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

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

ABELIANCA TEGORIES

(thist imeHarp er’sa

greedto mak ethem without

deductingfrom

my roy altie s;

the co rrection

of my left- righ t errors—scores of

them—forthe firstprin

tingha dcost meh

undredsof dollars).

Butfo rr ea sonsI never though

tto askab

out, Harp er’sdidn’t

mark the seco ndpr inting ass uch.

The copyr ight page, –8, isa l-

mostiden tical,e

ven thedate.

(WhenI need todetermine which

printing I’m holding—as,

forexa mple,

when findinga

copy for

thist hird“

reprinting

”—Ic heck thelast verb on page –3.

Int he

seco ndpr inting itis has insteadof

have).

Af ewother page-

spec ificcommen ts:

Pag e8 : Yikes!

In thefir st printing there’sno

definitionof

natural equivalenc

e.

Ma kingro om forit requiredm

uch short-

eningof thispa

ragraph fromthe

firstprin ting:

Oncethe definitions

ex iste dit was quick

lynot iced

that functors

and natural transfo

rmations had be-

comea majo rto ol inmo dernma

thematics.In 1952

Eilenb erg andSteenro dpublis

hedtheir Found

ations

ofA lgebraic

Top ology [7],an axiomat icappr

oach to

homolog y the ory . A homology

theory was defi ned

asa functor

from ato polog icalcateg

oryto analg e-

braic catego ryob eyingce

rtain axioms.

Among the

more striking result

sw astheir classification

ofsuc h

“the ories,”

animp ossibleta

sk withoutthe

notion of

natural equivalence

of functors.

Ina fairly explosive

manner,functor sand

natural trans

format ionsha

ve

permeated aw

idev ariety ofsub jects.

Such monu-

menta lw orksas Cartan

and Eilenb

erg’s Homologic

al

Algebr a [4], and Grothendieck’s

Elements of Alge-

braic Geometry [1]testify

tothe facttha

tfuncto rs

hav eb eco mea nes tablished co

ncept inm athematic

s.

Pag e21 :The term“difference

ker nel”

in1.6 was doomed, of

− 23

PETER J. FREYD

course, to be replaced by the word “equalizer”.

Pages 29–30: Exercise 1–D would have been much easier if it had been delayed until after the definitions of generator and pushout. The category [→] is best characterized as a generator for the category of small categories that appears as a retract of every other generator. The category [→→] is a pushout of the two maps from 1 to [→] and this characterization also simpli- fies the material in section 3: if a functor fixes the two maps from 1 to [→] then it will be shown to be equivalent to the identity functor; if, instead, it twists them it is equivalent to the dual-category functor. These characterizations have another ad- vantage: they are correct. If one starts with the the two-element monoid that isn’t a group, views it as a category and then for- mally “splits the idempotents” (as in Exercise 2–B, page 61) the result is another two-object category with exactly three endo- functors. And the supposed characterization of [→→] is coun- terexampled by the disjoint union of [→] and the cyclic group of order three.

Page 35: The axioms for abelian categories are redundant:

either A 1orA 1* suffices, that is, each in the presence of the

other axioms implies the other. The proof, which is not straight- forward, can be found on section 1.598 of my book with Andre

Scedrov¹, henceforth to be referred to asCats & Alligators. Sec-

tion 1.597 of that book has an even more parsimonious definition of abelian category (which I needed for the material described below concerning page 108): it suffices to require either prod- ucts or sums and that every map has a “normal factorization”, to wit, a map that appears as a cokernel followed by a map that appears as kernel.

Pages 35–36: Of the examples mentioned to show the in-

1Categories, Allegories, North Holland, 1990

−22

ABELIAN CATEGORIES

egory theory or on functorializing model theory¹¹. It uses the

strange subject ofτ-categories. More accessibly, it is exposed in

section 1.54 of Cats & Alligators.

Philadelphia November 18, 2003

Z –

11Mimeographed notes, Univ. Pennsylvania, Philadelphia, Pa., 1974

−13

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

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

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

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

paragr aphis

aline sh orter;hence theextr

aspace int

hesecond

printing.

Pag e 84:

After I learned

abo ut adjoin t functors

the main

theorems ofm yho norsthesis

mutat ed into ac hapter about

the

general adjoint

fun ctor theorems

inm yPh.D.

dissert ation

. 4

I

was stillthinking ,thou

gh,in termsof

reflective subca tegories

ands till defined the limit(o

r,if you insist,the

roo t)o

D→ f

A

asit srefle ction

inth es ubcat eg oryof consta nt functo rs.If

Ihad

reallycon ver tedto adjoint

functors Iw ould hav ekno wn that

limitso ffuncto

A rsin

shouldb D

edefined viathe

right adjoin t

ofthe functor

A→

D A

that delivers constan

tfunctor s.

Alas,

Ihad notto

tally co nv erted and Istuc kto my olddefinitio nin

Exercise 4–J.

Even if we allo w that the category

of cons tan t

functorscan be identified

A with

we’re intr oublewhen

D is

empty:

noempt ylimits.

Hence thep eculiar“

co nditionzero”

in

thesta tement

of theg eneraladjoin

tfunctor theoremand

any

num ber ofrequiremen tsto

comeab out zeroo bjects and suc h,

allo fw hich are redundant

when one usesthe

right definitiono

f

limit.

Thereis onegeneraliza tionof

theg eneral adjoint

functor the-

orem wor thmen tioninghere.

Let“w eak-”b

ethe opera

toron

definitions thatremo ves

uniqueness conditions.

Itsuffice sthat

allsmall diagrams

A in

hav ew eak limits and that T preserves

them.

See sectio n1 .8of Cats&

Alligators .

(Thew eaklycom-

pleteca tegorie

sof particula rin

ter est are inho motop

ytheory .A

more ca tegorical example

cosc is

anecof ,the categor

yof small

categories and natural

equivale ncec lass esof functors.)

Pag es8 5–86 : Only oncein

my lifeha ve Idecided toref

rain

fromfurther argumen

tab out ano n-ba roq ue matter inma th-

ematics and that was shortly after

the bo ok’s publication:

I

Princeton, 4

1960

− 19

PETER J.FREYD

category ofsets inHo

motop yIs Not Concrete

.I 8

was surprised,

whenreading page

108 forthis Forew ord, tosee how

simila rin

spiritits set-upis

tothe one Iused 5y

earsla terto demonstrat

e

theimp ossib

ility ofan em bedding of

thehomo topy

catego ry.

Pag e(1 08):

Pa renthetica llyI

wrote inExercise

4–I,

“The

only[non-trivia l]em

bedding theorem

fo rlar geab elianca

tegories

that we know of[requires]

both ag enera torand

acog enerator.”

Itt ook closet

ot enmor ey ears tofind ther

ight theorem:

an

abelian ca tegory isv eryab elianiff

itis well po wered (which

it

shouldb enot iced, follow

sfrom thereb

eingan yem bedding atall

into the category

ofs ets, indeed, allone needsis

af unctor

that

distinguishes zero maps

from non- zero maps).

See my pap er

Concreteness

.The 9

proof ispainful.

Pag es 118–

119:

The material in

small print (squeezed

in

whent hefirst printing

was readyfor

bed) was, sadto

relate,

directlydisb elieved.

Thepro ofs whoseexistence

are being as-

sertedare natural

extensionsof thea rguments

inExercise 3–O

onmo deltheory (pages

91–9 3)as sug gestedb

yt he“co nspicuo us

omission”men tionedab

ov e.

One needs tota ilorLo wenheim-

Sko lemt oa llow first-ordertheor

ieswith infinitesen

tences.

But

itis my experience thata

ny one who isc on versa nt inb othmo del

theory and theadjoin t-functo

rtheo remswill,

withminimal prod-

ding,come upwith

thepro ofs.

Pag es1 30–13 1:

TheThird Proo

fin thefirst

printing was

hopeless ly inadequat e

(and Saunders,

bless him,

noticed that

factin hisr eview).

Thepr oof that replacedit

for thes econd

printing

ok is

.Fitting itin

to theallo tedspace

was, ifI may say

so,a masterlyexa

mpleof compression.

Th 8

e Steenr od Algebr a and its Appli cations , Lecture Notes inMathe-

matics, Vol.

168S pringer, Berlin

1970 ⁹

J.of Pure andA pplie dA lgebra ,V ol.3, 1973

−16

PETER J. FREYD

refused to engage in the myriad discussions about the issues dis- cussed in the material that starts on the bottom of page 85. It was a good rule. I had (correctly) predicted that the contro- versy would evaporate and that, in the meantime, it would be a waste of time to amplify what I had already written. I should, though, have figured out a way to point out that the forgetful

functor for the category, B, described on pages 131–132 has all

the conditions needed for the general adjoint functor except for the solution set condition. Ironically there was already in hand a much better example: the forgetful functor from the category of complete boolean algebras (and bi-continuous homomorphisms) to the category of sets does not have a left adjoint (put another way, free complete boolean algebras are non-existently large).

The proof (albeit for a different assertion) was in Haim Gaif-

man’s 1962 dissertation⁵.

Page 87: The term “co-well-powered” should, of course, be

“well-co-powered”.

Pages 91–93: I lost track of the many special cases of Exercise 3–O on model theory that have appeared in print (most often in proofs that a particular category, for example the category of semigroups, is well-co-powered and in proofs that a particular category, for example the category of small skeletal categories, is co-complete). In this exercise the most conspicuous omission resulted from my not taking the trouble to allow many-sorted theories, which meant that I was not able to mention the easy

theorem that B^A is a category of models whenever A is small

and B is itself a category of models.

Page 107: Characteristic zero is not needed in the first half of Exercise 4–H. It would be better to say that a field arising as the ring of endomorphisms of an abelian group is necessar-

5Infinite Boolean PolynomialsI. Fund. Math. 54 1964

−18

ABELIAN CATEGORIES

ily a prime field (hence the category of vector spaces over any non-prime field can not be fully embedded in the category of abelian groups). The only reason I can think of for insisting on characteristic zero is that the proofs for finite and infinite charac- teristics are different—a strange reason given that neither proof is present.

Page 108: I came across a good example of a locally small abelian category that is not very abelian shortly after the second printing appeared: to wit, the target of the universal homol-

ogy theory on the category of connected cw-complexes (finite

dimensional, if you wish). Joel Cohen called it the “Freyd cat-

egory” in his book⁶, but it should be noted that Joel didn’t

name it after me. (He always insisted that it was my daugh- ter.) It’s such a nice category it’s worth describing here. To

construct it, start with pairs ofcw-complexeshX⁰, XiwhereX⁰

is a non-empty subcomplex ofX and take the obvious condition

on maps, to wit, f : hX⁰, Xi → hY⁰, Yi is a continuous map

f :X →Y such thatf(X⁰)⊆ Y⁰. Now impose the congruence

that identifies f, g :hX⁰, Xi → hY⁰, Yiwhen f|X⁰ and g|X⁰ are

homotopic (as maps to Y). Finally, take the result of formally

making the suspension functor an automorphism (which can, of course, be restated as taking a reflection). This can all be found

in Joel’s book or in my article with the same title as Joel’s⁷.

The fact that it is not very abelian follows from the fact that the stable-homotopy category appears as a subcategory (to wit,

the full subcategory of objects of the form hX, Xi) and that

category was shown not to have any embedding at all into the

6Stable Homotopy Lecture Notes in Mathematics Vol. 165 Springer-

Verlag, Berlin-New York 1970

7Stable Homotopy, Proc. of the Conference of Categorical Algebra,

Springer-Verlag, 1966

−17

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

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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