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*To*: categories@mta.ca*From*: Michael MAKKAI <makkai@math.mcgill.ca>*Date*: Wed, 15 Sep 1999 19:02:00 -0400*Sender*: cat-dist@mta.ca

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 ,http://mystic.biomed.mcgill.ca/M_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

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