Theory and Applications of Categories, Vol. 1, No. 1, 1995, pp. 1–9.

ORIENTED SINGULAR HOMOLOGY

MICHAEL BARR

Transmitted by Ronald Brown

ABSTRACT. We formulate three slightly different notions of oriented singular chain complexes and show that all three are naturally homotopic to ordinary singular chain complexes.

1. Introduction

A standard fact about simplicial homology theory on simplicial complexes is that the oriented chain group is homotopic to the ordered (that might more evocatively be called unoriented) chain group. This was first proved in [Eilenberg, 1944] as motivation for his new definition of ordered singular homology to replace the oriented singular homology as defined, for example, in [Lefschetz, 1942]. Curiously, Eilenberg does not seem to have raised, let alone answered, the question as to whether his definition led to the same homology groups as did Lefschetz’. For simplicial complexes, the two definitions coincide, since they give the same groups as simplicial homology, but nothing was said about what happens beyond that domain.

Eilenberg’s work was at least partly a response to a criticism by Čech of Lefschetz’ definition. Čech pointed out that the chain group used by Lefschetz (it is the group that will be denoted C/U below) has elements of order 2 and is thus not truly free. It is not altogether clear, at this remove, why this was considered such a disadvantage, but it evidently was (see [Eilenberg, Steenrod, 1952], page 206). Eilenberg observed, for example, that this torsion in the chain groups could cause difficulty in defining morphisms on the chains.

Eilenberg also pointed out in his 1944 paper that, although he had proved that oriented simplicial homology of simplicial complexes was isomorphic to ordered singular homology, even naturally so, he could not exhibit an arrow in either direction between the chain groups that induced this isomorphism. Instead he gave natural maps from the ordered simplicial chain groups to each of the other two that induced the isomorphism. It is an interesting consequence of the results here that we find natural maps in both directions between the ordered and oriented singular groups. One direction can be composed with the obvious natural map from the oriented simplicial to the oriented singular chain groups to give a natural map that is a homotopy equivalence from the oriented simplicial chain group to the ordered singular chains.

In the preparation of this paper, I have been assisted by a grant from the NSERC of Canada.

Received by the editors 9 March 1995 and, in revised form, 4 April 1995.

Published on 6 April 1995

AMS subject classification (1990): 55N10, 18G35 .

Key words and phrases: Oriented singular homology, acyclic models .

\begin{matrix}c \\ ◯ \end{matrix} Michael Barr 1995. Permission to copy for private use granted.

The main tool we use is the theorem on acyclic models which was formulated and proved in slightly different language in [Barr, Beck, 1966]. This version of the theorem was originally motivated by applications to homological algebra in which such things as standard simplexes do not exist. However, one of its virtues, crucial in the present application is that it does not require that the chain complex functors to be compared be free or even projective as does the usual version (see, for example, [Rotman, 1988], page 242, or [Dold, 1980], page 176).

The present paper was originally motivated by a question raised by Robert Milson about the equivalence of oriented and ordered singular homology, in connection with his research into cohomology rings. I am indebted to Jon Beck, who led me to the discussion of these definitions by Eilenberg and Steenrod in their book, which in turn led me to the papers of Eilenberg and Lefschetz cited above.

As usual, we identify the standard n-simplex Δ_{n} as

{(t_{0},…, t_{n}) ∈ R^{n+1} | t_{i} ≥ 0 and t_{0} + … + t_{n} = 1}

For a topological space X, a singular n-simplex in X is a continuous map σ : Δ_{n} ⟶ X. Incidentally, one does not realize until reading the old papers what a great simplification was the use of a standard simplex to define singular simplexes instead of allowing all possible homeomorphs of the n simplex modulo an equivalence relation.

