Subject:     two typos in Categories, Allegories
Date: Tue, 30 Oct 90 03:08:31 CST
From: koslowj@math.ksu.edu (Juergen Koslowski)

p 143, item 1.82(10). The theorem should read: A functor that preserves
       pre-limits preserves limits.

p 206, line 2: the capital P (denoting a point in a projective plane) should
       be a small p.

I only got the book today, but the first impression is certainly very promising.

J"urgen Koslowski
Department of Mathematics
Kansas State University

koslowj@math.ksu.edu

Subject:     EECS '91

                 EAST EUROPEAN CATEGORY SEMINAR '91

                Bulgaria, Predela, March 11-18, 1991

        The Sixth Annual East European Category Seminar (EECS '91), organised
by the Institute of Applied Mathematics and Computer Science of the Technical
University of Sofia, Bulgaria, will take place in Predela, Bulgaria, the week
of March 11 through 18, 1991 (Monday through Monday).

        Note that the timing is so arranged that APEX air fares requiring a
Saturday night abroad are perfectly eligible for use.  It is hoped, moreover,
that most or all of the time of the meeting may fall during "spring break"
periods at the participants' universities.

        The Organisers expect prospective participants to send a camera-ready
copy of a summary of their projected contribution on 8.5"x11" paper or on the
nearest DIN equivalent (circa 20x28.3 cm^2).

        Participants are expected to arrive in Sofia not later than late mid-
afternoon of March 11, in time to be able to board a complimentary chartered
bus to Predela departing from the parking lot of the Technical University at
5:30 pm March 11.  Participants arriving earlier that day may check luggage
in Room 2226, one flight up in the building known as Block 2, Durvenitsa,
which houses the Institute of Applied Mathematics and Computer Science.
The bus ride, of some two to three hours to the ski country south of Sofia,
will arrive in time for supper at Predela.  The return to Sofia on March 18
will be by the same bus, leaving Predela right after breakfast, so return
reservations from Sofia are most conveniently planned for no earlier than
noon of March 18, and preferably somewhat later.

        During the time in Predela, room and board are offered at utterly
nominal cost (by Western standards -- last year, the week cost under $30.00
-- should be approximately similar this year, too, I believe).

        Predela may be found on the map of Bulgaria by following the main
road south from Sofia to Simitli, just past Blagoevgrad, and then turning
east for some dozen kilometers to what should be the first (maybe the only)
saddle.  The meeting site is some 400 meters off the road, and some 500
meters from a local inn/tavern offering excellent wine, beer, rakia, and
yogurt when it's in stock.

        Bulgaria still has a visa requirement, at least for US citizens.
Citizens of other countries should check with their local Bulgaria Embassy
or Consulate, or consult a reputable visa-procurement agency fairly well in
advance.  The Embassy in Washington, D.C., issues visas with a maximum
validity period of three months.  Normal handling requires a three-week
turn-around period, but anyone who can get to the Embassy in person may
request "express" service: for a small additional fee, the visa is issued
on the spot, then and there.  (Ever since the USPS "lost" my freshly visaed
passport for over a month, not finding it until *after* I had returned from
EECS '89, I've only applied in person!)

        The organisers recommend attaching their Official Invitation letter
(copy available from me, flinton@eagle.Wesleyan.EDU, by regular mail, if you
send me your full name and postal address) to the Bulgarian visa application
(but I've always just declared "tourism -- late winter holiday" and never had
a problem).

        Those intending to participate should make their intentions -- and,
ultimately, their travel plans, specifying dates, times, and flight/train
numbers -- known as soon as possible.  Addresses to use:

        for preliminary intentions:     Prof. V. V. Topencharov
                                        Technical University, Sofia
                                        Institute of Applied  Mathematics
                                                and Computer Science
                                BG-1000 SOFIA, P.O. Box 384
                                        Bulgaria

                                or:     flinton@eagle.Wesleyan.EDU

        for last-minute arrival info:   Phone: +359 (02) 68-20-83
                                        TELEX: +865 22575 (EECS'91, LOZANOV)
(NB: there is another telex number,  22755 , in circulation, but as near as
I can make out that one's a typo, and it's the former, the 22575, that works.)

        Participant numbers for the previous five meetings have been roughly
20, 30, 40, 50, 30, in sequence (the drop back to thirty due most likely to
greater than usual uncertainty, back at the end of 1989, as to what the spring
would produce in the way of Eastern European unrest in Bulgaria).

                                    -- Fred

Fred E.J. Linton  Wesleyan U. Math. Dept.  649 Sci. Tower  Middletown, CT 06457
E-mail:  <FLINTON@eagle.Wesleyan.EDU>  or  <fejlinton@{att|mci}mail.com>
Tel.:         + 1 203 776 2210 (home)  or  + 1 203 347 9411 x2249 (work)

Subject:     TTT-CTCS Corrections
Date:        Tue, 30 Oct 90 22:36:25 EST
From:        "Michael Barr, Math Dept, McGill University"
             <INHB@MUSICB.MCGILL.CA>

Dear Bob

More corrections and clarifications to ttt and ctcs.  The former are all
from Fer-Jan de Vries and the latter are all from Al Vilcius (a former
student of Joan's who now works for CIBC and has read ctcs with an eagle
eye).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ttt %%%%%%%%%%%%%%%%%%%%%%%%%

\item\lb p. 139\rb  (F-J de Vries) l. -3:  "subststantially" should read like
"substantially"

\item\lb p. 140\rb  (F-J de Vries) l. 9:  "Kiesler" should be "Keisler"

\item\lb p. 168\rb  (F-J de Vries) l. -15: "Kiesler" should be "Keisler"

\item\lb p. 183\rb  (F-J de Vries) l. -7:  Kock \lb 1982\rb  should be
Kock \lb 1981\rb (SDG)

\item\lb p. 267\rb  (F-J de Vries) l. 267:  "Mikkelson" should be "Mikkelsen"

\item\lb p. 337\rb  (F-J de Vries) "Mikkelson" should be "Mikkelsen"

\item\lb p. 338\rb  (F-J de Vries) "J.Shonfield" should read as "J.R.
Shoenfield"

Two additional references (F-J de Vries):

H. Cartan and S. Eilenberg, {\bf Homological Algebra}.  Princeton University
Press, 1952.

Anders Kock, {\bf Synthetic Differential Geometry}.  London Mathematical
Society Lecture Note Series 51.  Cambridge University Press, 1981.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% to ctcs %%%%%%%%%%%%%%%%%
\item\lb p. 79\rb  (Al Vilcius). l. 7: replace ``$\id_i$'' by $\id_{D_i}$.

\item\lb p. 87\rb  (Al Vilcius).  In Theorem 4.2.20, l. 2 replace ``$\Csc$''
by ``$\Gsc$'.  In the last line replace ``$\Mod(\Gsc,\Csc)$'' by
``$\Mod(\Gsc,\Dsc)$''.

\item\lb p. 103\rb  (Al Vilcius).  Change ``set'' to ``collection'' in the
second line of 4.6.2 and the sixth line of 4.6.3.  (We are using
``collection'' as an informal synonym for ``class'', without getting
into set theory.)

At the end of the first paragraph of 4.6.3, add the following:
``We write $M:\Ssc\to\Csc$ for such a model.  This use of the same
symbol to denote both the sketch homomoprhism and the graph homomorphism
is a bit of notational overloading that in practice is always
disambiguated by context.''

\item\lb p. 108\rb  (Al Vilcius).  Change ``set'' to ``collection'' in line 3
of 4.7.2.

\item\lb p. 208\rb  (Al Vilcius).  Top diagram should have a $B$, not $C$ in
the SE corner.  The fourth line should say that $p_2$, not $p_1$ is the
pullback of $f$ along $g$.

\item\lb p. 228\rb  (Al Vilcius). last line:  Word ``be'' is missing.

\item\lb p. 255\rb  (Al Vilcius). l. -5.  Replace ``cleavage'' by
``opcleavage''.

\item\lb p. 256\rb  (Al Vilcius). l. -8, the expression has misplaced
brackets.  Should be ``$$Fg\lb Ff(u)\rb (Al
Vilcius).\o\kappa(g,Ff(X))\o\kappa(f,X)$$''

l. -2 should be ``functors $\Csc\op\to\Cat$.''

\item\lb p. 258\rb  (Al Vilcius).  Definition 11.2.3, GS--1 should be ``object
of $G_0(\Csc,F)$''.

\item\lb p. 259\rb  (Al Vilcius). l.8 should read ``$x'=Ff(x)$''.

In paragraph starting line 11, orientation of $G_0(\Csc,F)$ requires the
functor $G_0(F)$ to be the projection on the second coordinate, not
first.

\item\lb p. 261\rb  (Al Vilcius).  In Theorem 11.2.9 first projection should
be second as above.

\item\lb p. 264\rb  (Al Vilcius).  In Definition 11.3.4, FI--1, replace
``P:\Esc\to\Cat'' by ``$P:\Esc\to\Csc$''.

Regards, Mike

Subject:     Homotopy = homobject
Date: Tue, 13 Nov 90 15:55:21 PST
From: Vaughan Pratt <pratt@cs.stanford.edu>

Cellular geometry arises both with categories, starting with
Ehresmann's 1965 notion of an n-category, and with concurrency as per
my POPL-91 paper and also as per a paper that David Murphy just brought
to my attention, ``Deterministic Asynchronous Automata,'' Mike Shields,
Proc. Formal Models in Programming (Ed. E.Neuhold, G.Choust), Elsevier
1985.

I'm not sure who in category-land cares about homotopy in n-categories,
but it is the basis for distinguishing true from false nondeterminism
in my POPL paper.  As David points out to me, a special case of
homotopy can be found in Mazurkiewicz's independence relation: the
independence of a and b should be identified with the paths ab and ba
being homotopic, as in a|b.  In ab+ba however these two paths are not
homotopic: one has to decide which of the ab or ba paths one is going
to follow.

While the following is obviously too cryptic for general consumption, I
am mentioning the idea here for two reasons: to mumble my obscure
thought processes concerning true nondeterminism out loud on the
concurrency and category lists, and to find out if this definition of
homotopy as homobject rings a bell with anyone.  It seems so obvious
that I am fully expecting it to have been around for decades, at least
somewhere.  It just isn't in the places I've looked so far.  If it is
spelled out somewhere, any attempt on my part to expand on the mumbling
below may not be necessary.

Here's the idea.  It seems to me that a very natural definition of
homotopy is arrived at by identifying homotopy with homobject, in the
enriched category sense.  That is, the homotopy of the paths from x to
y is the homobject ?x,y?, or d(x,y) in the notation of Casley et al,
CTCS-89, Manchester, LNCS 389, the "distance" from point x to point y.
(The basic law governing homobjects is the abstract triangle
inequality, which is why it is appropriate to think of the homobject
?x,y? as an abstract distance d(x,y).  This view is due to Lawvere
1974.)

Hence homotopy is governed principally by the triangle inequality, the
basic law of enriched category theory.  In this sense the homotopy from
x to y and the distance from x to y become the same thing.

The homotopy of an ordinary category is discrete because its homobjects
are sets.  The homotopy of a set is nonexistent because sets don't have
homobjects worth mentioning (all points are equidistant).  The homotopy
of a poset is trivial because its homobjects contain either no elements
(i.e. paths) or one.

The intuitive notion of homotopy as an equivalence relation on paths
arises for categories whose homobjects are equivalence relations; then
?x,y? is a set (X,^) of paths and an equivalence ^ on paths whose
blocks are the homotopy classes.  However it would seem nicer to take
arbitrary categories for homobjects, the homotopy of a 2-category.

The homotopy of an order-enriched category lies between that of
categories and 2-categories.  The simplest case of this arises for a
monoid (1-object category), the basis for my recently developed "action
logic" ACT (pub/jelia.{tex,dvi} via ftp from boole.stanford.edu).
Action logic is accessible to anyone who understands lattice theory,
and employs no categorical language or explicit categorical concepts,
yet it contains interesting homotopy in the above sense, in a way that
Boolean logic and intuitionistic logic as cartesian closed posets do
not.

In a closed category homotopy is internalized just like a homobject,
via exponentiation/implication.  That is, the entire homotopy ?x,y? can
be compressed into the single point b?a or a=>b as its internal
representation.  The homotopy so coded can then be recovered as the
homotopy from I (the unit of the closed category) to that point, via
the isomorphism between ?I,a=>b? and ?a,b?.  Thus isomorphic copies of
all homotopy present in a closed category can be found radiating out
from its unit.

In the case of action logic the homotopies so radiating out from I
(called 1 there) are exactly the theorems of action logic.

I know the above must look to many of you rather unrelated to the
traditional geometry of triangles, circles, and squares.  Hopefully
someone will someday volunteer to draw enough pretty pictures of this
really very simple notion of homotopy to dispel any remaining mystery
about it.  You will find a few such pictures at the end of the action
logic paper, of paths with fixed endpoints sweeping across surfaces,
which should fit right in with any prior intuition you had about
homotopy.

        Vaughan Pratt

Subject:     homotopy = homobject
Date: Wed, 14 Nov 90 14:32 GMT
From: MAS010@vaxa.bangor.ac.uk

This is something of a reply to Vaughan Pratt's letter of 13 Nov
on homotopies as hom-objects.
 This is fairly well used in the case of groupoids. It is
mentioned in my survey `From groups to groupoids' Bull London
Math. Soc. 19, 113-134, 1987.
 In the case of multiple categories, the situation is more
complicated. Ehresmann and Ehresmann studied this in papers in the
Cahiers.
 Brown and Higgins studied the cubical version of mutiple
categories and the internal tensor product and hom in JPAA 47,
1-33, 1987. The main interest in that paper was, however, carrying
this structure over, in the groupoid case, to the equivalent cate-
gory of crossed complexes. The aim in any case is that the
internal hom carries information on homotopies and higher
homotopies.
 Infinity categories were defined by Brown and Higgins in
Cah. Top. Geom. Diff. Cat. 22, 371-386, 1981. There, infinity
groupoids were shown equivalent to crossed complexes.
 Richard Steiner (Glasgow) and F.Al-Agl (Saudi Arabia, ex
Bangor) have recently studied the relations between infinity
categories and various forms of cubical or simplicial infinity or
multiple category.  The aim is to obtain equivalences of
categories, so that there is transfer of information from one
category to the other. This has been important in the applications
of the groupoid case to homotopy theory. This area is summarised
in R.Brown, `Some problems in non-Abelian homotopical and
homological algebra', Springer LNM 1418, 1990, 105-129.  The
cubical case is an easy one for defining homotopies and higher
homotopies, and the adjoint tensor product. There is work to be
done on say the simplicial case. Crossed complexes are equivalent
to simplicial T-complexes, (Ashley, Diss. Math. 265, 1988), but
no-one has produced the internal hom and tensor product in that
category.
Ronnie Brown
