SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES Dedicated to Dominique Bourn on the occasion of his 60th birthday

Transcription

1 Theory and Applications o Categories, Vol. 23, No. 4, 21, pp SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES Dedicated to Dominique Bourn on the occasion o his 6th birthday TAMAR JANELIDZE Abstract. The purpose o this paper is to prove a new, incomplete-relative, version o Non-abelian Snake Lemma, where relative reers to a distinguished class o normal epimorphisms in the ground category, and incomplete reers to omitting all completeness/cocompleteness assumptions not involving that class. 1. Introduction The classical Snake Lemma known or abelian categories (see e.g. [5]) has been extended to homological categories by D. Bourn; see F. Borceux and D. Bourn [1], and reerences there. In [3], we extended it to the context o relative homological categories: here relative reers to a distinguished class E o regular epimorphisms in a ground category C satisying certain conditions which, in particular, make (C, E) relative homological when (a) C is a homological category and E is the class o all regular epimorphisms in C; (b) C is a pointed initely complete category satisying certain cocompleteness conditions, and E is the class o all isomorphisms in C. In this paper we drop the completeness/cocompleteness assumption and extend Snake Lemma urther to the context o what was called an incomplete relative homological category in [4] (see, however, the correction below Deinition 2.1). Let us recall the ormulation o Snake Lemma rom [1] (see Theorem o [1]). It Partially supported by the International Student Scholarship o the University o Cape Town, and Georgian National Science Foundation (GNSF/ST6/3-4). Received by the editors and, in revised orm, Published on in the Bourn Festschrit. 2 Mathematics Subject Classiication: 18G5, 18G25, 18A2, 18B1. Key words and phrases: incomplete relative homological category, homological category, normal epimorphism, snake lemma. c Tamar Janelidze, 21. Permission to copy or private use granted. 76

2 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 77 says: Let C be a homological category. Consider the diagram K g K u K K v K w k u k v X Y g v u X Y g u Z Z k w w d q v q w v g w where all squares o plain arrows are commutative and all sequences o plain arrows are exact. There exists an exact sequence o dotted arrows making all squares commutative. The proo given in [1] contains explicit constructions o the dotted arrows, which makes the result ar more precise. This seems to suggest that these constructions should be included in the ormulation o the lemma, but there is a problem: while the constructions o K, g K,, and g as induced morphisms are straightorward, the several-step construction o the connecting morphism d : K w u is too long and technical. A natural solution o this problem, well-known in the abelian case, involves partial composition o internal relations in C and goes back at least to S. Mac Lane [6] (see also e.g. [2] or the so-called calculus o relations in regular categories): one should simply deine d as the composite vg k w, where g is the relation opposite to g, etc. We do not know i the non-abelian version o d = vg k w is mentioned anywhere in the literature, but we use its relative version in our relative Snake Lemma (Theorem 3.1), which is the main result o this paper. For the reader s convenience we are mostly using the same notation as in [1]. 2. Incomplete relative homological categories Throughout the paper we assume that C is a pointed category and E is a class o morphisms in C containing all isomorphisms. Let us recall rom [4]: 2.1. Deinition. A pair (C, E) is said to be an incomplete relative homological category i it satisies the ollowing conditions: (a) Every morphism in E is a normal epimorphism;

3 78 TAMAR JANELIDZE (b) The class E is closed under composition; (c) I E and g E then g E; (d) I E then ker() exists in C; (e) A diagram o the orm A A B g g has a limit in C provided and g are in E, and either (i) = g and = g, or (ii) and g are in E, (, g) and (, g ) are relexive pairs, and and g are jointly monic; B () I A B A A is a pullback and is in E, then is also in E; π 1 A (g) I h 1 : H A and h 2 : H B are jointly monic morphisms in C and i α : A C and β : B D are morphisms in E, then there exists a morphism h : H X in E and jointly monic morphisms x 1 : X C and x 2 : X D in C making the diagram H A α C x 1 h 1 X h B h 2 B x 2 β commutative (it easily ollows rom the act that every morphism in E is a normal epimorphism, that such actorization is unique up to an isomorphism); D

