ABEL IAN CATEGORIES

Rep rints inTh eor yand Appli

cati onsof Categorie

s, No.

3,2003.

ABEL IAN CATEGORIES

PETER J.FREYD

Forew ord

Theearly 60s

wa sa grea ttime inAmerica

fora yo ungma th-

ematician.

Washingto n

had responded to

Sputnik with

a lot

ofmoney forscienc

eeducatio na

ndthe scientists,

blessth em,

saidtha tthey co uldnot doa

nything until

stude nts kne wma th-

ematics.What Sputnikpr

ov ed, incrediblyenoug

h,w asthat the

country needed

moremat hema

ticians.

Publishers got the messa

ge.

At annual AMSmee

tings you

couldsp enden tireev eningsc

ra wlingpublishers’

cock tailpar ties.

They weren’t look

ing for bo ok buyers, they

were look ing for

writersand some

how the yha dconcluded that

theb estw ay to

get mathematicia ns

to write

elemen tary texts

was to publish

theira dvanced texts.

Wor dhad goneo

uttha tI wa sw riting

atext ons

omething ca

lled“

ca tegory the ory”

and whatever

it

was, somebig

namesse emedto

be interested.

Ilost coun

to f

theb ookmen whovisited

my officeb

earingg ift copieso

ftheir

advanced texts.

Ic hoseHa rpe r&

Ro wb eca usethey promise

d

Origin allypu blis he das : Abe lian Categor ies, Harper andRo w,1964.

Receiv ed by thee ditors 2003-11-10.

Tran smit ted by M.Barr.

Reprin tpu blish ed on2003- 12-17.

Footnot ed

references adde dto the Forew ordand pos te d2004-01-20.

2000Math ematics Sub

jectClassification :

18-01, 18B15.

Keyw ords and phrase s:Ab eli anc ategorie

s,e xacte mb ed ding.

c

Peter J.F re yd, 1964.P erm ission tocop yfor priv ateu se granted .

− 25

PETER J.FREYD

course,to be replaced by thew ord“equa lizer”.

Pag es29 –30:

Exercise 1–D would hav

eb eenm uch easierif

itha db een delay edun tilafter the

definit ionsof

gene rato rand

pushout. The categor

→ y[

]is be stc haract eriz eda sa generator

forthe ca tegory of sma llcateg oriestha

tapp ears

as aretr acto

f

every otherg

en erato r.

Thec atego

→→]is ry[

apusho utof

the

tw omaps from

[→]and 1to

thisc haract erizatio nalso

simpli-

fiesthe material

ins ection 3:

ifa functorfixes

the tw oma ps

from 1

[→] to

then it will be sho wn to be equivalen

t to the

identit yf unctor;

if,ins tead, itt wists themit

isequiv alent

tot he

dual-categ oryf unctor.

These chara cterizations

hav ea nothera

d-

van tage: theya

recorr ect.

Ifone starts

witht hethe tw o-elemen t

monoid that isn’ta group, views

itas acat eg oryand thenfo

r-

mally“ split sthe idemp

oten ts”(a sin Ex erc ise2–B, page61

)t he

resultis another tw

o-obje ctcat eg orywith exact

lythree endo

-

functors. Andthe

supposed cha racterization

→→]is of[

coun-

terexampled by

thedisjoin tun

ionof [→]and

the cyclicg

roup

oforder three.

Pag e3 5:

Thea xiomsf

or abelian categories

are redundan

t:

either A1 or A1*

suffices, thatis, each int hepresenc eof

the

othe rax iomsimplies

theot her.The proof,

which isno tstr aight-

forwa rd,can be foundon

section1 .598 ofm yb ook withAndr

e

Scedrov

,hencef 1

orth to be referred toas Cats&

Alligator s.Se

c-

tion 1.597 oftha tb ook hasa nev enmor eparsimonio usdefinition

ofab eliancatego

ry(whic hI

needed forthe materia

ldescrib ed

belo wconcerning page

108):

its uffices torequ ireeither

pro d-

uctsor sumsa

ndthat every

map hasa

“normal factorizat

ion”,

towit, amap that app ea rsas ac ok ernelfollo wed by amap that

appear sa sk ernel.

Pag es 35–

36:

Of the examples

mentio ned to show the in-

Ca 1

tegor ies,A lle gories ,North Holland,

1990

−22

PETER J. FREYD

a low price (≤$8) and—even better—hundreds of free copies to

mathematicians of my choice. (This was to be their first math publication.)

