Announcing intention of conducting

	Seminar on Higher Dimensional Categories (HDC's);

and Progress Report.

Now that Sjoerd Crans and Paddy McCrudden are among our midst, I feel that
there should be a systematic seminar on HDC's. My selfish purpose with it
is that I would like to have an outlet for my recent work (see Progress
Report below), the writing up (and working out in some cases) of which
would be greatly helped by being able to talk about it to, and discuss it
with, an expert and/or interested audience. However, I do not mean that
the Seminar would only be about things that I am directly involved in. The
above mentioned gentlemen will surely like to talk about and discuss their
own ongoing and extant works. Also, if anyone out there wants to join the
fray now, and needs instruction from scratch, I am very much ready to
provide such instruction, in the framework of such a Seminar. And then
too, suggestions for looking, in the framework of the Seminar, at specific
things in the literature are appreaciated. There may be some overlap with
the CRTC Seminar, whenever that venue is available and agreeable to
accomodating the extremely talkative HDC-ists. 

Suggestions for timing, and comments in general, are hereby being invited. 

I'd like to outline now what I would like to expose in said Seminar -- if
that indeed materializes. This will take the form of a 

		Preliminary Report

on work done and work underway. 


1. The universals in the multitopic categories

The break-through work [B/D]: "Higher-Dimensional Algebra III.
n-Categories and the Algebra of Opetopes"  by J. C. Baez and J. Dolan has
two ingredients. One is opetopes and opetopic sets; the other is the
universal arrows. The first has been reworked in [H/M/P]: "On weak higher
dimensional categories I" by C. Hermida, myself, and J. Power -- although
the exact relationship of [B/D] and [H/M/P] is still to be worked out.
However, [H/M/P] does something more explicitly than [B/D]: [H/M/P] gives
the complete description of the category of multitopes, which should be
compared to the category of opetopes; the latter is not given in [B/D]
or elsewhere (?) in detail. 

This lack of precise detail does not prevent one from adapting the
[B/D]-definition of universals for the multitopic context. The authors of
[H/M/P] saw clearly (at least it seemed to them that they saw it clearly)
how to do that; however, they did not incorporate universals into
[H/M/P]. 

I am now announcing that I have a way of defining "universals" in the
multitopic context, one that is more "inclusive" than the one directly
suggested by [B/D]. The relations of this to the [B/D]-style concept is
still to be worked out -- but it is clear that the basic character of the
relationship is similar to that of the Mac Lane-style definition of
bicategory vs the original Benabou-style definition of same; the former
gives a small finite set of coherence-arrows and laws, the latter takes up
"all possible such" into the definition. The essential equivalence of the
two definitions is the Mac Lane coherence theorem. My proposal for
"universals" is like the "Benabou definition"; the [B/D] is like the "Mac
Lane definition", at least for the purposes of this comparison. The analog
of the Mac Lane coherence theorem is not proved yet (assuming it is true);
but many details are known, mainly thanks to the recent work of Victor
Harnik on the [B/D]-definition and variants of it, which point to the
truth of said analog.

Here is an outline of the proposal. I am now using the  concept of
"multitope" which is the main notion of [H/M/P]. Roughly speaking, a
multitope is the "shape" of single cell with domain a finite pasting
diagram of arbitrarily high dimension, which enjoys the fundamental
simplifying property that the target of each cell in it is a single cell,
as opposed to an arbitrary pasting diagram in which the target of a cell
can be an "arbitary"  pasting diagram. Obviously, this is a recursive
definition when made precise (as is done in [H/M/P]). In what follows, I
will talk about the *cells in a multitope  t  *. These include all cells
"involved" in t; there is one "top", or *main* cell in the multitope. The
precise definition says that a cell in a multitope is given by an arrow,
possibly the identity arrow, whose domain is the given multitope in the
category of multitopes according to [H/M/P]. The main cell is the identity
arrow. 

I now take a multitope t , and mark certain cells in it with o ;  o here
is to be read "universal". Furthermore, I mark exactly one cell, which is
of codimension 1, by #. These markings have to be done "correctly"; below
I will return to this. 

Let M be a multitopic set, that is, a Set-valued functor on the category
of multitopes. A cell of type s , for s any multitope, is an element of
M(s). I have to explain what is M[t^--], for t the correctly marked
multitope . M[t^--] is a system of cells in M, one for each cell in t ,
except the main cell in t and the one marked with # ; this system has to
be compatible in the way t shows; the two minuses in the superscript
position indicate that two places in t are "empty" (this is like a "niche"
in [B/D]). A precise way of defining M[t^--] can be given in my FOLDS
(=First Order Logic with Dependent Sorts; still unpublished and under
revision and extension; available electronically in the unrevised form);  
[t^--] is a particular *context* in the category of contexts, M is
regarded as a (lex) functor on the category of contexts, and thus M[t^--]
is just the value of that functor.

Now, take the totality of all correctly marked mutitopes. Take a
multitopic set M . A set U of cells of various dimensions and shapes is a
*system of universals for M * if the following condition is satisfied:
every time t is a correctly marked multitope, and I have a system S in
M[t^--], and the cells in S corresponding to the o-marked cells in t are
all in U , then there are cells a and u "filling in" for the one marked #
and for the main cell in t , respectively, so that u is in particular in
M(t) and in fact, u is related to S correctly; and also,  u is in  U .

This is a "horn-filling"-type condition --  but of course, the "horns" are
a lot more general than the original ones.



Also, the main part of the [B/D]-definition of "universal" (and of
"balanced") can be seen to be of the same form. But, the difference is
that the present definition is more inclusive; there are more conditions,
the notion of "correctly marked" multitope being a very general, and in
fact purely geometrical, concept. Roughly speaking, t is correctly marked
if iteratively collapsing along the arrows marked o results in the mere
cell marked # . I should emphasize (this was omitted above) that the main
cell of t is always marked o (consistent with the fact that we will
require it to be filled by something in U ).

Finally, a multitopic category (I mean multitopic omega-category) is
defined as a multitopic set M together with a system of universals for M.
When M is (essentially) finite dimensional, then having a system of
universals means that there is a canonical system of universals, the
maximal one, which is definable in the multitopic set-structure (by
"universal properties", just like in [B/D]); thus, being a multitopic
(weak) n-category for a finite n is a property of an n-dimensional
(truncated) multitopic set.


2. The multitopic category of all multitopic categories. 

I have a proposal for said entity. First, some introduction.

Already the e-mail letter (dated Nov 25, 1995) of Baez and Dolan to Ross
Street that was the announcement of their approach (although I cannot see
the relevant passage in the printed version I now have) raises the
question of what the (weak) n+1-category of all (weak) n-categories is.
Upon seeing that announcement, I thought right away that the right
definition should proceed by defining "functors", "natural
transformations", "modifications",... in a style that is similar to the
definition of "categories", in all dimensions. Notice that the
n+1-category of  n-categories must have all those things in it. 

I now, first, give the outline of the definition of the multitopic set of
all (small) multitopic categories; more precisely, I give the definition
of a particular (large) multitopic set MULTCAT whose 0-cells are the small
multitopic categories; and second, I will assert that MULTCAT has in fact
a canonical system of universals, making it a multitopic category. The
definition will look, perhaps, strange; the only way to understand it
would be through the study of examples.

Let t be a multitope. Let dim(t)=n. I define what is a correct t-coloring
of another, variable, multitope s . Let |t| be the set of the cells in t
(see above);  |s| is the set of the cells in s . Said correct coloring is
a map f:|s|-->|t| satisfying some conditions. First, I require that f
should not raise -- but it may lower! -- dimension: dim(f(x))<=dim(x). The
main, second, requirement is that "collapsing" iteratively [this of course
has to be defined precisely] along arrows z:x-->y (here both x and y are
single arrows; thus, z is a special arrow for which not only the target,
but also the source is a single arrow of dimension one less than that of z
; however, the lower dimensional source of x and y can still be
arbitrarily complex) for which all three of x,y,z are colored by the same
color c in t results in the whole or a part of t . This second requirement
can also be stated more simply. Let  I^d_t  ( I  for incidence;  d  for
domain) be the relation, written temporarily just  I , on the set  |t|  
for which  aIb  if  a  is "in" the domain of  b ;  I^c_t  has  a equal the
codomain of  b. The "correctness" condition for  f  above is:

	a(I^d_t)b  implies  [ (fa)(I^d_s)(fb)  or  fa = fb ] and
	a(I^c_t)b  implies  [ (fa)(I^c_s)(fb)  or  fa = fb ] . 


Fix t as above. A * t-colored multitopic set* M is a multitopic set with
an additional structure that assigns to each cell u in M a color k(u) , a
cell in t, in a "compatible" way. The latter means the following. Look at
the type of u : the multitope s for which u is in M(s). Now, for each x in
|s| , we have that  x  is an arrow  x:s-->r  in the category of
multitopes. Therefore, we have  M(x):M(s)-->M(r) , and we can define
 x*=M(x)(u) , and  x*  is in  M(r) , a cell in  M  of type  r . Now,
consider the coloring of  s  by  t  in which the color of any  x  is
k(x*) . The requirement is that said coloring of the multitope  s  should
be correct. 

Having defined  t-colored multitopic sets, I now define  t-colored
multitopic categories. The condition is that it should have a t-colored
system of universals. The definition of the latter is just like it was in
the "uncolored" case, except that now each "horn-filling" condition is
given by a t-colored -- and also marked--  multitope.

MULTCAT(t)  is the class of all (small) t-colored multitopic categories.
It is easy to see what the multitopic-set structure of MULTCAT should be.
As I said, in fact MULTCAT is a multitopic category, in fact, with a
canonical system of universals.

The above was put down here just to give a flavor of what is going on.
The definition becomes comprehensible only by studying its low-dimensional
cases -- and by proving expected facts about it. Some of the latter has
been done, but most of it is in progress; hopefully, if the Seminar I am
trying to bring together will in fact come together, much of the
significance of the definition will be clarified in talks in it.

3. Very briefly, I want to say that the above are to be brought together
with the "protocategory" approach, of which I briefly talked in a previous
abstract, and in lectures. This "bringing together" is an integral part of
the "validation" of the concepts involved.


With the best wishes: Michael Makkai