4 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 79 (h) The E-Short Five Lemma holds in C, i.e. in every commutative diagram o the orm K k A B K k w A B with and in E and with k = ker() and k = ker( ), the morphism w is an isomorphism; (i) I in a commutative diagram K k A B u w K k A B,, and u are in E, k = ker() and k = ker( ), then there exists a morphism e : A M in E and a monomorphism m : M A in C such that w = me. Let us use this opportunity to make a correction to conditions 2.1(c) and 3.1(c) o [4]. They should be replaced, respectively, with: (a) A diagram o the orm A A B g g has a limit in C provided, g,, and g are in E, and either (i) = g and = g, or (ii) (, g) and (, g ) are relexive pairs (that is, h = 1 B = gh and h = 1 B = g h or some h and h ), and and g are jointly monic. (b) Condition 2.1(e) o the present paper. Note that this replacement will not aect any results/arguments o [4], except that without it a pair (C, Iso(C)) would be an incomplete relative homological category only when C (is pointed and) admits equalizers o isomorphisms Remark. As easily ollows rom condition 2.1(g), i a morphism : A B in C actors as = em in which e is in E and m is a monomorphism, then it also actors (essentially uniquely) as = m e in which m is a monomorphism and e is in E. B

5 8 TAMAR JANELIDZE 2.3. Lemma. I a pair (C, E) satisies conditions 2.1(a)-2.1(d), then (C, E) satisies conditions 2.1(h) and 2.1(i) i and only i in every commutative diagram o the orm K k A B u w K k A B (2.1) with k = ker(), k = ker( ), and with,, and u in E, the morphism w is also in E. Proo. Suppose (C, E) satisies conditions 2.1(a)-2.1(d), 2.1(h) and 2.1(i). Consider the commutative diagram (2.1) with k = ker(), k = ker( ), and with,, and u in E. By condition 2.1(i), w = me where e : A M is in E and m : M A is a monomorphism. Consider the commutative diagram: K u K ek m M m A Since u is in E and m is a monomorphism, by condition 2.1(a) there exists a unique morphism m : K M with mu = ek and m m = k ; m is a monomorphism since so k. Since m m =, m is a monomorphism, and k = ker( ), we conclude that m = ker( m). By condition 2.1(c), m is in E, thereore we can apply the E-Short Five Lemma to the diagram K m M m k m B K k A B and conclude that m is an isomorphism. Hence, by condition 2.1(b) w is in E, as desired. Conversely, suppose or every commutative diagram (2.1) with k = ker(), k = ker( ), and with and in E, i u is in E then w is also in E. It is a well know act that under the assumptions o condition 2.1(h), ker(w) = ; moreover, since E contains all isomorphisms and and are in E, w : A A is also in E. Since every morphism in E is a normal epimorphism, we conclude that w is an isomorphism, proving condition 2.1(h). The proo o condition 2.1(i) is trivial.

6 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES Lemma. Let (C, E) be a pair satisying conditions 2.1(a)-2.1(d) and 2.1(g). Consider the commutative diagram: A B α A B β (2.2) (i) I α : A A and β : B B are in E and i : A B actors as = me in which e is in E and m is a monomorphism, then : A B also actors as = m e in which e is in E and m is monomorphism. (ii) I α : A A and β : B B are monomorphisms and i : A B actors as = m e in which e is in E and m is a monomorphism, then : A B also actors as = me in which e is in E and m is a monomorphism. Proo. (i): Consider the commutative diagram (2.2) and suppose α and β are in E and = me in which e : A C is in E and m : C B is a monomorphism. Since β is in E and m is a monomorphism, by Remark 2.2 there exists a morphism γ : C C in E and a monomorphism m : C B such that βm = m γ. Consider the commutative diagram: α A γe e A C m B Since α is in E and m is a monomorphism, conditions 2.1(a) and 2.1(d) imply the existence o a unique morphism e : A C with e α = γe and m e =. Since α, e, and γ are in E, conditions 2.1(b) and 2.1(c) imply that e is also in E. Hence, = m e in which e is in E and m is a monomorphism, as desired. (ii) can be proved similarly. Let (C, E) be an incomplete relative homological category. We will need to compose certain relations in C: Let R = (R, r 1, r 2 ) : A B be a relation rom A to B, i.e. a pair o jointly monic morphisms r 1 : R A and r 2 : R B with the same domain, and let S = (S, s 1, s 2 ) : B C be a relation rom B to C. I the pullback (R B S, π 1, ) o r 2 and s 1 exists in C, and i there exists a morphism e : R B S T in E and a jointly monic pair o

7 82 TAMAR JANELIDZE morphisms t 1 : T A and t 2 : T C in C making the diagram R B S π 1 e R T r S 1 s 2 t r 2 1 s 1 t 2 A B C (2.3) commutative, then we will say that (T, t 1, t 2 ) : A C is the composite o (R, r 1, r 2 ) : A B and (S, s 1, s 2 ) : B C. One can similarly deine partial composition or three or more relations satisying a suitable associativity condition. Omitting details, let us just mention that, say, a composite RR R might exist even i neither RR nor R R does Convention. We will say that a relation R = (R, r 1, r 2 ) : A B is a morphism in C i r 1 is an isomorphism Deinition. Let (C, E) be an incomplete relative homological category. A sequence o morphisms is said to be:... A i 1 i 1 A i i Ai+1... (i) E-exact at A i, i the morphism i 1 admits a actorization i 1 = me, in which e E and m = ker( i ); (ii) an E-exact sequence, i it is E-exact at A i or each i (unless the sequence either begins with A i or ends with A i ). As easily ollows rom Deinition 2.6, the sequence A g B is E-exact i and only i = ker(g); and, i the sequence C A B g C is E-exact then g = coker() and g is in E. In the next section we will oten use the ollowing simple act: 2.7. Lemma. (Lemma 4.2.4(1) o [1]) I in a commutative diagram K k A B u K k A v B w (2.4) in C, k = ker() and w is a monomorphism, then k = ker( ) i and only i the let hand square o the diagram (2.4) is a pullback.

8 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES The Snake Lemma This section is devoted to our main result which generalizes Theorem o [1] and its relative version mentioned in [3]. Formulating it, we use the same notation as in [1] Theorem. [Snake Lemma] Let (C, E) be an incomplete relative homological category. Consider the commutative diagram K K u g K K v K w k w k u X k v Y v u w X Y g g Z Z (3.1) u d q v v g w q w in which all columns, the second and the third rows are E-exact sequences. I the morphism g actors as g = g 2g 1 in which g 1 is in E and g 2 is a monomorphism, then: (i) The composite d = vg k w : K w u is a morphism in C. (ii) The sequence K u K v K w d u v w (3.2) where d = vg k w, is E-exact. Proo. (i): Under the assumptions o the theorem, let (Y Z K w, π 1, ) be the pullback o g and k w (by condition 2.1(e) this pullback does exist in C); since g is in E, by condition 2.1() the morphism is also in E, and since k w is a monomorphism so is π 1. Since = ker(g ) and g vπ 1 =, there exists a unique morphism ϕ : Y Z K w X with

9 84 TAMAR JANELIDZE vπ 1 = ϕ (see diagram (3.4) below). Using the act that (Y Z K w, π 1, ) is the pullback o g and k w and that k w = ker(w), an easy diagram chase proves that (Y Z K w, π 1, ϕ) is the pullback o v and. Thereore, we obtain the commutative diagram 1 P 1 P P P 1 P 1 P 1 P P P 1 P π 1 π 1 1 P P Y P P 1 P 1 P ϕ X π 1 1 Y 1 Y π 1 ϕ 1 X 1 X K w Y 1 Y Kw X k w g X 1 Y 1 Y v 1 X 1 X K w Z Y Y X u where P = Y Z K w, and all the diamond parts are pullbacks. Since and are in E, by condition 2.1(g) we have the actorization (unique up to an isomorphism) P K w Y Z K w 1 Y Z Kw ϕ r Y Z K w R r 1 X r 2 u (3.3) where r : Y Z K w R is a morphism in E and r 1 : R K w and r 2 : R u are jointly monic morphisms in C. As ollows rom the deinition o composition o relations, (R, r 1, r 2 ) is the composite relation vg k w rom K w to u (Note, that since the pullback (Y Z K w, π 1, ) o k w and g, and the pullback (Y Z K w, π 1, ϕ) o v and exists in C, the composite relations g k w : K w Y and v : Y X also exist. Moreover, since and are in E, the composite ( v)(g k w ) o the three relations g k w, v, and also exists and we have ( v)(g k w ) = vg k w ). To prove that vg k w : K w u is a morphism in C, consider the commutative

10 diagram SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 85 K K u g K K v K w k u X u 1 u X v h d k v Y Z K w θ π 1 g Y Z ϕ 2 X Y g Z k w w q v q w u v g w in which: (3.4) - All the horizontal and vertical arrows are as in diagram (3.1). - π 1 : Y Z K w Y, : Y Z K w K w, and ϕ : Y Z K w X are deined as above, i.e. (Y Z K w, π 1, ) is the pullback o g and k w, and ϕ : Y Z K w X is the unique morphism with vπ 1 = ϕ. - = 2 1 where 1 : X X is a morphism in E and 2 : X Y is the kernel o g (such actorization o does exist in C since the second row o diagram (3.1) is E-exact).

11 86 TAMAR JANELIDZE - Since (Y Z K w, π 1, ) is the pullback o g and k w, there exists a unique morphism θ : X Y Z K w with π 1 θ = 2 and θ =. Since is in E and 2 = ker(g), we conclude that = coker(θ). - Since = ker(g ) and g v 2 =, there exists a unique morphism u : X X with u = v 2. It easily ollows that u 1 = u, = coker(u ), and ϕθ = u. - h : K v Y Z K w is the canonical morphism and an easy diagram chase proves that h = ker(ϕ) (we do not need this act now, but we shall need it below when proving (ii)). Since ϕθ = u = and = coker(θ), there exists a unique morphism d : K w u with ϕ = d. We obtain the ollowing actorization: K w Y Z K w 1 Y Z Kw ϕ Y Z K w K w 1 Kw X d u (3.5) Comparing diagrams (3.3) and (3.5), we conclude that the relation (K w, 1 Kw, d) can be identiied with the relation (R, r 1, r 2 ). Thereore, r 1 is an isomorphism, proving that vg k w is a morphism in C. (ii): To prove that the sequence (3.2) is E-exact, we need to prove that it is E-exact at K v, K w, u, and v. E-exactness at K v : It ollows rom the act that the irst column o the diagram (3.4) is E-exact at X, that the kernel o u exists in C. Indeed, consider the commutative diagram X X 1 u u 1 u u 1 u 2 M X u (3.6) where u = u 2 u 1 is the actorization o u with u 2 = ker( ) and u 1 E (which does exists since the irst column o the diagram (3.4) is E exact at X ), and u 1 is the induced morphism. Since 1 and u 1 are in E, u 1 is also in E, and thereore the kernel o u 1 exists in C. Since u 2 is a monomorphism we conclude that Ker(u ) Ker(u 1).

12 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 87 Consider the ollowing part o the diagram (3.4) K K u K v k u e K u k u m k v X Y 1 u X 2 v g K g K w u X Y Z g Z k w w (3.7) in which: - k u = ker(u ); - Since k v = ker(v) and v 2 k u = u k u =, there exists a unique morphism m : K u K v with k v m = 2 k u ; - Since k u = ker(u ) and u 1 k u = uk u =, there exists a unique morphism e : K u K u with k u e = 1 k u ; since k v is a monomorphism, we conclude that me = K. The E-exactness at K v will be proved i we show that e E and m = ker(g K ). The latter, however, easily ollows rom Lemma 2.7. Indeed: by Lemma 2.7, the squares 1 k u = k u e and 2 k u = k v m are pullbacks; thereore, since 1 is in E we conclude that e also is in E, and since k w is a monomorphism, by the same lemma m = ker(g K ). E-exactness at K w : Consider the commutative diagram (3.4), we have: dg K = d h = ϕh =. To prove that the sequence (3.2) is E-exact at K w, it suices to prove that the kernel o d exists in C and that the induced morphism rom K v to the kernel o d is in E. It easily ollows rom Lemma 2.4 that there exists a actorization d = d 2 d 1 where d 2 is a monomorphism and d 1 is in E. Indeed: since the second column o the Diagram (3.4) is E-exact at Y, there exists a actorization v = v 2 v 1, where v 2 = ker(q v ) is a monomorphism and v 1 is in E. Applying Lemma 2.4(ii) to the diagram Y Z K w ϕ X π 1 Y v Y

13 88 TAMAR JANELIDZE we conclude that ϕ = ϕ 2 ϕ 1 where ϕ 2 is a monomorphism and ϕ 1 is in E, and then applying Lemma 2.4(i) to the diagram Y Z K w ϕ X K w d u we obtain the desired actorization o d. Since d 1 is in E and d 2 is a monomorphism, we conclude that the kernel o d exists in C (precisely, Ker(d) Ker(d 1 )). Let k d : K d K w be the kernel o d and let e d : K v K d be the induced unique morphism with k d e d = g K, it remains to prove that e d is in E. For, consider the commutative diagram K v e d K d h h k d X θ Y Z K w ϕ u 1 θ K w s π X 2 u K w d π 1 K qu u2 X u in which: - π 1, π 2 are the pullback projections, and s = ϕ,, θ = u 2,, and h =, k d are the induced morphisms (the pullback (X u K w, π 1, π 2) o and d does exist in C since is in E); since is in E so is π 2. - u 1 : X K qu and u 2 : K qu X are as in diagram (3.6). - Since u 2 = ker( ) and k d = ker(d), we conclude that θ = ker(π 2) and h = ker(π 1). Since θ = ker( ), θ = ker(π 2), and the morphisms, π 2, and u 1 are in E, by Lemma 2.3, s : Y Z K w X u K w is also in E. Thereore, since h = ker(ϕ) and h = ker(π 1), by Lemma 2.7 and condition 2.1() the morphism e d : K v K d is also in E, as desired. E-exactness at u : Consider the commutative diagram (3.4), we have: d = ϕ = q v ϕ = q v vπ 1 =, and since is an epimorphism we conclude that d =. To prove that the sequence (3.2) is E-exact at u, it suices to prove that the kernel o exists in C and that the induced morphism rom K w to the kernel o is in E. It easily ollows rom Lemma 2.4(i) that there exists a actorization = 2 1 where 2 is a monomorphism and 1 is in E. Indeed: since is a monomorphism and

14 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 89 E contains all isomorphisms, applying Lemma 2.4(i) to the diagram X Y u v q v we obtain the desired actorization o ; since 1 is in E and 2 is a monomorphism, we conclude that the kernel o exists in C. Let k : K u be the kernel o and let e : K w K be the induced unique morphism with e k = d, it remains to prove that e is in E. Since is in E, the pullback (K u X, p 1, p 2 ) o k and exists in C and p 1 is in E; thereore, we have the commutative diagram p Y v 2 K u X p 2 X Y p 1 q v K k u v in which v 2 = ker(q v ) (recall, that since the second column o the diagram (3.4) is E- exact at Y, we have v = v 2 v 1 where v 1 E and v 2 = ker(q v )) and p : K u X Y is the induced morphism. Using the act that v 2 and p 2 are monomorphisms and that k = ker( ), an easy diagram chase proves that the square p 2 = v 2 p is the pullback o and v 2. Next, consider the commutative diagram Y π 1 Y Z K w K w v 1 ϕ ψ e v Y p K u X p 1 K d v 2 p 2 Y X k u were ψ = ϕ, v 1 π 1 ; since k is a monomorphism and the equalities k p 1 ψ = p 2 ψ = ϕ = d = k e

15 9 TAMAR JANELIDZE hold, we conclude that p 1 ψ = e. Recall, that the square ϕ = vπ 1 is a pullback (see the proo o (i)), thereore, (Y Z K w, π 1, ψ) is the pullback o p and v 1, yielding that ψ is in E. Since p 1 ψ = e, and ψ, p 1, and are in E, we conclude that e is also in E, as desired. E-exactness at v : Consider the commutative diagram (3.4). According to the assumptions o the theorem, we have g = g 2g 1 were g 1 is a morphism in E and g 2 is a monomorphism. Then, by Lemma 2.4(i) there exists a actorization g = g 2 g 1 were g 1 is a morphism in E and g 2 is a monomorphism, hence, the kernel o g exists in C. Since g = q w g = and is an epimorphism, we conclude that g =, thereore, to prove that the sequence (3.2) is E-exact at v it suices to prove that the induced morphism rom u to the kernel o g is in E. For, consider the commutative diagram u X X e 1 K g e g u Y Z v 1 w 1 Ȳ ȳ Z v w v 2 w 2 z Y g Z e 2 v w Z p 2 q v k g p 1 v w g g q w (3.8) in which: - k g = ker(g ) and e g : u K g is the induced unique morphism with k g e g =. - ( v w Z, p 1, p 2) is the pullback o g and q w (this pullback does exist since q w is in E), and e 2 = q v, g and = k g, are the canonical morphisms; since k g = ker(g ) we conclude that = ker(p 2). - Since = ker(p 2) and p 2e 2 = there exists a unique morphism e 1 : X K g with e 1 = e 2 ; it easily ollows that e g = e 1. - Since the second and the third columns o the diagram (3.8) are E-exact at Y and Z respectively, we have the actorizations v = v 2 v 1 and w = w 2 w 1, where v 1, w 1 E and v 2 = ker(q v ) and w 2 = ker(q w ).

16 SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 91 - z =, w 2, and since w 2 = ker(q w ) we conclude that z = ker(p 1). - Since z = ker(p 1) and p 1e 2 v 2 =, there exists a unique morphism ȳ : Ȳ Z with e 2 v 2 = zȳ. Since w 1, g, and v 1 are in E, we conclude that ȳ is in E; thereore, by Lemma 2.3, e 2 is also in E. Then, Lemma 2.7 implies that e 1 is in E, and since e g = e 1 we conclude that e g is also in E, as desired. Reerences [1] F. Borceux, D. Bourn, Mal cev, protomodular, homological and semi-abelian categories, Mathematics and its Applications, Kluwer, 24. [2] A. Carboni, G. M. Kelly, and M. C. Pedicchio, Some remarks on Mal cev and Goursat categories, Applied Categorical Structures 1, 1993, [3] T. Janelidze, Relative homological categories, Journal o Homotopy and Related Structures 1, 26, [4] T. Janelidze, Incomplete relative semi-abelian categories, Applied Categorical Structures, available online rom March 29. [5] S. Mac Lane, Categories or the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, [6] S. Mac Lane, An algebra o additive relations, Proc. Nat. Acad. Sci. USA 47, 1961, Department o Mathematics and Applied Mathematics, University o Cape Town, Rondebosch 771, Cape Town, South Arica tamar.janelidze@uct.ac.za This article may be accessed at or by anonymous tp at tp://tp.tac.mta.ca/pub/tac/html/volumes/23/4/23-4.{dvi,ps,pd}

17 THEORY AND APPLICATIONS OF CATEGORIES (ISSN X) will disseminate articles that signiicantly advance the study o categorical algebra or methods, or that make signiicant new contributions to mathematical science using categorical methods. The scope o the journal includes: all areas o pure category theory, including higher dimensional categories; applications o category theory to algebra, geometry and topology and other areas o mathematics; applications o category theory to computer science, physics and other mathematical sciences; contributions to scientiic knowledge that make use o categorical methods. Articles appearing in the journal have been careully and critically reereed under the responsibility o members o the Editorial Board. Only papers judged to be both signiicant and excellent are accepted or publication. Full text o the journal is reely available in.dvi, Postscript and PDF rom the journal s server at and by tp. It is archived electronically and in printed paper ormat. Subscription inormation. Individual subscribers receive abstracts o articles by as they are published. To subscribe, send to tac@mta.ca including a ull name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca. Inormation or authors. The typesetting language o the journal is TEX, and L A TEX2e strongly encouraged. Articles should be submitted by directly to a Transmitting Editor. Please obtain detailed inormation on submission ormat and style iles at Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca Assistant TEX editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne: gavin seal@astmail.m Transmitting editors. Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.r Richard Blute, Université d Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.r Ronald Brown, University o North Wales: ronnie.probrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Cuill Inc.: valeria@cuill.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University o Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University o Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University o Aarhus: kock@im.au.dk Stephen Lack, University o Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University o New York at Bualo: wlawvere@acsu.bualo.edu Tom Leinster, University o Glasgow, T.Leinster@maths.gla.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.r Ieke Moerdijk, University o Utrecht: moerdijk@math.uu.nl Susan Nieield, Union College: nieiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University o Illinois at Chicago: bshipley@math.uic.edu James Stashe, University o 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 o Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca