Theory and Applications of Categories, Vol. 20, No. 6, 2008, pp. 74–84.

COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED

F. MARMOLEJO AND R. J. WOOD

Abstract. In this paper we show that eight coherence conditions suffice for the definition of a pseudodistributive law between pseudomonads.

1. Introduction

The definition of a pseudodistributive law of one pseudomonad over another is given in [Marmolejo, 1999]. We find there the four familiar diagrams given in the original definiton of a distributive law of [Beck, 1969], but commutativity is replaced by invertible 2-cells. These are required to satisfy coherence conditions; precisely the issue that [Marmolejo, 1999] addressed. There we find nine coherence conditions and a justification of why they should suffice. In the thesis [Tanaka, 2005] it is suggested that the nine coherence axioms given in that paper are incomplete, in the sense that one of the coherence axioms is missing, a concern echoed in [Tanaka, Power, 2006], now in the form one axiom may be missing. This kind of criticism casts doubt on the whole integrity of [Marmolejo, 1999] without pointing out where the mistake might be. We find this unacceptable. It is our contention that [Marmolejo, 1999] is fundamentally correct, if a little conservative in its efforts to provide a complete set of axioms. For here we show that in fact eight of the nine axioms of [Marmolejo, 1999] suffice.

We show here that the contentious condition (H-2) of [Tanaka, 2005] and, by duality, (coh 8) of [Marmolejo, 1999] are redundant. Our technique involves showing that given pseudomonads 𝔻 on A and 𝕌 on B, a lifting of a 2-functor F : A → B to the corresponding 2-categories of pseudoalgebras is classified by a strong transformation r : UF → F D together with two invertible modifications subject to two coherence conditions. Readers familiar with [Street, 1972] will at once recognize (F, r, ⋯) as a pseudo version of the notion of morphism of monads. When we apply this to the particular case A = B and 𝔻 = 𝕌, the corresponding classifying transformation r : UF → F U is what was defined in [Tanaka, 2005] as a pseudo-distributive law of 𝕌 over F — except that the condition (H-2) in that paper turns out to be redundant.

Incidentally, while it is true that [Marmolejo, 1999] does not give the definition of pseudoalgebra for a pseudomonad, such a definition does appear in [Marmolejo, 1997]. Perhaps this reference should have been more prominent. We also point out [Marmolejo, 2004], where we find a study of the algebras for the composite pseudomonad resulting from a distributive law.

The first author acknowledges financial support from the DGAPA of UNAM, which made possible his sabbatical stay at Dalhousie University, and also the hospitality of this latter institution. The second author thanks NSERC Canada for financial support.

Received by the editors 2008-01-10 and, in revised form, 2008-03-13.

Transmitted by Ross Street. Published on 2008-03-13.

2000 Mathematics Subject Classification: 18C15,18D05.

Key words and phrases: pseudomonads, pseudodistributive laws, coherence.

\begin{matrix}c \\ ◯ \end{matrix} F. Marmolejo and R. J. Wood, 2008. Permission to copy for private use granted.

For simplicity, we work in the context of 2-categories, 2-functors, strong transformations and modifications, but the results are true in the general context of bicategories, homomorphisms of bicategories, etc. Also for simplicity, we refrain in this paper from producing higher dimensional structures as in [Marmolejo, 1999] or [Tanaka, 2005], but we do feel that the paper would be incomplete without a proof of the equivalence between liftings and the transformations that classify them, so we include this in the paper.

2. From transitions to liftings to algebras

Let A and B be 2-categories, 𝕌 = (U, u, n, β_{𝕌}, η_{𝕌}, μ_{𝕌}) a pseudomonad on A, and 𝔻 = (D, d, m, β_{𝔻}, η_{𝔻}, μ_{𝔻}) a pseudomonad on B. Thus

[figure: U; U^{2}; U; U; uU; 1; n; Uu; 1; β_{𝕌}; η_{𝕌}; U^{3}; U^{2}; U^{2}; U,; Un; nU; n; n; μ_{𝕌}]

with β_{𝕌}, η_{𝕌}, and μ_{𝕌} invertible modifications satisfying the coherence conditions found in, say, [Marmolejo, 1997] and similary for 𝔻. Let F : A → B be a 2-functor.

2.1. Definition. A transition from 𝕌 to 𝔻 along F : A → B is a strong transformation r: DF → F U together with invertible modifications

[figure: F; DF; F U,; dF; F u; r; ω_{1}; D^{2}F; DF U; F U^{2}; DF; F U; Dr; rU; mF; F n; r; ω_{2}]  (1)

that satisfy the following coherence conditions:

[figure: DF; D^{2}F; DF U; F U^{2}; DF; F U; DdF; Dr; mF; rU; 1; F n; r; η_{𝔻}F^{−1}; ω_{2}] = [figure: DF; D^{2}F; DF U; F U; F U^{2}; F U; DdF; DF u; Dr; r; rU; F Uu; F n; 1; Dω_{1}; r_{u}^{−1}; F η_{𝕌}^{−1}]  (2)

[figure: D^{3}F; D^{2}F U; DF U^{2}; D^{2}F; D^{2}F; DF U; F U^{2}; DF; F U; D^{2}r; mDF; DmF; DrU; DF n; mF; Dr; mF; rU; F n; r; Dω_{2}; ω_{2}; μ_{𝔻}F] = [figure: D^{3}F; D^{2}F U; DF U^{2}; D^{2}F; DF U; DF U; F U^{2}; DF; F U; F U^{3}; F U^{2}; D^{2}r; mDF; mF U; DrU; DF n; rU^{2}; mF; Dr; rU; rU; F n; r; F Un; F nU; F n; ^{m_{r}^{−1}}; ω_{2}U; r_{n}^{−1}; F μ_{𝕌}; ω_{2}] (3)

In the case A = B a transition from 𝕌 to 𝕌 is the same thing as a distributive law of 𝕌 over F in the sense of Definition 5.1 in [Tanaka, 2005] without the coherence condition (H-2). We will show shortly that this condition follows from the other two.

A transition induces a lifting of F to algebras, F̂, whose definition is given by the following proposition. (See [Marmolejo, 1997] for the definition of algebras.)

2.2. Proposition. Let (r, ω_{1}, ω_{2}) be a transition from 𝕌 to 𝔻 along F : A → B. We define F̂ : A^{𝕌} → B^{𝔻} as follows. For an object

( [figure: A; UA; A; uA; 1; a; a_{1}], [figure: U^{2}A; UA; UA; A; Ua; nA; a; a; a_{2}] )

in A^{𝕌}, we define F̂(A, a, a_{1}, a_{2}) as

( [figure: F A; DF A; F UA; F A; dF A; 1; F uA; rA; F a; ω_{1}; F a_{1}] , [figure: D^{2}F A; DF UA; DF A; F U^{2}A; F UA; DF A; F UA; F A.; DrA; mF A; rUA; r; DF a; F Ua; F a; F n; rA; F a; ω_{2}A; r_{a}^{−1}; F a_{2}] )

For (f, f_{1}) : (A, a, a_{1}, a_{2}) → (B, b, b_{1}, b_{2}) in A^{𝕌}, we define F̂(f, f_{1}) as

(F f, [figure: DF A; DF B; F UA; F UB; F A; F B; DF f; F Uf; F f; rA; F a; rB; F b; r_{f}^{−1}; F f_{1}] ) ,

For ξ: (f, f_{1}) → (g, g_{1}) in A^{𝕌}, define F̂ξ = F ξ. Then F̂ : A^{𝕌} → B^{𝔻} is a 2-functor such that the diagram

\xymatrix{A^{𝕌} \ar[d] \ar[r]^{F̂} & B^{𝔻} \ar[d] \\ A \ar[r]_{F} & B}  (4)

commutes, where the vertical arrows are the usual forgetful functors.

Proof. We prove that F̂ is well defined on objects and leave the rest to the reader. Thus, we must show that F̂(A, a, a_{1}, a_{2}) is a 𝔻-algebra. Paste η_{𝔻}F A^{−1} on the left of the second coordinate of the above pasting. Use (2). Since (A, a, a_{1}, a_{2}) is a 𝕌-algebra, we can replace the pasting of F η_{𝕌}A^{−1} and F a_{2} by F a ∘ F Ua_{1}. Then replace the pasting of r_{uA}^{−1}, r_{a}^{−1} and F Ua_{1} by rA ∘ DF a_{1}. This gives us one of the equations.

For the other, paste D of the second coordinate of the above pasting with the second coordinate of the above pasting, and with μ_{𝔻}F A. Use (3). Using that (A, a, a_{1}, a_{2}) is a 𝕌-algebra, replace the pasting of DF a_{2}, r_{a}^{−1}, r_{nA}^{−1}, F a_{2} and F μ_{𝕌}A by the pasting of r_{a}^{−1}, r_{Ua}^{−1}, F a_{2}, F n_{a}^{−1} and F a_{2}. Conclude the calculation replacing the pasting of Dr_{a}^{−1}, r_{Ua}^{−1}, F n_{a}^{−1} and ω_{2}UA by the pasting of ω_{2}A, m_{F a}^{−1} and r_{a}^{−1}.

We show now the missing condition.

2.3. Theorem.

[figure: DF; D^{2}F; DF; DF U; F U^{2}; F U; dDF; 1; mF; Dr; r; rU; F n; β_{𝔻}F; ω_{2}] = [figure: DF; D^{2}F; F U; DF U; F U^{2}; F U; dDF; r; Dr; dF U; F uU; rU; F n; 1; d_{r}; ω_{1}S; F β_{𝕌}]  (5)

Proof. Lemma 9.1 of [Marmolejo, 1997], applied to F̂(UA, nA, β_{𝕌}A, μ_{𝕌}A) for every possible A gives us

[figure: DF U; D^{2}F U; DF U; DF U^{2}; F U^{3}; F U^{2}; DF U; F U^{2}; F U; dDF U; 1; mF U; DrU; rU; rU^{2}; F nU; DF n; F Un; F n; rU; F n; β_{𝔻}F U; ω_{2}U; r_{n}^{−1}; F μ_{𝕌}] = [figure: DF U; D^{2}F U; F U^{2}; DF U^{2}; F U; DF U; F U^{2}; F U.; dDF U; rU; DrU; dF U^{2}; F n; DF n; dF U; F uU; rU; F n; 1; d_{rU}; d_{F n}; ω_{1}U; F β_{𝕌}]

Precompose both sides with DF u. On top of both sides paste first with d_{DF u}, then with Dr_{u}, and then with DF η_{𝕌}. On the left and down of both sides paste first with r_{u}^{−1} and then with F η_{𝕌}^{−1}.

On the left replace the pasting of Dr_{u}, d_{DF u}, ω_{2}U, β_{𝔻}F U and r_{u}^{−1} by the pasting of r_{Uu}^{−1}, F n_{u}^{−1}, ω_{2} and β_{𝔻}F. Observe that the pasting of all the 2-cells left with the exception of β_{𝕌}F and ω_{2} is an identity.

On the right, replace the pasting of Dr_{u}, d_{DF u}, d_{rU} and r_{u}^{−1} by the pasting of d_{r} and d_{F Uu}. Now the pasting of DF η_{𝕌}, d_{F Uu}, d_{F n} and F η_{𝕌}^{−1} cancels out.

3. Transitions and liftings are essentially the same

3.1. Definition. Transitions (r, ω_{1}, ω_{2}) and (s, π_{1}, π_{2}) from 𝕌 to 𝔻 along F are said to be coherently isomorphic if there is an invertible α: r → s such that

ω_{1} = [figure: F; DF; F U,; α; dF; F u; s; r; π_{1}] ω_{2} = [figure: D^{2}F; DF U; F U^{2}; DF; F U.; Ds; Dr; sU; rU; mF; F n; r; s; π_{2}; α^{−1}; Dα; αU]

3.2. Proposition. If α : (r, ω_{1}, ω_{2}) → (s, π_{1}, π_{2}) is a coherent isomorphism between transitions from 𝕌 to 𝔻 along F, and G, H: A^{𝕌} → B^{𝔻} are the corresponding induced liftings, then there is a 2-isomorphism ψ : G → H such that

[figure: A^{𝕌}; B^{𝔻}; B; G; H; ψ]

is the identity, where the rightmost arrow is the usual forgetful functor.

Proof. For any (a_{1}, a_{2}) = (A, a, a_{1}, a_{2}) ∈ A^{𝕌}, define ψ(a_{1}, a_{2}) = (Id_{F A}, F a ∘ αA).

We now produce a transition from a lifting. Assume G: A^{𝕌} → B^{𝔻} such that

\xymatrix{A^{𝕌} \ar[d] \ar[r]^{G} & B^{𝔻} \ar[d] \\ A \ar[r]_{F} & B}

commutes. For every A ∈ A we have (β_{𝕌}A, μ_{𝕌}A) = (UA, nA, β_{𝕌}A, μ_{𝕌}A) ∈ A^{𝕌}. Thus we have the 𝔻-algebra G(β_{𝕌}A, μ_{𝕌}A) = (F UA, G(β_{𝕌}A, μ_{𝕌}A)_{0}, G(β_{𝕌}A, μ_{𝕌}A)_{1}, G(β_{𝕌}A, μ_{𝕌}A)_{2}):

( [figure: F UA; DF UA; F UA; dDUA; 1; G(β_{𝕌}A,μ_{𝕌}A)_{0}; G(β_{𝕌}A,μ_{𝕌}A)_{1}] , [figure: D^{2}F UA; DF UA; DF UA; F UA; DG(β_{𝕌}A,μ_{𝕌}A)_{0}; mF UA; G(β_{𝕌}A,μ_{𝕌}A)_{0}; G(β_{𝕌}A,μ_{𝕌}A)_{0}; G(β_{𝕌}A,μ_{𝕌}A)_{2}] ) .

For f : A → B in A, we have (Uf, n_{f}^{−1}) : (β_{𝕌}A, μ_{𝕌}A) → (β_{𝕌}B, μ_{𝕌}B) in A^{𝕌}. Applying G we obtain

(F Uf, [figure: DF UA; DF UB; F UA; F UB; DF Uf; G(β_{𝕌}A,μ_{𝕌}A)_{0}; G(β_{𝕌}B,μ_{𝕌}B)_{0}; F Uf; G(Uf,n_{f}^{−1})] ) : G(β_{𝕌}A, μ_{𝕌}A) → G(β_{𝕌}B, μ_{𝕌}B)

in B^{𝔻}. Given A ∈ A, we define

rA := ( \xymatrix{DF A \ar[r]^{DF uA} & DF UA \ar[r]^{G(β_{𝕌}A,μ_{𝕌}A)_{0}} & F UA,} )

and, for f : A → B a 1-cell in A, we define

r_{f} := [figure: DF A; DF B; DF UA; DF UB; F UA; F UB; DF f; DF uB; DF uA; DF Uf; G(β_{𝕌}A,μ_{𝕌}A)_{0}; G(β_{𝕌}B,μ_{𝕌}B)_{0}; F Uf; G(Uf,n_{f}^{−1})^{−1}; ^{DF u^{f}}]

For any φ: f → g: A → B in A,

Dφ: (Df, Dn_{f}^{−1}) → (Dg, Dn_{g}^{−1}) : (β_{𝕌}A, μ_{𝕌}A) → (β_{𝕌}B, μ_{𝕌}B)

is a 2-cell in A^{𝕌}. Applying G to Dφ it follows that F Uφ is a 2-cell in B^{𝔻}. It is now easy to see that with the given definitions:

3.3. Lemma. r: DF → F U is a strong transformation.

For A ∈ A, define

ω_{1}A := [figure: F A; DF A; F UA; DF UA; F UA; dF A; F uA; DF uA; dF UA; 1; G(β_{𝕌}A,μ_{𝕌}A)_{0}; d_{F uA}; G(β_{𝕌}A,μ_{𝕌}A)_{1}]

Observe that (nA, μ_{𝕌}A) : (β_{𝕌}UA, μ_{𝕌}UA) → (β_{𝕌}A, μ_{𝕌}A) in A^{𝕌}. Apply G to (nA, μ_{𝕌}A) and define ω_{2}A as the pasting

[figure: D^{2}F A; D^{2}F UA; DF UA; DF U^{2}A; F U^{2}A; DF UA; DF A; DF UA; F UA.; D^{2}F uA; mF A; DG(β_{𝕌}A,μ_{𝕌}A)_{0}; mF UA; DF uUA; G(β_{𝕌}UA,μ_{𝕌}UA)_{0}; F nA; G(β_{𝕌}A,μ_{𝕌}A)_{0}; 1; DF nA; DF uA; G(β_{𝕌}A,μ_{𝕌}A)_{0}; DF β_{𝕌}A; m_{F uA}^{−1}; G(nA,μ_{𝕌}A)^{−1}; G(β_{𝕌}A,μ_{𝕌}A)_{2}]

3.4. Proposition. If G: A^{𝕌} → B^{𝔻} is a lifting of F : A → B to algebras and r, ω_{1}, and ω_{2} are defined as above, then (r, ω_{1}, ω_{2}) is a transition from 𝕌 to 𝔻 along F.

3.5. Theorem. Let (r, ω_{1}, ω_{2}) be a transition from 𝕌 to 𝔻 along F and write G: A^{𝕌} → B^{𝔻} for the corresponding lifting. If (s, π_{1}, π_{2}) is the transition induced by G, then (r, ω_{1}, ω_{2}) and (s, π_{1}, π_{2}) are coherently isomorphic. Let G: A^{𝕌} → B^{𝔻} be a lifting to algebras of F : A → B and write (r, ω_{1}, ω_{2}) for the transition from 𝕌 to 𝔻 along F induced by G. If H is the lifting induced by (r, ω_{1}, ω_{2}), then there is a 2-natural isomorphism ψ: G → H such that U^{𝔻} ∘ ψ is the identity.

Proof. In one direction, define α: r → s as the pasting

[figure: DF; F U; DF U; F U^{2}; F U.; r; DF u; F Uu; 1; F n; rU; ω_{u}; F η_{𝕌}]

In the other, start with (A, a, a_{1}, a_{2}) ∈ A^{𝕌}. Now (a, a_{2}) : (β_{𝕌}A, μ_{𝕌}A) → (a_{1}, a_{2}) is a 1-cell in A^{𝕌}. We define ψ(a_{1}, a_{2}) as

(Id_{F A}, [figure: DF UA; F UA; DF A; DF A; F A; G(β_{𝕌}A,μ_{𝕌}A)_{0}; DF a; F a; DF uA; 1; G(a_{1},a_{2})_{0}; DF a_{1}; G(a,a_{2})^{−1}] ) .

COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED  81

4. Op-transitions

Dually, we have op-transitions: 4.1. Definition. An op-transition from 𝕌 to 𝔻 along F is a strong transformation r: F U → DF together with invertible modifications

[figure: F; F U; DF,; F u; dF; r; ω_{1}; F U^{2}; DF U; D^{2}F; F U; DF; rU; Dr; F n; mF; r; ω_{2}]  (6)

that satisfy the following coherence conditions:

[figure: F U; F U^{2}; DF U; D^{2}F; F U; DF; F uU; rU; F n; Dr; 1; mF; r; F β_{𝕌}; ω_{2}] = [figure: F U; F U^{2}; DF U; DF; D^{2}F; DF; F uU; dF U; rU; r; Dr; dDF; mF; 1; ω_{1}U; d_{r}; β_{𝔻}F]  (7)

[figure: F U^{3}; DF U^{2}; D^{2}F U; F U^{2}; F U^{2}; DF U; D^{2}F; F U; DF; rU^{2}; F Un; F nU; DrU; mF U; F n; rU; F n; Dr; mF; r; ω_{2}U; ω_{2}; F μ_{𝕌}^{−1}] = [figure: F U^{3}; DF U^{2}; D^{2}F U; F U^{2}; DF U; DF U; D^{2}F; F U; DF; D^{3}F; D^{2}F; rU^{2}; F Un; DF n; DrU; mF U; D^{2}r; F n; rU; Dr; Dr; mF; r; mDF; DmF; mF; r_{n}; Dω_{2}; m_{r}; μ_{𝔻}F^{−1}; ω_{2}] (8)

We obtain the dual of Theorem 2.3 4.2. Proposition.

[figure: F U; F U^{2}; F U; DF U; D^{2}F; DF; F Uu; 1; F n; rU; r; Dr; mF; F η_{𝕌}^{−1}; ω_{2}] = [figure: F U; F U^{2}; DF; DF U; D^{2}F; DF; F Uu; r; rU; DF u; DdF; Dr; mF; 1; r_{u}^{−1}; Dω_{1}; η_{𝔻}F^{−1}]  (9)

5. Distributive laws

We are now in a position to explain why there are eight coherence conditions for a pseudodistributive law.

5.1. Proposition. A distributive law of 𝕌 over 𝔻 consists of a transition (r: UD → DU, ω_{1}, ω_{3}) from 𝕌 to 𝕌 along D, together with an op-transition (r, ω_{2}, ω_{4}) from 𝔻 to 𝔻 along U that satisfy the following coherence conditions:

[figure: 1; U; D; DU; UD; u; d; dU; Du; Ud; r; d_{u}^{−1}; ω_{2}] = [figure: 1; U; D; UD; DU,; u; d; Ud; uD; Du; r; u_{d}; ω_{1}]  (10)

[figure: U^{2}; U^{2}D; UDU; DU^{2}; U; DU; U^{2}d; UdU; dU^{2}; n; Ur; rU; Dn; dU; Uω_{2}; ω_{2}U; d_{n}] = [figure: U^{2}; U^{2}D; U; UD; UDU; DU^{2}; DU,; U^{2}d; n; nD; Ur; Ud; dU; r; rU; Dn; n_{d}; ω_{3}; ω_{2}]  (11)

[figure: D^{2}; UD^{2}; DUD; D^{2}U; D; DU; uD^{2}; DuD; D^{2}u; m; rD; Dr; mU; Du; ω_{1}D; Dω_{1}; m_{u}^{−1}] = [figure: D^{2}; UD^{2}; D; UD; DUD; D^{2}U; DU,; uD^{2}; m; Um; rD; uD; Du; r; Dr; mU; u_{m}; ω_{4}; ω_{1}]  (12)

and

[figure: U^{2}D^{2}; UDUD; UD^{2}U; U^{2}D; UDU; DUDU; D^{2}U^{2}; DU^{2}; UD; DU; UD^{2}; DU^{2}D; UrD; U^{2}m; nD^{2}; UDr; rUD; UmU; rDU; Ur; nD; rU; DrU; mU^{2}; Dn; r; Um; DUr; Uω_{4}; ω_{4}U; ω_{3}; n_{m}^{−1}; r_{r}; =; U^{2}D^{2}; UDUD; DU^{2}D; UD^{2}; DUD; DUDU; D^{2}U^{2}; D^{2}U; DU^{2}; DU; UD; UrD; nD^{2}; rUD; DnD; DUr; rD; Um; Dr; DrU; Dn; mU^{2}; mU; Dn; r; ω_{3}D; Dω_{3}; m_{n}; ω_{4}] (13)

Proof. The references (coh1), …, (coh9) are to the paper [Marmolejo, 1999]. The coherence conditions for the transition of 𝕌 to 𝕌 along D correspond to the coherence conditions (coh2) and (coh4). The coherence conditions for the op-transition from 𝔻 to 𝔻 along U correspond to the coherence conditions (coh7) and (coh9). The coherence condition (coh8) follows from these last two according to Proposition 4.2. (10), (11), (12) and (13) are (coh1), (coh3), (coh5) and (coh6) rewritten.

References

[Beck, 1969] J. Beck, Distributive Laws, in Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics 80, 119-140.

84  F. MARMOLEJO AND R. J. WOOD

[Marmolejo, 1999] F. Marmolejo, Distributive Laws for Pseudomonads, Theory and Applications of Categories

5 (1999), No. 5, 91-147

[Marmolejo, 2004] F. Marmolejo, Distributive Laws for Pseudomonads II, Journal of Pure and Applied Algebra

194, No. 1-2, 2004, 169-182.

[Marmolejo, 1997] F. Marmolejo, Doctrines Whose Structure Forms a Fully Faithful Adjoint String, Theory

and Applications of Categories 3 (1997), No. 2, 24-44.

[Street, 1972] R. Street, The Formal Theory of Monads, Journal of Pure and Applied Algebra 2, 1972, 149-168. [Tanaka, 2005] Miki Tanaka, Pseudo-Distributive Laws and a Unified Framework for Variable Binding, Ph.D.

Thesis, University of Edinburgh, 2005.

[Tanaka, Power, 2006] Miki Tanaka and John Power, Pseudo-Distributive Laws and Axiomatics for Variable Binding, Higher-Order and Symbolic Computation, Vol. 19, No. 2-3, Springer Netherlands, 2006, 305-337.

Instituto de Matemáticas,

Universidad Nacional Autónoma de México,

Área de la Investigación Científica,

Ciudad Universitaria, México D.F. 04510,

México.

Department of Mathematics and Statistics,

Dalhousie University,

Chase Building, Halifax, Nova Scotia, Canada B3H 3J5.

     Email: quico@matem.unam.mx
           rjwood@dal.ca

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     ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/6/20-06.{dvi,ps,pdf}

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. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Transmitting editors.

Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Xerox Palo Alto Research Center: paiva@parc.xerox.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au 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 Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu 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: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca