I am announcing a paper, and enclose an (somewhat) extended abstract. The
paper is available at the site
ftp://ftp.math.mcgill.ca/pub/makkai ,
the name of the file is mltomcat.zip . It is a ZIPPED package of 8
POSTSCRIPT files. When accessed through NETSCAPE, there was no difficulty
getting it; but with ordinary ftp-ing, we couldn't get to it. The problems
with the ftp sites here at McGill are being looked at, but they are not
solved yet.
The multitopic omega-category of all multitopic omega-categories
by M. Makkai (McGill University)
September 2, 1999
Abstract
The paper gives two definitions: that of "multitopic omega-category" and
that of "the (large) multitopic set of all (small) multitopic
omega-categories". It also announces the theorem that the latter is a
multitopic omega-category. (The proof of the theorem will be contained in
a sequel to this paper.)
The work has two direct sources. One is the paper [H/M/P] (for the
references, see at the end of this abstract) in which, among others, the
concept of "multitopic set" was introduced. The other is the present
author's work on FOLDS, First Order Logic with Dependent Sorts. The
latter was reported on in [M2]. A detailed account of the work on FOLDS is
in [M3]. For the understanding of the present paper, what is contained in
[M2] suffices. In fact, section 1 of the present paper gives the
definitions of all that's needed in this paper; so, probably, there won't
be even a need to consult [M2].
The concept of multitopic set, the main contribution of [H/M/P], was, in
turn, inspired by the work of J. Baez and J. Dolan [B/D]. Multitopic sets
are a variant of opetopic sets of loc. cit. The name "multitopic set"
refers to multicategories, a concept originally due to J. Lambek [L], and
given an only moderately generalized formulation in [H/M/P]. The earlier
"opetopic set" of [B/D] is based on a concept of operad. I should say that
the exact relationship of the two concepts ("multitopic set" and "opetopic
set") is still not clarified. The main aspect in which the theory of
multitopic sets is in a more advanced state than that of opetopic sets is
that, in [H/M/P], there is an explicitly defined category Mlt of
*multitopes*, with the property that the category of multitopic sets is
equivalent to the category of Set-valued functors on Mlt, a result given a
detailed proof in [H/M/P]. The corresponding statement on opetopic sets
and opetopes is asserted in [B/D], but the category of opetopes is not
described. In this paper, the category of multitopes plays a basic role.
Multitopic sets and multitopes are described in section 2 of this paper;
for a complete treatment, the paper [H/M/P] should be consulted.
The indebtedness of the present work to the work of Baez and Dolan goes
further than that of [H/M/P]. The second ingredient of the Baez/Dolan
definition, after "opetopic set", is the concept of "universal cell". The
Baez/Dolan definition of weak n-category achieves the remarkable feat of
specifying the composition structure by universal properties taking place
in an opetopic set. In particular, a (weak) opetopic (higher-dimensional)
category is an opetopic set with additional properties ( but with no
additional data), the main one of the additional properties being the
existence of sufficiently many universal cells. This is closely analogous
to the way concepts like "elementary topos" are specified by universal
properties: in our situation, "multitopic set" plays the "role of the
base" played by "category" in the definition of "elementary topos". In
[H/M/P], no universal cells are defined, although it was mentioned that
their definition could be supplied without much difficulty by imitating
[B/D]. In this paper, the "universal (composition) structure" is supplied
by using the concept of FOLDS-equivalence of [M2].
In [M2], the concepts of "FOLDS-signature" and "FOLDS-equivalence" are
introduced. A (FOLDS-) signature is a category with certain special
properties. For a signature L , an *L-structure* is a Set-valued functor
on L. To each signature L, a particular relation between two variable
L-structures, called L-equivalence, is defined. Two L-structures M, N, are
L-equivalent iff there is a so-called L-equivalence span M<---P--->N
between them; here, the arrows are ordinary natural trasnformations,
required to satisfy a certain property called "fiberwise surjectivity".
The slogan of the work [M2], [M3] on FOLDS is that *all meaningful
properties of L-structures are invariant under L-equivalence*. As with all
slogans, it is both a normative statement ("you should not look at
properties of L-structures that are not invariant under L-equivalence"),
and a statement of fact, namely that the "interesting" properties of
L-structures are in fact invariant under L-equivalence. (For some slogans,
the "statement of fact" may be false.) The usual concepts of "equivalence"
in category theory, including the higher dimensional ones such as
"biequivalence", are special cases of L-equivalence, upon suitable, and
natural, choices of the signature L; [M3] works out several examples of
this. Thus, in these cases, the slogan above becomes a tenet widely held
true by category theorists. I claim its validity in the generality stated
above.
The main effort in [M3] goes into specifying a language, First Order Logic
with Dependent Sorts, and showing that the first order properties
invariant under L-equivalence are precisely the ones that can be defined
in FOLDS. In this paper, the language of FOLDS plays no role. The concepts
of "FOLDS-signature" and "FOLDS-equivalence" are fully described in
section 1 of this paper.
The definition of *multitopic omega-category* goes, in outline, as
follows. For an arbitrary multitope SIGMA of dimension >=2, for a
multitopic set S, for a pasting diagram ALPHA in S of shape the domain of
SIGMA and a cell a in S of the shape the codomain of SIGMA, such that a
and ALPHA are parallel, we define what it means to say that a is a
*composite* of ALPHA. First, we define an auxiliary FOLDS signature
L<SIGMA> extending Mlt, the signature of multitopic sets. Next, we define
structures S<a> and S<ALPHA>, both of the signature L<SIGMA>, the first
constructed from the data S and a , the second from S and ALPHA, both
structures extending S itself. We say that a is a composite of ALPHA if
there is a FOLDS-equivalence-span E between S<a> and S<ALPHA> that
restricts to the identity equivalence-span from S to S . Below, I'll refer
to E as an *equipment* for a being a composite of ALPHA. A multitopic
set is a *mulitopic omega-category* iff every pasting diagram ALPHA in it
has at least one composite.
The analog of the universal arrows in the Baez/Dolan style definition is
as follows. A *universal arrow* is defined to be an arrow of the form
b:ALPHA-----> a where a is a composite of ALPHA via an equipment E that
relates b with the identity arrow on a : in turn, the identity arrow on
a is any composite of the empty pasting diagram of dimension dim(a)+1
based on a . Note that the main definition does *not* go through first
defining "universal arrow".
A new feature in the present treatment is that it aims directly at weak
*omega*-categories; the finite dimensional ones are obtained as truncated
versions of the full concept. The treatment in [B/D] concerns finite
dimensional weak categories. It is important to emphasize that a
multitopic omega category is still just a multitopic set with additional
properties, but with no extra data.
The definition of "multitopic omega-category" is given is section 5; it
uses sections 1, 2 and 4, but not section 3.
The second main thing done in this paper is the definition of MltOmegaCat.
This is a particular large multitopic set. Its definition is completed
only by the end of the paper. The 0-cells of MltOmegaCat are the samll
multitopic omega-categories, defined in section 5. Its 1-cells, which we
call 1-transfors (thereby borrowing, and altering the meaning of, a term
used by Sjoerd Crans [Cr]) are what stand for "morphisms", or "functors",
of multitopic omega-categories. For instance, in the 2-dimensional case,
multitopic 2-categories correspond to ordinary bicategories by a certain
process of "cleavage", and the 1-transfors correspond to homomorphisms of
bicategories [Be]. There are n-dimensional transfors for each n in N . For
each multitope (that is, "shape" of a higher dimensional cell) PI, we
have the *PI-transfors*, the cells of shape PI in MltOmegaCat.
For each fixed multitope PI, a PI-transfor is a *PI-colored multitopic
set* with additional properties. "PI-colored multitopic sets" are defined
in section 3; when PI is the unique zero-dimensional multitope, PI-colored
multitopic sets are the same as ordinary multitopic sets. Thus, the
definition of a transfor of an arbitrary dimension and shape is a
generalization of that of "multitopic omega-category"; the additional
properties are also similar, they being defined by FOLDS-based universal
properties. There is one new element though. For dim(PI)>=2 , the concept
of PI-transfor involves a universal property which is an
omega-dimensional, FOLDS-style generalization of the concept of right
Kan-extension (right lifting in the terminology used by Ross Street).
This is a "right-adjoint" type universal property, in contrast to the
"left-adjoint" type involved in the concept of composite (which is a
generalization of the usual tensor product in modules).
The main theorem, stated but not proved here, is that MltOmegaCat is a
multitopic omega-category.
The material in this paper has been applied to give formulations of
omega-dimensional versions of various concepts of homotopy theory;
details will appear elesewhere.
I thank Victor Harnik and Marek Zawadowski for many stimulating
discussions and helpful suggestions. I thank the members of the Montreal
Category Seminar for their interest in the subject of this paper, which
made the exposition of the material at a time when it was still in an
unfinished state a very enjoyable and useful process for me.
References:
[B/D] J. C. Baez and J. Dolan, Higher-dimensional algebra III.
n-categories and the algebra of opetopes. Advances in Mathematics 135
(1998), 145-206.
[Be] J. Benabou, Introduction to bicategories. In: Lecture Notes in
Mathematics 47 (1967), 1-77 (Springer-Verlag).
[Cr] S. Crans, Localizations of transfors. Macquarie Mathematics
Reports no. 98/237.
[H/M/P] C. Hermida, M. Makkai and J. Power, On weak higher dimensional
categories I. Accepted by: Journal of Pure and Applied Algebra. Available
electronically (when the machines work ...).
[L] J. Lambek, Deductive systems and categories II. Lecture Notes in
Mathematics 86 (1969), 76-122 (Springer-Verlag).
[M2] M. Makkai, Towards a categorical foundation of mathematics. In:
Logic Colloquium '95 (J. A. Makowski and E. V. Ravve, editors). Lecture
Notes in Logic 11 (1998) (Springer-Verlag).
[M3] M. Makkai, First Order Logic with Dependent Sorts. Research
momograph, accepted by Lecture Notes in Logic (Springer-Verlag). Under
revision. Original form available electronically (when the machines
work ...).
Cheers: M. Makkai