On the day I arrived at Harper’s with the finished manuscript I was introduced, as a matter of courtesy, to the Chief of Pro- duction who asked me, as a matter of courtesy, if I had any preferences when it came to fonts and I answered, as a matter of courtesy, with the one name I knew, New Times Roman.

It was not a well-known font in the early 60s; in those days one chose between Pica and Elite when buying a typewriter—not fonts but sizes. The Chief of Production, no longer acting just on courtesy, told me that no one would choose it for something like mathematics: New Times Roman was believed to be maximally dense for a given level of legibility. Mathematics required a more spacious font. All that was news to me; I had learned its name only because it struck me as maximally elegant.

The Chief of Production decided that Harper’s new math series could be different. Why not New Times Roman? The book might be even cheaper than $8 (indeed, it sold for $7.50).

We decided that the title page and headers should be sans serif

and settled that day on Helvetica (it ended up as a rather non- standard version). Harper & Row became enamored with those particular choices and kept them for the entire series. (And—

coincidently or not—so, eventually, did the world of desktop publishing.) The heroic copy editor later succeeded in convinc- ing the Chief of Production that I was right in asking for nega- tive page numbering. The title page came in at a glorious –11 and—best of all—there was a magnificent page 0.

The book’s sales surprised us all; a second printing was or- dered. (It took us a while to find out who all the extra buyers were: computer scientists.) I insisted on a number of changes

−24

ABELIAN CATEGORIES

(this time Harper’s agreed to make them without deducting from my royalties; the correction of my left-right errors—scores of them—for the first printing had cost me hundreds of dollars).

But for reasons I never thought to ask about, Harper’s didn’t mark the second printing as such. The copyright page, –8, is al- most identical, even the date. (When I need to determine which printing I’m holding—as, for example, when finding a copy for this third “reprinting”—I check the last verb on page –3. In the

second printing it is hasinstead of have).

A few other page-specific comments:

Page 8: Yikes! In the first printing there’s no definition of natural equivalence. Making room for it required much short- ening of this paragraph from the first printing:

Once the definitions existed it was quickly noticed

that functors and natural transformations had be-

come a major tool in modern mathematics. In 1952

Eilenberg and Steenrod published theirFoundations

of Algebraic Topology [7], an axiomatic approach to

homology theory. A homology theory was defined

as a functor from a topological category to an alge- braic category obeying certain axioms. Among the more striking results was their classification of such

“theories,” an impossible task without the notion of natural equivalence of functors. In a fairly explosive manner, functors and natural transformations have permeated a wide variety of subjects. Such monu-

mental works as Cartan and Eilenberg’sHomological

Algebra [4], and Grothendieck’s Elements of Alge-

braic Geometry [1] testify to the fact that functors

have become an established concept in mathematics.

Page 21: The term “difference kernel” in 1.6 was doomed, of

−23

ABELIANCA TEGORIES

depe ndenceo

f A 3 and A 3*

oneis clear,

theo therrequires

wor k:it isnot exactly

trivial that epimorphismsin

thecateg ory

ofgro ups(

abelian ornot

)a reo nto

—oneneeds the“

amalgama -

tion lemma”.

(Giv enthe symmetry

ofthe axioms

eitheron eo f

theexa mplesw

ould,note, hav

es uffic ed.)F orthe independence

of A 2 (hence,

by taking its dual,

alsoof A 2*) let R be a

ring, commut

ative for conv enience.

Thefull subc

atego F ry,

,o f

finitelypres ented

R-modules iseasily

seen tob ec lose dunder

theforma tiono

fc ok ernelsof

arbitra rymaps—quite

enough for

A2*

and A3 .Wit ha littlew

ork one ca nsho wthat thek

ernel

ofa ny epiin F isfinitely gene

rated which

guara ntees thatit

is

theima geo fa map F in andthat’s enough

for A3*

.The nec-

essa rya ndsuffic ient

condition F that

satisfy A2 istha t R be

“cohe ren t”,tha tis ,a llo fits finitelygenerat

ed idealsb

efinitely

presented asmo dules.

For presen tpur po ses we don’tneed

the

necessar yand sufficien

tco ndition.So:

let K be afi eld and R be

ther esult ofa djoininga

sequence of

elemen ts X

subject n

to the

condition that X X i

=0 j

all i,j . Thenm ultiplic ation by, say,

X

definesa 1

ne ndomor phism

on R,the kernel ofwhic

his not

finitelygenerat ed.

More tothe po int, itfails toha

ve ak ernelin

F . Pag e60 : Exercise2–

Aon additive

categories wa

se ntirely

redonefo rthe secondpr

inting.

Amongthe problems inthe

first

printing were thew

ord“m onoidal”

in place of

“pre-addit ive”

(clashingwith themo

dernse nseo fmo noidalcateg

ory)an d—

would yo ub elieve it!—theabsenc

eof thedistr ibutive

law.

Pag e7 2:

Areview er

mentio neda sa nexa mpleo

fone ofm

y

private jokes

the size of the font for the title of sectio n 3.6,

bifuncto

. rs

Goo d heav ens.

I was not really aw are of how

many jok es(pr ivate or otherwise) had

acc umulated inth

etext;

I must hav e been aw are of each one of them in its time but

− 21

PETER J.FREYD

refused toengag ein

the myria ddiscuss ions

abo utthe issues dis-

cussed inthe material

that starts

on the bot tomof page

85.

It

was ag oo drule.

Ihad (correc

tly) predictedthat

theco ntro -

versy would evap

orate andtha

t,in them ea ntime, itw

ould be a

waste oft imeto amplif ywhat Iha

da lreadywritten.

Ishould,

though, hav

efig uredout

aw ay top oin to uttha tt hef org etf ul

functorf or thecat eg B ory,

,describ ed

on pages 131 –132 has all

thecondit ionsnee

dedfor thegeneral

adjoin tf unctor

exceptfor

thesolutio nset

condition.

Ironically therew

as already inhand

a

muc hb ettere xample:

the fo rgetfulfuncto

rfrom the ca tegory of

completeb oolean algebras

(and bi-co

ntin uous homomor

phisms)

tothe ca tegory of setsdo esno tha ve ale ftadjo int (put another

way , freecomplete bo

oleanalg ebr asa reno n-existen tly

large).

Thepro of(

albeit for adifferen ta

ssertio n)w

asin HaimGaif-

man’s19 62dissertat

5 ion

.

Pag e87 : Theterm

“co -w ell-po wered”

sho uld,of course,b

e

“well-co-p ow

ered”.

Pag es91 –93:

Ilost track

oft heman ysp ecialcases

ofExercis e

3–O onmo delt heo ryt hath av eapp earedin

prin t(

mosto fte n

inpr oofs that apar ticularc

atego ry, fo rexa mplethe

categor yo f

semigroups ,is well-c o-p ow ereda ndin pro ofstha ta particular

category ,for example

thecatego ryo

fs mall skeletal catego

ries,

isco-c omplete).

In this exercisethe

mostconspicuous omission

resultedfrom my

nottak ingthe

trouble toallo

wma ny-sort ed

theories ,whic hmea nt that Iw asno ta bleto mention

theeasy

theorem B that

isa A

category ofmo delswhenev

A er issmall

B and isit self ac atego ryof models .

Pag e1 07:

Characteristic zero

isnot needed

inthe firsthalf

ofE xer cise 4–H.

Itw ouldb eb etterto

sa ythat afield

arising

asthe ringo

fendomo rphismso

fa nab eliangro

upis necessa

r-

Infini 5

teBo oleanP

olynomials I.F

und.

Mat h.

541964

−18

PETER J. FREYD

I kept no track of their number. So now people were seeking the meaning for the barely visible slight increase in the size of

the word bifunctors on page 72. If the truth be told, it was

from the first sample page the Chief of Production had sent me for approval. Somewhere between then and when the rest of

the pages were done the size changed. But bifunctors didn’t

change. At least not in the first printing. Alas, the joke was removed in the second printing.

Pages 75–77: Note, first, that a root is defined in Exercise 3–B not as an object but as a constant functor. There was a month or two in my life when I had come up with the no- tion of reflective subcategories but had not heard about adjoint functors and that was just enough time to write an undergrad-

uate honors thesis². By constructing roots as coreflections into

the categories of constant functors I had been able to prove the equivalence of completeness and co-completeness (modulo, as I then wrote, “a set-theoretic condition that arises in the proof”).

The term “limit” was doomed, of course, not to be replaced by

“root”. Saunders Mac Lane predicted such in his (quite favor-

able) review³, thereby guaranteeing it. (The reasons I give on

page 77 do not include the really important one: I could not

for the life of me figure out how A×B results from a limiting

process applied to Aand B. I still can’t.)

Page 81: Again yikes! The definition of representable func- tors in Exercise 4–G appears only parenthetically in the first printing. When rewritten to give them their due it was nec-

essary to remove the sentence “To find A, simply evaluate the

left-adjoint of S on a set with a single element.” The resulting

2Brown University, 1958

3The American Mathematical Monthly, Vol. 72, No. 9. (Nov., 1965),

pp. 1043-1044.

−20

ABELIAN CATEGORIES

paragraph is a line shorter; hence the extra space in the second printing.

Page 84: After I learned about adjoint functors the main theorems of my honors thesis mutated into a chapter about the

general adjoint functor theorems in my Ph.D. dissertation⁴. I

was still thinking, though, in terms of reflective subcategories

and still defined the limit (or, if you insist, the root) of D → A

as its reflection in the subcategory of constant functors. If I had really converted to adjoint functors I would have known that

limits of functors in A^D should be defined via the right adjoint

of the functor A → A^D that delivers constant functors. Alas,

I had not totally converted and I stuck to my old definition in Exercise 4–J. Even if we allow that the category of constant

functors can be identified with A we’re in trouble when D is

empty: no empty limits. Hence the peculiar “condition zero” in the statement of the general adjoint functor theorem and any number of requirements to come about zero objects and such, all of which are redundant when one uses the right definition of limit.

There is one generalization of the general adjoint functor the- orem worth mentioning here. Let “weak-” be the operator on definitions that removes uniqueness conditions. It suffices that

all small diagrams in A have weak limits and that T preserves

them. See section 1.8 of Cats & Alligators. (The weakly com-

plete categories of particular interest are in homotopy theory. A

more categorical example iscoscanecof, the category of small

categories and natural equivalence classes of functors.)

Pages 85–86: Only once in my life have I decided to refrain from further argument about a non-baroque matter in math- ematics and that was shortly after the book’s publication: I

4Princeton, 1960

−19

ABELIANCA TEGORIES

ilya primefield

(hencethe categor

yof vec tor space so ver an y

non-primefield ca

nnot be fullye mb eddedin

thecatego ryo

f

abelian groups).

Theonly reasonI

can thinko

ffor insistingon

char acteristicz

ero istha tthe proofs

forfinite andinfinite

char ac-

teristicsar ed ifferen t—

astra ngereaso

ngiv en that neithe

rpro of

ispr esen t.

Pag e10 8:

Icame across

ag oo de xample ofa

locally small

abelian category

thatis notv eryab eliansho

rtlya fterthe second

printing appear

ed:

tow it, theta rgeto

fthe univers

al homol-

ogy theory ont heca tegory

ofcon nected

-co cw

mplexes(

finite

dimensional,if you wish).

Jo elCohen calle

dit the

“Freyd cat-

egory”

in his bo

6 ok

, but it sho uld be noted that

Joe l didn’t

nameit afterm

e.

(Hea lway sinsisted that

itw as my daugh-

ter.) It’

ss uch anic ec atego ryit’s wor thdescribing here

. To

constructit, startwith

pairs cw of -co mplexes

0 hX i ,X where

0 X

isa non-empt ysub

co mplexo

f X andta ke theob vious condition

onma ps,

towit, f

hX : ,X 0

i→

0 hY i ,Y isa contin uousma

p

f :

→ X

Y such that f

0 (X

⊆ )

0 Y . Now imp ose thecong ruence

that identifies f, g hX : ,X 0

i→

0 hY i ,Y when

|X f and 0 0 g|X

are

homotop ic(a smaps to

Y ).

Finally, take theresult of

formally

making the susp ensionf

unctor ana utomorphism

(which ca n,o f

course,b erestat eda

sta kinga reflection).

Thiscan allb

efound

inJo el’sb ook or inm ya rticlewith

thesame titlea

sJo

7 el’s

.

Thefact thatit

isno tv erya belia nfollo wsfrom

thefact that

thesta ble-homo topy

category appears

as asub category

(to wit,

the full subca tegory of ob jects of the hX, form

i) X and that

category was sho wnno tto hav ean yem bedding atall

into the

Stable 6

Homotop y Lectur e Notes in Mathematics

Vol . 165 Spri nger-

Verlag ,Berlin -New

York

1970 ⁷

Stable Homotopy,

Proc.

of the Confer ence of Categori cal Algebr a ,

Sprin ger-Ver

lag,1966

− 17

PETER J.FREYD

Pag e15 9:T heY one dalemma turnsout

notto be inY oneda’s

paper. When,some

timea fterb oth printing

so fthe bo oka p-

peared, thisw

asbro ught

tom y(m uch cha grined) attention,

I

brough tit thea ttentio no fthe person whohad

told methat it

was theY oneda lemma.

Heconsulted hisnotes

anddisco ver

ed

that itapp eared

ina lecturetha

tMac Lanega

ve onY oneda’s

treatmen tof thehigher Ext

functors.The name

“Yoneda lemma”

was notdo omed tob

ereplaced.

Pag es163 –164:

Allows and

Generating were

missingin the

indexo fthe firstpr

inting asw

as page1 29for Mitchell

. Still

missingin these

cond prin tingar e Natural equivalenc e,

8 and

Pre-additive cate

gory,60 . Notmissing, alas,

is Monoidalc

ate-

gory .

FINALLY, aco mment

onwhat I“

hoped tob ea geo desic

course”to the fullem bedding theorem(men

tioned onpage

10).

Ithink thehop

ew asj ustifi ed forthe fullem

bedding theorem,

but if one settles

for the exact emb edding theorem

the n the

geodesic courseomitted

an impo rtan tdev elopment.

Bybroa d-

eningt hepr oblemto

regular categories onecan

finda cho ice-free

theorem which—aside

fromits widerapplica

bility ina top os-

theoretic setting—has

theadv antage

ofna turality.

Thepro of

requiresconstructio nsin

thebroa der

con textbut ifo nea pplies

theg eneral cons truction tothe

spe cial caseo fab elianc

atego ries,

we obta in:

There isa construction thatassigns

toe achsmal la belian cat-

egory anexact A

embedding intothe

cate goryof

abelian groups

→G A suchthat

forany exactfunctor

→ A there B isa nat-

ural assignment

of a natural tran sformation A from

→ G to

→ A

→G B . A When

B → is an embedding then

so is the

transfor mation.

Thepro ofis sug gestedin

my pamphlet

Onc anonizingc

at-

−14

PETER J. FREYD

category of sets in Homotopy Is Not Concrete⁸. I was surprised,

when reading page 108 for this Foreword, to see how similar in spirit its set-up is to the one I used 5 years later to demonstrate the impossibility of an embedding of the homotopy category.

Page (108): Parenthetically I wrote in Exercise 4–I, “The only [non-trivial] embedding theorem for large abelian categories that we know of [requires] both a generator and a cogenerator.”

It took close to ten more years to find the right theorem: an abelian category is very abelian iff it is well powered (which it should be noticed, follows from there being any embedding at all into the category of sets, indeed, all one needs is a functor that distinguishes zero maps from non-zero maps). See my paper

Concreteness⁹. The proof is painful.

Pages 118–119: The material in small print (squeezed in when the first printing was ready for bed) was, sad to relate, directly disbelieved. The proofs whose existence are being as- serted are natural extensions of the arguments in Exercise 3–O on model theory (pages 91–93) as suggested by the “conspicuous omission” mentioned above. One needs to tailor Lowenheim- Skolem to allow first-order theories with infinite sentences. But it is my experience that anyone who is conversant in both model theory and the adjoint-functor theorems will, with minimal prod- ding, come up with the proofs.

Pages 130–131: The Third Proof in the first printing was hopelessly inadequate (and Saunders, bless him, noticed that fact in his review). The proof that replaced it for the second

printing isok. Fitting it into the alloted space was, if I may say

so, a masterly example of compression.

8The Steenrod Algebra and its Applications, Lecture Notes in Mathe-

matics, Vol. 168 Springer, Berlin 1970

9J. of Pure and Applied Algebra, Vol. 3, 1973

−16

ABELIAN CATEGORIES

Pages 131–132: The very large categoryB (Exercise 6–A)—

with a few variations—has been a great source of counterexam- ples over the years. As pointed out above (concerning pages 85–86) the forgetful functor is bi-continuous but does not have either adjoint. To move into a more general setting, drop the

condition that G be a group and rewrite the “convention” to

becomef(y) = 1G fory /∈S (and, of course, drop the condition

that h:G→G⁰ be a homomorphism—it can be any function).

The result is a category that satisfies all the conditions of a Grothendieck topos except for the existence of a generating set.

It is not a topos: the subobject classifier, Ω, would need to be the size of the universe. If we require, instead, that all the values of

allf :S →(G, G) be permutations, it is a topos and a boolean

one at that. Indeed, the forgetful functor preserves all the rel- evant structure (in particular, Ω has just two elements). In its

category of abelian-group objects—just as inB—Ext(A, B) is a

proper class iff there’s a non-zero group homomorphism fromA

toB(it needn’t respect the actions), hence the only injective ob-

ject is the zero object (which settled a once-open problem about whether there are enough injectives in the category of abelian groups in every elementary topos with natural-numbers object.) Pages 153–154: I have no idea why in Exercise 7–G I didn’t cite its origins: my paper, Relative Homological Algebra Made

Absolute¹⁰.

Page 158: I must confess that I cringe when I see “A man learns to think categorically, he works out a few definitions, per- haps a theorem, more likely a lemma, and then he publishes it.”

I cringe when I recall that when I got my degree, Princeton had never allowed a female student (graduate or undergraduate). On the other hand, I don’t cringe at the pronoun “he”.

10Proc. Nat. Acad. Sci., Feb. 1963

−15

ABELIANCA TEGORIES

egory theory

oron functorializing

model theory

. 11

Ituses the

strange sub ject of τ -ca tegories.

More acc essibly,

itis exposed in

section 1.54of

Cats&

Alligator s.

Philadelphia

Nov emb er18, 2003

Z –

Mim 11

eographed notes,Univ.

Pen nsylvani a,P hiladelp hia,P

a., 1974

− 13

F.Wi lliamLa

wv ere,S.U.

N.Y.at Bu ffalo:

wlawvere@acsu.buffalo.edu

Jean-Louis Lod ay, Univ ersit

´ ede Strasb ourg:

loday@math.u-strasbg.fr

Ieke Moerd ijk, Universit yof Utrech

t:

moerdijk@math.uu.nl

Susan Niefield,Un

ionCollege niefiels@union.edu :

Robe rt Par

´ e, Dalhou sie Univ ersit y:

pare@mathstat.dal.ca

Andrew Pitts, Univ

ersit yof Camb ridge:

Andrew.Pitts@cl.cam.ac.uk

Robe rt Rosebr ugh, Moun tAl lisonUn ivers ity:

rrosebrugh@mta.ca, Man-

agingEditor Jiri Rosic ky, Mas aryk Univ ers ity:

rosicky@math.muni.cz

James Stasheff,Un

iversit yof NorthC

arolin a:

jds@math.unc.edu

RossStr eet, Macquarie

Univ ersity:

street@math.mq.edu.au

Walter Th olen,Y ork Universit tholen@mathstat.yorku.ca y:

Myle sT ie rney ,Ru tgers Univ ersit y:

tierney@math.rutgers.edu

Robe rt Walters, Universit

yof Insubri a:

robert.walters@uninsubria.it

R.J. Wo od, Dalhou sie Universit y:

rjwood@mathstat.dal.ca

This reprin tm ay be acces sedfrom

http://www.tac.mta.ca/tac/tacreprints

REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles from the body of important literature in Category The- ory and closely related subjects which have never been published in journal form, or which have been published in journals whose narrow circulation makes access very difficult. Publication in ‘Reprints in Theory and Ap- plications of Categories’ will permit free and full dissemination of such documents over the Internet.

Articles appearing have been critically reviewed by the Editorial Board

ofTheory and Applications of Categories. Only articles of lasting signifi-

cance are considered for publication.

Distribution is via the Internet toolsWWW/ftp.

Subscription information. Individual subscribers receive (by e- mail) abstracts of articles as they are published. To subscribe, send e-mail

totac@mta.caincluding a full name and postal address. For institutional

subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors. After obtaining written permission from any copyright holder, an author, or other member of the community, may propose an article to a TAC Editor for TAC Reprints. TAC Editors are listed below. When an article is accepted, the author, or other proposer, will be required to provide to TAC either a usable TeX source for the article or a PDF document acceptable to the Managing Editor that reproduces a typeset version of the article. Up to five pages of corrections, commentary and forward pointers may be appended by the author. Please obtain de- tailed information on TAC format and style files from the journal’s WWW

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Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu

Michael Barr, McGill University: barr@barrs.org, Associate Managing

Editor

Lawrence Breen, Universit´e 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, Universit`a dell Insubria:aurelio.carboni@uninsubria.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