It is clear that the permutation group Σ_{n+1} acts on the set of singular n simplexes by the formula

(pσ)(t_{0},…, t_{n}) = σ^{∘} p^{−1}(t_{0},…, t_{n}) = σ(t_{p^{−1}0},…, t_{p^{−1}n})

for p ∈ Σ_{n+1} and σ : Δ_{n} ⟶ X. The singular n-chain functor C_{n} assigns to each space X the free abelian group generated by the singular n-chains and is a module over Σ_{n+1} as well. One way of defining an oriented chain group, the one that gives Lefschetz’ definition, is to factor out of C_{n} the subgroup U_{n} generated by all chains of the form pσ − sgn(p)σ where σ is an n-simplex and p ∈ Σ_{n+1}. A second way is to factor U_{n} by the subgroup V_{n} generated by U_{n} and all singular n-simplexes σ such that pσ = σ for some transposition p ∈ Σ_{n+1}. Of course, if σ is such a simplex, then 2σ ∈ U_{n} so the only difference from the first definition is some 2-torsion. Finally, one might note that there remain other 2-torsion simplexes mod U_{n}, namely all those σ for which pσ = σ for some odd permutation σ. We let W_{n} denote the subgroup generated by these together with U_{n}. There is no analog to these in the simplicial theory, since one easily sees that in that theory, a simplex that is fixed under any permutation is fixed under some transposition.

One might reasonably call any of C_{n}/U_{n}, C_{n}/V_{n} or C_{n}/W_{n} the oriented singular complex functor. It follows from the theorem below that you can use whichever of the definitions is convenient.

1.1. Theorem. Each of the maps C ⟶ C/U ⟶ C/V ⟶ C/W is a homotopy equivalence.

This theorem states that this equivalence is in the functor category, which means that both the homotopy inverses and the homotopies themselves are natural transformations.

2. The oriented singular complex

2.1. FDP complexes. Our first job is to show that the three subfunctors U_{n}, V_{n} and W_{n} are the nth components of subcomplexes of C_{n}. We begin with the observation that any map (not just an order preserving one) from the finite ordinals f : m + 1 ⟶ n + 1 induces a map of singular chain functors C_{n} ⟶ C_{m}. In fact, if σ is such a singular n-simplex, σ^{∘} f : Δ_{m} ⟶ X is a singular m-simplex.

It is clear that any function m+1 ⟶ n+1 can be represented as a permutation of m+1 followed by an order preserving function. Since the order-preserving maps are composites of faces and degeneracies, it follows that the faces, degeneracies and permutations generate the action of all the maps. Accordingly, we define an FDP complex to be a functor on the category of finite non-empty sets and an augmented FDP complex to be a functor on the category of finite sets. This is equivalent to an (augmented) simplicial set {X_{n} | n ≥ 0} (resp. n ≥ −1), equipped with involutions p^{i} = p_{n}^{i} : X_{n} ⟶ X_{n} for 0 ≤ i ≤ n − 1 that satisfy the identities listed below. These involutions correspond to transposition of adjacent elements and are known to generate the symmetric group ([Sayers, 1934]). In stating these identities, we use the usual notation for face operators d^{i} = d_{n}^{i} : X_{n} ⟶ X_{n−1} and degeneracy operators s^{i} = s_{n}^{i} : X_{n} ⟶ X_{n+1} for 0 ≤ i ≤ n, satisfying the usual simplicial identities. The additional identities that the p^{i} are p^{i}p^{i+1}p^{i} = p^{i+1}p^{i}p^{i+1} and the following commutations with respect to the face and degeneracy operators:

d^{i}p^{j} = \{ \begin{matrix}p^{j−1}d^{i} & if i < j \\ d^{i+1} & if i = j \\ d^{i−1} & if i = j + 1 \\ p^{j}d^{i} & if i > j + 1 \end{matrix}

s^{i}p^{j} = \{ \begin{matrix}p^{j+1}s^{i} & if i < j \\ p^{i+1}p^{i}s^{i+1} & if i = j \\ p^{i−1}p^{i}s^{i−1} & if i = j + 1 \\ p^{j}s^{i} & if i > j + 1 \end{matrix}

2.2. Proposition. The subfunctors U_{n}, V_{n} and W_{n} of C_{n} are the nth components of subcomplexes U, V and W, resp. of C. Moreover, C/W is the torsion free quotient of both C/U and C/V.

Proof. Case of U_{n}. U_{n} is generated by the images of the 1 + p^{j}, j = 0,…, n − 1. We have

d^{∘}(1 + p^{j}) = ∑_{i=0}^{n}(−1)^{i}d^{i} ^{∘}(1 + p^{j})

= ∑_{i=0}^{j−1}(−1)^{i}d^{i} ^{∘}(1 + p^{j}) + (−1)^{j}d^{j}(1 + p^{j})

+ (−1)^{j+1}d^{j+1}(1 + p^{j}) + ∑_{i=j+2}^{n}(−1)^{i}d^{i} ^{∘}(1 + p^{j})

= ∑_{i=0}^{j−1}(−1)^{i}(1 + p^{j−1})^{∘} d^{i} + (−1)^{j}(d^{j} + d^{j+1})

+ (−1)^{j+1}(d^{j+1} + d^{j}) + ∑_{i=j+2}^{n}(−1)^{i}(1 + p^{j})^{∘} d^{i}

= (1 + p^{j−1})^{∘} ∑_{i=0}^{j−1}(−1)^{i}d^{i} + (1 + p^{j}) ∑_{i=j+2}^{n}(−1)^{i} ^{∘} d^{i}

⊆ U_{n−1}

Case of V_{n}. First I claim that V_{n} is generated by U_{n} and adjacent transpositions. Temporarily let V_{n} be the subgroup of C_{n} generated by U_{n} and adjacent transpositions. Suppose that σ is a simplex such that σ = σ^{∘} ( i j) for some i < j. If j − i = 1, then σ ∈ V_{n} by definition. Otherwise, make the inductive hypothesis that τ ∈ V_{n} whenever τ = τ ^{∘} ( i + 1 j). Then

σ^{∘} ( i i + 1 ) ( i + 1 j) = σ^{∘} ( i j) ( i i + 1 ) = σ^{∘} ( i i + 1 )

so that σ^{∘} ( i i + 1 ) ∈ V_{n}. But σ + σ^{∘} ( i i + 1 ) ∈ U_{n} ⊆ V_{n} and hence σ ∈ V_{n}.

Since d(U_{n}) ⊆ U_{n−1} we need only show that if σ = p^{i}σ, then d(σ) ∈ V_{n−1}. But d^{i}σ = d^{i}p^{i}σ = d^{i+1}σ so that the two terms (−1)^{i}d^{i}σ and (−1)^{i+1}d^{i+1}σ cancel. The remaining terms are of the form d^{j}σ for j < i or j > i + 1. In the first case, we have d^{j}σ = d^{j}p^{i}σ = p^{i−1}d^{j}σ so that d^{j}σ is fixed by p^{i−1} or by p^{i} and hence belongs to V_{n−1} and therefore so does dσ.

Case of W_{n}. What we will establish in this case is that C_{n}/W_{n} is the torsion-free quotient of C_{n}/U_{n}. Since the torsion subgroup is invariant under any homomorphism, it follows that any homomorphism that takes U_{n} to U_{n−1} also takes W_{n} to W_{n−1}.

Now let S be a set on which Σ_{m} acts for some m and suppose that C is the free abelian group generated by S and that U and W are subgroups of C defined as follows. U is generated by all ps − sgn(p)s for s ∈ S and p ∈ Σ_{m} and W is the subgroup generated by U as well as all s ∈ S such that ps = s for some odd permutation p ∈ Σ_{m}. I claim that under those circumstances, W/U is the torsion subgroup of C/U. It is sufficient to show that every element of W is torsion mod U, which is obvious as noted above, and that C/W is torsion free. It is sufficient to do this when S is transitive, since C, U and W all split into direct sums corresponding to orbits. In that case, C/U is generated by the class containing any element of S (which I suppose is non-empty). If s ∈ S is fixed by an odd permutation, then 2s ∈ U and C/U ≅ Z_{2}, while W = C. If not, then the isotropy group of S is included in the alternating group and then C/U = C/W = Z. In each case, W/U is the torsion subgroup of C/U.

It follows that we have chain complex functors C/U, C/V and C/W.

3. Acyclic models

Let G : X ⟶ X be a functor and ϵ : G ⟶ Id a natural transformation. We say that an augmented chain complex functor K is ϵ-presentable if for each n ≥ 0, there is a natural transformation θ_{n} : K_{n} ⟶ K_{n}G such that K_{n}ϵ^{∘} θ_{n} = id. We say that the augmented chain complex functor L ⟶ L_{−1} ⟶ 0 is G-contractible if LG ⟶ L_{−1}G ⟶ 0 has a natural contracting homotopy. 3.1. Theorem. Suppose X is a category and K ⟶ K_{−1} ⟶ 0 and L ⟶ L_{−1} ⟶ 0 are augmented chain complex functors from X to an abelian category. Suppose G is an endofunctor on X and ϵ : G ⟶ Id a natural transformation. If K is ϵ-presentable and L ⟶ L_{−1} ⟶ 0 is G-contractible, then any natural transformation K_{−1} ⟶ L_{−1} can be extended to a natural chain map K ⟶ L and any two such extensions are naturally homotopic.

Of course, it follows that if both K and L are ϵ-presentable and both K ⟶ K_{−1} ⟶ 0 and L ⟶ L_{−1} ⟶ 0 are G-contractible, then an isomorphism K_{−1} ⟶ L_{−1} extends to a homotopy equivalence K ⟶ L.

We will be applying this theorem with the following functor G which was called C and used for similar purposes in [Kleisli, 1974] where all details may be found.

For a space X and element x ∈ X, let I −∘_{x} X denote the space of paths π : I ⟶ X such that π(0) = x, topologized with the compact/open topology. Define GX = ∑_{x∈X} I −∘_{x} X. Of course, the point set of GX is just the set of paths in X, but the topology is not that of the path space, since paths starting at distinct points are in different components. We define ϵX : GX ⟶ X as evaluation at 1.

4. Proof of 1.1

In order to understand the next theorem, we will think of Σ_{n+1} as embedded as the subgroup of Σ_{n+2} consisting of those permutations of {0,…, n + 1} that fix n + 1. 4.1. Proposition. There is a natural chain contraction s on CG ⟶ C_{−1} ⟶ 0 such that for an n-simplex σ and p ∈ Σ_{n+1}, s(pσ) = ps(σ).

Proof. Let X be any space and suppose that σ : Δ_{n} ⟶ GX is a singular n-simplex. Then σ is a continuous function Δ_{n} ⟶ I −∘_{x} X for some x ∈ X since Δ_{n} is connected and GX is the union of those connected components. We can think of this as a continuous map we still call σ : Δ_{n} × I ⟶ X and write σ(t_{0},…, t_{n}; u) for (t_{0},…, t_{n}) ∈ Δ_{n} and u ∈ I such that σ(t_{0},…, t_{n}; 0) = x for all (t_{0},…, t_{n}) ∈ Δ_{n}. First we consider the case of dimensions −1 and 0. A path component (−1-simplex) of GX is determined by an x ∈ X and we define sx(u) = x, the constant path at x. A singular 0-simplex is just a path in X. However, to be consistent with our previous notation, we write σ(t; u) with t = 1 the only allowed value for t. Then define (sσ)(t_{0}, t_{1}; u) = σ(\frac{t_{0}}{1−t_{1}} ; (1 − t_{1})u). Of course, t_{0} + t_{1} = 1, so this is really σ(1; t_{0}u).

(d^{1}sσ)(1; u) = (sσ)(1, 0; u) = σ(1; u)

while

(d^{0}sσ)(1; u) = (sσ)(0, 1; u) = σ(1; 0)

while sd^{0}σ(1; u) is the constant path at d^{0}σ. Now d^{0} : C_{0}(GX) ⟶ H_{0}(GX) assigns to each path its path component and the path components of GX are in one-one correspondence with the elements of X by the map that takes π to π(0). Thus sd^{0}σ is the constant path at σ(1; 0) and so we have d^{1}sσ = σ and d^{0}sσ = sd^{0}σ. Next we turn to dimension n > 0. Define

(sσ)(t_{0},…, t_{n+1}; u) = \{ \begin{matrix}σ(\frac{t_{0}}{1−t_{n+1}} ,…, \frac{t_{n}}{1−t_{n+1}} ; (1 − t_{n+1})u) & if t_{n+1} ≠ 1 \\ x & if t_{n+1} = 1 \end{matrix}

We see that sσ is an n + 1-simplex in GX and that

\begin{aligned}d^{n+1}sσ(t_{0},…, t_{n}; u) & = (sσ)(t_{0},…, t_{n}, 0; u) \\ & = σ(t_{0},…, t_{n}; u) \end{aligned}

while for i ≤ n,

\begin{aligned}d^{i}(sσ)(t_{0},…, t_{n}; u) & = sσ(t_{0},…, t_{i−1}, 0, t_{i},…, t_{n}; u) \\ & = σ (\frac{t_{0}}{1 − t_{n+1}} ,…, \frac{t_{i−1}}{1 − t_{n+1}} , 0, \frac{t_{i}}{1 − t_{n+1}} ,…, \frac{t_{n−1}}{1 − t_{n+1}} ; u) \end{aligned}

and

\begin{aligned}s(d^{i}σ)(t_{0},…, t_{n}; u) & = d^{i}σ (\frac{t_{0}}{1 − t_{n+1}} ,…, \frac{t_{n−1}}{1 − t_{n+1}} ; u) \\ & = σ (\frac{t_{0}}{1 − t_{n+1}} ,…, \frac{t_{i−1}}{1 − t_{n+1}} , 0, \frac{t_{i}}{1 − t_{n+1}} ,…, \frac{t_{n−1}}{1 − t_{n+1}} ; u) \end{aligned}

Thus d^{n+1}s = id and for i ≤ n, d^{i}s = sd^{i} and we have a contraction at the simplicial level, and it is standard that the sequence of (−1)^{n}s_{n} is a contraction in the associated chain complex. It is evident that for any permutation p ∈ Σ_{n+1}, s(σ^{∘} p) = (sσ)^{∘} p. Since pσ = σ^{∘} p^{−1}, it follows that s(pσ) = ps(σ).

4.2. Proposition. For all n ≥ 0, there is a natural transformation θ_{n} : C_{n} ⟶ C_{n}G such that C_{n}ϵ^{∘} θ_{n} = id and such that for each p ∈ Σ_{n+1}, θ_{n}(pσ) = pθ_{n}(σ).

Proof. Let X be a topological space. Define θ_{n}X : C_{n}X ⟶ C_{n}GX by

θ_{n}(σ)(t_{0},…, t_{n})(u) = σ (ut_{0} + \frac{1 − u}{n + 1},…, ut_{n} + \frac{1 − u}{n + 1})

which is a simplex in the component of GX based at σ(\frac{1−u}{n+1},…, \frac{1−u}{n+1}). It is clear that θ_{n}(σ)(t_{0},…, t_{n})(1) = σ(t_{0},…, t_{n}) and that θ(σ^{∘} p) = θ(σ)^{∘} p, from which the second claim follows.

Now we can apply acyclic models to C as well as C/U, C/V and C/W. We have shown that C is ϵ-presentable and that C ⟶ C_{−1} ⟶ 0 is G-contractible and both natural transformations commute with the action of the symmetric groups. This implies that s(U_{n}) ⊆ U_{n+1}, s(V_{n}) ⊆ V_{n+1} and s(W_{n}) ⊆ W_{n+1} and similarly that θ_{n}(U_{n}) ⊆ U_{n}G, θ_{n}(V_{n}) ⊆ V_{n}G and θ_{n}(W_{n}) ⊆ W_{n}G and so C/U, C/V and C/W are also ϵ-presentable and the complexes augmented over C_{−1} ⟶ 0 are G-contractible. Since all four complexes have the same augmentation term, it follows that all the maps and composites in C ⟶ C/U ⟶ C/V ⟶ C/W are homotopy equivalences.

5. An explicit computation

The acyclic models theorem gives an explicit construction and you can work out what the homotopy inverse to the projection C ⟶ C/W is in low dimensions. Basically, what is needed is a map f_{n} : C_{n} ⟶ C_{n} such that for each transposition p ∈ Σ_{n+1} and singular n-simplex σ, f_{n}(σ^{∘} p) = −f_{n}(σ) and if p is an arbitrary odd permutation such that σ^{∘} p = σ, then f_{n}(σ) = 0. If you follow the construction given in the proof, here is what happens in dimension 1. Suppose [a_{0}, a_{1}] is a 1-simplex. This means there is a simplex σ : Δ_{1} ⟶ X such that σ(0) = a_{0} and σ(1) = a_{1}. Let a_{01} = σ(1/2) be the barycenter. Then f_{1}([a_{0}, a_{1}]) = [a_{0}, a_{01}] − [a_{1}, a_{01}]. Both simplexes on the left use σ to interpolate between the given endpoints. Compare this with the simplicial subdivision map given in [Rotman, 1988], page 113 or [Dold, 1980], page 40, where Sd([a_{0}, a_{1}]) = [a_{01}, a_{1}]−[a_{01}, a_{0}]. In degree 2, following analogous conventions, we get the following map:

\begin{aligned}f_{2}([a_{0}, a_{1}, a_{2}]) & = [a_{0}, a_{01}, a_{012}] − [a_{1}, a_{01}, a_{012}] + [a_{1}, a_{12}, a_{012}] \\ & − [a_{2}, a_{12}, a_{012}] + [a_{2}, a_{02}, a_{012}] + [a_{0}, a_{02}, a_{012}] \end{aligned}

It seems clear what the general pattern is, but I have not actually verified this in any higher dimension.

6. Appendix: some historical notes

I had never understood the reason for the names “oriented” and “ordered” in connection with the chain groups nor could recall which was which. After looking at the references [Eilenberg] and [Lefschetz], it became clear. Originally, all chain groups were oriented, that is based on oriented simplexes. An oriented chain is simply a chain, simplicial or singular, based on an oriented simplex, that is a simplex with a chosen orientation. If you changed the orientation, you changed the sign of the chain. This was necessary to make d^{2} = 0 and doubtless went back to the earliest days of algebraic topology. They didn’t use standard simplexes, but any oriented simplex and identified two simplexes if they differed by an orientation preserving “barycentric map”, meaning one that was the linear extension of a map that takes vertices to vertices. An orientation reversing barycentric map just changed the sign of the chain. What Eilenberg did was to replace the oriented simplex by an ordered simplex, that is one with a linear ordering of its vertices. Now in identifying two chains, only order preserving maps were permitted. This gave a larger chain group (fewer identifications could be made) since now, for example, the 1-simplex [v_{0}, v_{1}] is not related to the simplex [v_{1}, v_{0}] since the barycentric map that mediates between them does not preserve the order on the vertices. In this way, the group of chains based on ordered simplexes became the ordered chain group.

After this paper was submitted to the editor, he suggested I look in the book by Seifert and Threlfall [1980]. This book, originally published in 1934, used a definition of singular homology that was essentially that of C/W. In effect, their definition took as basis for the chain group the set of non-degenerate singular simplexes, where a singular simplex σ is degenerate if there is an odd permutation p such that pσ = σ. This follows from the second sentence in the statement of Proposition 2.2 since the degenerate simplexes generate the torsion subgroup. (Like the other early definitions, this was complicated by the fact that they didn’t use standard simplexes, but any space isomorphic to a simplex.) Thus the definition in this book actually solves the problem of torsion. Eilenberg writes, “There is also the possibility of leaving out the elements of order 2 as degenerate, but this would make the use of [the complex] cumbersome.” This makes it seem that Eilenberg was unaware of Seifert and Threlfall, which had appeared in Germany 10 years earlier.

References

M. Barr and J. Beck (1966), Acyclic models and triples. S. Eilenberg et. al., eds. Proc. Conf. Categorical Algebra, La Jolla, 1965. Springer-Verlag, 336-343.

A. Dold (1980), Lectures on Algebraic Topology. Springer-Verlag, Berlin, Heidelberg, New York.

S. Eilenberg (1944), Singular homology theory. Ann. Math., 45, 207–247.

H. Kleisli (1974), On the construction of standard complexes. J. Pure Applied Algebra, 4, 243-260.

S. Lefschetz (1942), Algebraic Topology. Amer. Math. Soc. Colloquium Publ., 27.

J. J. Rotman (1988), An Introduction to Algebraic Topology. Springer-Verlag, Berlin, Heidelberg, New York.

Theory and Applications of Categories, Vol. 1, No. 1 9

D. L. Sayers (1934), The Nine Tailors. Victor Gollancz, London. H. Seifert and W. Threlfall (1980), A Textbook of Topology, Academic Press, 1980.

Translation of Lehrbuch der Topologie, 1934.

Department of Mathematics and Statistics McGill University Montreal, Quebec, Canada

Email: barr@math.mcgill.ca

This article is may be accessed via WWW at http://www.tac.mta.ca/tac/ or by

anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/volumes/1995/v1n1.{dvi,ps}

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/gopher/ftp. The journal is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi and Postscript format. Details will be emailed to new subscribers and are available by WWW/gopher/ftp. 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 LaTEX 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@triples.math.mcgill.ca Lawrence Breen, Universite de Paris 13: breen@dmi.ens.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.ik Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, University of Genoa: carboni@vmimat.mat.unimi.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: mthfwl@ubvms.cc.buffalo.edu Jean-Louis Loday, Universite de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@gar.union.edu Robert Pare, Dalhousie University: pare@cs.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@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca R. W. Thomason, Universite de Paris 7: thomason@mathp7.jussieu.fr Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca