the paper is devoted to preserving of the projectivity and injectivity by such a tensor product of functors from any small category ℂ to R-modules, called Rℂ-modules.

Section 2 deals with contravariant functors from a small category ℂ to R-modules. We start from the illuminating counterexample showing that the object-wise tensor product of two projective Rℂ-modules is not projective even for the category ℂ associated with a finite partially ordered set. We observe that the tensor product of two such projective functors is projective if and only the functor R(ℂ(C, −) × ℂ(D, −)) is projective for any objects C, D in the category ℂ. By [2], it is sufficient that the components of the comma category Y ↓ ℂ(C, −) × ℂ(D, −) have right zeros, where Y : ℂ → 𝕊et^{ℂ^{op}} is the Yoneda imbedding and 𝕊et the category of sets. Then we deduce that the object-wise tensor product of two projective RO(G, X)-modules is projective, where O(G, X) denotes the category of canonical orbits of a discrete group G, over a G-set X.

Received by the editors 2000 March 14.

Published on 2000 September 1.

2000 Mathematics Subject Classification: Primary 18G05; secondary 16W50, 55P91.

Key words and phrases: category of canonical orbits, injective (projective) Rℂ-module, linearly

compact k-module, tensor product.

\begin{matrix}c \\ ◯ \end{matrix} Marek Golasiński, 2000. Permission to copy for private use granted.

In section 3 we move to covariant functors from ℂ to R-modules and examine the dual problem, preserving of the injectivity by the object-wise tensor product of Rℂmodules. We show that such a tensor product of two injective RO(G, X)-modules is injective provided that the group G is finite, X a finite G-set and the tensor product of any two injective R-modules is injective. Then we deduce that the exterior, symmetric and tensor power constructions preserve injectivity of RO(G, X)-modules, for R being a field of zero characteristic. At the end, for R being a field, we conclude that the projectivity (resp. injectivity) of functors from O(G, X) to linearly-compact vector Rspaces are preserved. That is the crucial fact in [5] to extend the equivariant homotopy theory on the disconnected case.

The author is deeply grateful to Professor A.K. Bousfield for his communication and Example 2.1.

2. Projective functors

Let _{R}Mod be the category of left R-modules over a commutative ring R with identity and ℂ a small category with the set of objects |ℂ|. A covariant functor ℂ → _{R}Mod is called a left Rℂ-module and the functor category _{Rℂ}Mod of all left such modules is called the category of left ℂ-modules.

The object of this section is the category Mod_{Rℂ} of contravariant functors ℂ → _{R}Mod, alias right Rℂ-modules, called in the sequel simply Rℂ-modules. Notions like coproducts, products, injective, projective etc. are defined as usual. In particular, an Rℂ-module P is projective if the following problem

\xymatrix{ & P \ar[d] \ar@{-->}[dl] \\ M \ar[r] & N \ar[r] & 0}

has a solution. For a set X, let R(X) be the free R-module generated by X. Thus, a contravariant functor F : ℂ → 𝕊et to the category 𝕊et of sets gives a rise to the right Rℂ-functor RF such that (RF)(C) = R(F (C)), for any object C ∈ |ℂ|. In particular, for any object C ∈ |ℂ|, we get the representable Rℂ-modules Rℂ(−, C)). Here ℂ(D, C) is the set of morphisms from D to C in the category ℂ. An Rℂ-module is called to be free if it is isomorphic to a coproduct of representable functors. Then the notion of finitely generated has its usual meaning. For C ∈ |ℂ|, let T_{C} be the evaluation functor, i.e., T_{C}(M) = M(C) for any Rℂ-module M. Then the left adjoint of T_{C} is the Rℂ-module given by

S_{C}(M)(D) = ⨁_{D→C} M = R(ℂ(D, C)) ⊗_{R} M

for any R-module M. Observe that S_{C}(M) is a projective Rℂ-module provided that M is a projective R-module. For an Rℂ-module M and x ∈ M(C) denote C by |x|. Them there is an epimorphism

⨁_{x∈M} R(ℂ(−, |x|)) ⟶ M → 0.

In particular, it follows that a projective Rℂ-module is a direct summand of a free Rℂmodule.

If ℂ is an EI-category (i.e., any endomorphism in ℂ is an isomorphism) then by [8, Theorem 9.39] any projective Rℂ-module can be split into a direct sum of projective Rℂ-modules living over various group rings R[Aut(C)] for C ∈ |ℂ|, where Aut(C) is the automorphism group of the object C. In particular, it follows the result stated in [9]: if ℂ is a one way category and finite from below then any projective ℂ-module is of the form ⨁_{C∈|ℂ|} S_{C}(P_{C}) for some projective R-modules P_{C}.

If M_{1} and M_{2} are Rℂ-modules then their object-wise tensor product M_{1} ⊗_{R} M_{2} is defined to be the composition

ℂ ⟶^{Δ} ℂ × ℂ ⟶^{M_{1}×M_{2}} _{R}Mod ×_{R}Mod ⟶^{⊗_{R}} _{R}Mod,

where Δ : ℂ → ℂ × ℂ is the diagonal functor. As it was observed by Bousfield [1], the object-wise tensor product M_{1} ⊗_{R} M_{2} is not projective in general, for any projective Rℂ-modules M_{1} and M_{2}.

2.1. Example. Let ℂ be the category associated with the partially order set ({a, b, c, d, e}, <) with the relations: a < b, a < c, b < d, c < d, b < e and c < e. This set can be also given by the following oriented graph

\xymatrix{ & e \\  & d \\ b \ar[uur] \ar[ur] &  & c \ar[uul] \ar[ul] \\  & a. \ar[ul] \ar[ur]}

Consider the projective Rℂ-modules S_{d}(R) and S_{e}(R) for a nonzero ring R which can be also presented by the commutative diagrams

[figure: 0; R; R; R; R]  and  [figure: R; 0; R; R; R.]

Then their object-wise tensor product S_{d}(R) ⊗_{R} S_{e}(R) is given by the commutative diagram

[figure: 0; 0; R; R; R.]

Take two other projective Rℂ-modules S_{b}(R) and S_{c}(R) given by the commutative diagrams

[figure: 0; 0; R; 0; R]  and  [figure: 0; 0; 0; R; R.]

Let β : S_{b}(R) → S_{d}(R) ⊗_{R} S_{e}(R) and γ : S_{c}(R) → S_{d}(R) ⊗_{R} S_{e}(R) be the maps ofRℂ-modules determined by the identity on 0 and R, respectively. Then the map p = (β, γ) : S_{b}(R) ⊕ S_{c}(R) → S_{d}(R) ⊗_{R} S_{e}(R) is surjective for which does not exist any splitting q : S_{d}(R) ⊗_{R} S_{e}(R) → S_{b}(R) ⊕ S_{c}(R). To show this observe that any map φ : S_{d}(R) ⊗_{R} S_{e}(R) → S_{b}(R) is determined by its values φ(a), φ(b) and φ(c). From the commutativity of the diagram

[figure: 0; 0; 0; R; φ(b); R; 0; R; φ(c); R; R; φ(a)]

it follows that φ(c) = φ(a) = 0. Thus φ(b) = 0 and finally we get that φ = 0. In the same way one can show that any map ψ : S_{d}(R) ⊗_{R} S_{e}(R) → S_{c}(R) is also trivial and the result follows.

Since a projective Rℂ-module is a direct summand of a free Rℂ-module we can state

2.2. Lemma. The object-wise tensor product of any two projective Rℂ-modules is projective if and only if the Rℂ-module S_{C}(R) ⊗_{R} S_{D}(R) = R(ℂ(−, C) × ℂ(−, D)) is projective, for all C, D ∈ |ℂ|.

By [8, p. 186] a functor F : 𝔸 → 𝔹 between EI-categories is called to be admissible if the induced restriction functor Res_{F} : R𝔹-modules → R𝔸-modules sends finitely generated free resp. projective R𝔹-modules to finitely generated free resp. projective R𝔸modules for any commutative ring R with unit. Therefore, Lemma 2.2 yields

2.3. Remark. The object-wise tensor product of any two projective Rℂ-modules is projective if and only if the diagonal functor Δ : ℂ → ℂ × ℂ is admissible.

Let Y : ℂ → 𝕊et^{ℂ^{op}} be the Yoneda imbedding. For a functor F : ℂ^{op} → 𝕊et, let Y ↓ F denote the associated comma category. In the light of [2] the sufficient condition for the Rℂ-module S_{C}(R) ⊗_{R} S_{D}(R) = R(ℂ(−, C) × ℂ(−, D)) to be projective is that the components of the comma category Y ↓ ℂ(−, C) × ℂ(−, D) have right zeros.

Let now G be a discrete group and O(G) the category of its canonical orbits. For any subgroups H_{1}, H_{2} ⊆ G, the G-set G/H_{1} × G/H_{2} is in one-one correspondence with a disjoint union ⋃_{α∈A} G/L_{α}, for some subgroups L_{α} ⊆ G. Thus, the set O(G)(G/K, G/H_{1}) × O(G)(G/K, G/H_{2}) is in one-one correspondence with the disjoint union ⋃_{α∈A} O(G)(G/K, G/L_{α}) and there is an isomorphism R(O(G)(−, G/H_{1}))⊗_{R}(O(G)R(−, G/H_{2})) ≈ ⨁_{α∈A} R (O(G)(−, G/L_{α})) of RO(G)-modules.

More generally, for a G-set X, consider the category O(G, X) of canonical orbits over X. Objects in O(G, X) are G-maps x : G/H → X for an arbitrary subgroup H ⊆ G. A morphism from x : G/H → X to y : G/K → X is a G-map σ : G/H → G/K such that yσ = x. Thus objects in O(G, X) correspond to points in fixed point subsets of the G-set X.

2.4. Proposition. If G is a discrete group and X is a G-set then the set O(G, X) ((G/K, y), (G/H_{1}, x_{1})) × O(G)((G/K, y), (G/H_{2}, x_{2})) is in one-one correspondence with the disjoint union ⋃_{α∈A} O(G, X)((G/K, y), (G/L_{α}, gx_{1} = g′x_{2})), where the union runs over all distinct isotropy subgroups L_{α} of the points (gH_{1}, g′H_{2}) ∈ G/H_{1} × G/H_{2} with gx_{1} = g′x_{2}.

Proof. Let G/H_{1} × G/H_{2} = ⋃_{α∈A} G(a_{α}H_{1}, b_{α}H_{2}) be the disjoint union and L_{α} the isotropy subgroup of the point (a_{α}H_{1}, b_{α}H_{2}) ∈ G/H_{1} × G/H_{2}, for α ∈ A. Consider two maps φ_{1} : (G/K, y) → (G/H_{1}, x_{1}), φ_{2} : (G/K, y) → (G/H_{2}, x_{2}) in the category O(G, X) with φ(K) = g_{1}H_{1}, φ_{2}(K) = g_{2}H_{2}. Then g_{1}H_{1} = ga_{α}H_{1} and g_{2}H_{2} = gb_{α}H_{2}, for some α ∈ A. Hence y = g_{1}x_{1} = ga_{α}x_{1}, y = g_{2}x_{2} = gb_{α}x_{2} and thus a_{α}x_{1} = b_{α}x_{2} = g^{−1}y. If L_{α} is the isotropy group of the point (a_{α}H_{1}, b_{α}H_{2}) in the G-set G/H_{1} × G/H_{2} then a_{α}^{−1}L_{α}a_{α} ⊆ H_{1}, b_{α}^{−1}L_{α}b_{α} ⊆ H_{2} and g^{−1}Kg ⊆ L_{α}. Consequently the maps φ_{1} and φ_{2} determine a map ψ : (G/K, y) → (G/L_{α}, a_{α}x_{1} = b_{α}x_{2}) with ψ(K) = gL_{α}.

Conversely, let L_{α} be the isotropy group of the point (a_{α}H_{1}, b_{α}H_{2}) ∈ G/H_{1} × G/H_{2} and ψ : (G/K, y) → (G/L_{α}, a_{α}x_{1} = b_{α}x_{2}) a map in the category O(G, X). Then we can consider the maps η_{1} : (G/L_{α}, a_{α}x_{1} = b_{α}x_{2}) → (G/H_{1}, x_{1}) and η_{2} : (G/L_{α}, a_{α}x_{1} = b_{α}x_{2}) → (G/H_{2}, x_{2}) with η_{1}(L_{α}) = a_{α}H_{1} and η_{2}(L_{α}) = b_{α}H_{2}. Hence, we get the maps φ_{1} = η_{1}ψ : (G/K, y) → (G/H_{1}, x_{1}) and φ_{2} = η_{2}ψ : (G/K, y) → (G/H_{2}, x_{2}) and the result follows.

Consequently, in the light of Lemma 2.2, we may state

2.5. Theorem. If G is a discrete group, X a G-set and M_{1}, M_{2} are projective RO(G, X)- modules then their object-wise tensor product M_{1}⊗_{R}M_{2} is projective, for any commutative ring R.

In particular, O(G, ∗) = O(G), the orbit category of the discrete group G for a single point set ∗. Thus, the object-wise tensor product M_{1} ⊗_{R} M_{2} of projective RO(G)-modules M_{1} and M_{2} is projective.

3. Injective functors

The object of this section is the category _{Rℂ}Mod alias left Rℂ-modules. For C ∈ |ℂ|, the right adjoint of the evaluation functor T′_{C} is the Rℂ-module given by

S′_{C}(M)(D) = ∏_{D→C} M = Hom_{R}(R(ℂ(D, C)), M),

for any R-module M and D ∈ |ℂ|. Observe that S′_{i}(M) is an injective Rℂ-module provided that M is an injective R-module. Given an Rℂ-module M, fix an R-monomorphism 0 → M(C) → Q_{C} for any object C ∈ |ℂ|, where Q_{C} is an injective R-module. Then there is a monomorphism of Rℂ-modules

0 → M ⟶ ∏_{x∈M} Hom_{R}(R(ℂ(−, |x|)), Q_{C}).

In particular, it follows that an injective Rℂ-module is a direct summand of an injective Rℂ-module ∏_{x∈M} Hom_{R}(R(ℂ(C, −)), Q_{C}), where Q_{C} are injective R-modules. A full characterization of Rℂ-modules for an EI-category ℂ, has been presented in [6].

If ℂ is a finite category (i.e., with a finite set of morphisms) then Hom_{R}(R(ℂ(C, −)), M) ⊗_{R} Hom_{R}(R(ℂ(D, −)), N) ≈ Hom_{R}(R(ℂ(C, −) × ℂ(D, −)), M ⊗_{R} N) for any Rmodules M and N. One can easily check that the tensor product of two injective Rmodules is injective for R being an integral domain or more generally, the product of a finite number of integral domains. But (even for a commutative ring R) this is not the case, in general. This problem, for noetherian commutative rings R, has been intensively studied in [3]: The tensor product of two injective R-modules is injective if and only if the localization R_{𝔭} is a quasi-Frobenius ring, for any prime ideal 𝔭 in the ring R. Consequently, in the light of the above facts and Section 2 we can state

3.1. Proposition. Let R be a commutative ring with unit and such that the tensor product of any two injective R-modules in injective. If G is a finite group, X a finite G-set and M_{1}, M_{2} are injective RO(G, X)-modules then the object-wise tensor product M_{1} ⊗_{R} M_{2} is also injective. In particular, if R is an integral domain, then the tensor product of injective RO(G, X)-modules is injective.

Let G be a finite group, X a finite G-set, k a field and M = {M_{i}}_{i≥0} a graded left kO(G, X)-module. Then for any (G/H, x) ∈ |O(G, X)| we get a graded left k-module M(G/H, x) = {M_{i}(G/H, x)}_{i≥0}. Write |m| = i, for m ∈ M_{i}(G/H, x). We define graded left kO(G, X)-modules T^{n}M and S^{n}M (called the n’th tensor and symmetric power, respectively) as follows:

(T^{n}M)_{i}(G/H, x) = ⨁_{i_{1}+···i_{n}=i} M_{i_{1}}(G/H, x) ⊗_{k} ⋯ ⊗_{k} M_{i_{n}}(G/H, x)

and

(S^{n}M)_{i}(G/H, x) = (T^{n}M)_{i}(G/H, x)/(R^{n}M)_{i}(G/H, x)

for i, n ≥ 0, where (R^{n}M)_{i}(G/H, x) is the homogeneous k-submodule of (T^{n}M)_{i}(G/H, x) generated by elements m_{1} ⊗ ⋯ ⊗ m_{n} − (−1)^{|m_{l}||m_{l+1}|}m_{1} ⊗ ⋯ ⊗ m_{l+1} ⊗ m_{l} ⊗ ⋯ ⊗ m_{n} for m_{l} ∈ M_{|m_{l}|}(G/H, x) and l = 1, …, n.

Let π_{(G/H,x)} : T^{n}M(G/H, x) → T^{n}M(G/H, x)/R^{n}M(G/H, x) be the natural canonical map. If the characteristic of k is zero then there is a natural map

σ_{(G/H,x)} : S^{n}M(G/H, x) ⟶ T^{n}M(G/H, x)

such that

σ_{(G/H,x)}([m_{1} ⊗ ⋯ ⊗ m_{n}]) = \frac{1}{n!} ∑_{τ∈S_{n}} ϵ(τ)m_{τ(1)} ⊗ ⋯ ⊗ m_{τ(n)}

for m_{l} ∈ M_{|m_{l}|}(G/H, x) with (G/H, x) ∈ |O(G, X)| and l = 1, …, n, where S_{n} is the n’th symmetric group and ϵ : S_{n} → {−1, +1} the sign map. Then π_{(G/H,x)} ∘ σ_{(G/H,x)} = id_{S^{n}M(G/H,x)} for (G/H, x) ∈ |O(G, X)| and consequently S^{n}M is a direct summand of T^{n}M. Moreover, we define T M and SM, the graded tensor and symmetric left kO(G, X)- algebra, where (T M)_{i} = ⨁_{n≥0}(T^{n}M)_{i} and (SM)_{i} = ⨁_{n≥0}(S^{n}M)_{i} for i ≥ 0. Observe that SM = T M/RM, where RM is the homogeneous ideal of T M generated by elements x ⊗ y − (−1)^{|x||y|}y ⊗ x for x, y ∈ T M. Then from Proposition 3.1 we can deduce

3.2. Corollary. Let G be a finite group and X a finite G-set. If M = {M_{i}}_{i≥0} is a graded component-wise injective left kO(G, X)-module, then the graded left kO(G, X)- modules T^{n}M, S^{n}M and T M = {(T^{n}M)}_{n≥0}, SM = {(S^{n}M)}_{n≥0} are component-wise injective kO(G, X)-modules, for n ≥ 0.

For n ≥ 0, let Λ^{n}M denote the n’th exterior power and ΛM the exterior algebra, respectively of a graded left kO(G, X)-module M = {M_{i}}_{i≥0}. We can proceed in the same way as above to get

3.3. Remark. If M = {M_{i}}_{i≥0} is a graded component-wise injective left kO(G, X)- module, then the graded kO(G, X)-modules Λ^{n}M and ΛM = {(Λ^{n}M)}_{n≥0} are componentwise injective left kO(G, X)-modules, for n ≥ 0.

At the end, we move to the dual category (_{k}Mod)^{op} which, in view of [7], is isomorphic to the category _{k}Mod^{c} of linearly compact k-modules. Recall some properties of these modules (see [5, 7] for the details).

(1) A linearly topological k-module M is linearly compact if and only if its topological dual M^{∗} is discrete.

(2) For a linearly compact (resp. discrete) k-modules M and N their complete tensor product M ˆ⊗_{k}N is linearly compact (resp. discrete) and there is a topological isomorphism

M^{∗} ˆ⊗_{k}N^{∗} →^{≈} (M ˆ⊗_{k}N)^{∗}.

(3) If {M_{i}}_{i∈I} and {N_{j}}_{j∈J} are collections of linearly compact k-modules then there exists a topological isomorphism

∏_{i∈I} M_{i} ˆ⊗_{k} ∏_{j∈J} N_{j} →^{≈} ∏_{i∈I, j∈J} M_{i} ˆ⊗_{k}N_{j}.

Then we may dualize Theorem 2.5 and Proposition 3.1 to summarize our considerations by the following conclusion being the crucial fact in [5] to extend the equivariant homotopy theory on the disconnected case.

3.4. Corollary. (1) Let G be a discrete group and X a G-set. If M_{1}, M_{2} : O(G, X) → kMod^{c} are injective kO(G, X)-modules then the object-wise tensor product M_{1} ˆ⊗_{k}M_{2} is also an injective kO(G, X)-module.

(2) Let G be a finite group and X a finite G-set. If M_{1}, M_{2} : O(G, X) → _{k}Mod^{c} are projective kO(G, X)-modules then the object-wise tensor product M_{1} ˆ⊗_{k}M_{2} is also a projective kO(G, X)-module.

References

[1] A.K. Bousfield, Private communication.

[2] Ch. Ching-an Cheng, B. Mitchell, Flatness and projectivity of modules that come from ℂ-sets, Algebra, Topology and Category Theory, Academic Press, New York (1976), 33-44.

[3] Edgar E. Enochs, Overtoun M.G. Jenda, Tensor and torsion products of injective modules, J. Pure Appl. Algebra 76 (1991), 143-149.

[4] B.L. Fine, Disconnected equivariant rational homotopy theory and formality of compact G-Kähler manifolds, Ph. D. Thesis, Univ. Chicago (1992).

[5] , M. Golasiński, Injectivite models of G-disconnected simplicial sets, Ann. Inst. Fourier (Grenoble) 47 (1997), 1491-1522.

[6] ————- , Injectivity of functors to modules and DGA’s, Comm. Algebra 27(8),(1999), 4027-4038.

[7] S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloq. Publ. vol. XXVII (1942).

[8] W. Lück, Transformation Groups and Algebraic K-theory, Lecture Notes in Math. 1408, Springer-Verlag (1989).

[9] B. Mitchell, Some categories of cohomological dimensions one and two, Istituto Nazionale di Alta Matematica, Symposia Mathematica, vol. IV (1970), 243-254.

[10] L.S. Scull, Rational 𝕊^{1}-equivariant homotopy theory, Ph. D. Thesis, Univ. Chicago, Chicago Il. (1999).

[11] G.V. Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. 274 (1982), 509-532.

Faculty of Mathematics and Informatics Nicholas Copernicus University 87-100 Toruń, Chopina 12/18 Poland

Email: marek@mat.uni.torun.pl

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Editorial board.

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