From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/368 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: weak \omega-categories Date: Mon, 28 Apr 1997 07:28:01 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016918 25408 80.91.229.2 (29 Apr 2009 14:55:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:55:18 +0000 (UTC) To: categories Original-X-From: cat-dist Mon Apr 28 07:29:28 1997 Original-Received: by mailserv.mta.ca; id AA09760; Mon, 28 Apr 1997 07:28:01 -0300 Original-Lines: 92 Xref: news.gmane.org gmane.science.mathematics.categories:368 Archived-At: Date: Fri, 25 Apr 1997 14:56:43 +1100 From: Olga Batanin The preprint version of my paper "Monoidal globular categories as a natural environment for the theory of weak n-categories" is now available. The dvi file is at http://www-math.mpce.mq.edu.au/~mbatanin/coh0.dvi Please, contact me if you have any difficulties with printing it out. I can mail hard copies. Michael Batanin. Abstract. The paper is devoted to the problem of defining weak $\omega$-categories. The definition presented here is based on a nontrivial generalization of the apparatus of operads and their algebras, originally developed by P.May \cite{May} for the needs of algebraic topology. Yet, for the purposes of higher order category theory, a higher dimensional notion of operad is required. Briefly, the idea of a higher operad may be explained as follows. An ordinary non-symmetric operad in $Set$ associates a set $A_{n}$ to every integer $n$. The set of integers may be interpreted as the set of $1$-cells in the free category generated by one object and one nonidentity endomorphism of this object. To find a higher order generalization of the notion of operad we have to describe the free strict $\omega$-category generated by one object and one nonidentity endomorphism of this object and one nonidentity endomorphism of this endomorphism and so on (so, for example, the set of integers is the one-dimensional part of this category). The required $\omega$-category $Tr$ will be the category of planar trees of a special type. The $k$-th composition of cells will be given by the colimit of the diagram of trees over a special tree $M_{n}^{k}$. The other component of the theory of operads is an appropriate monoidal category (with some extrastructure like braiding or symmetry) where one can consider the notion of operad. We need also a monoidal category (perhaps, with extrastructure as well) where one can define the notion of algebra for an operad. Finally, the corresponding coherence theorems for both types of monoidal categories are required. I call all these components a natural environment for a given theory of operads. One of my main goals was to find a natural environment for the theory of higher order operads. For this I introduce monoidal globular categories and show they are suitable for the development of the theory of higher order operads. The crucial point here is a coherence theorem for monoidal globular categories (section 4) which includes as special cases the coherence theorems for monoidal, symmetric monoidal, and braided monoidal categories and a sort of pasting theorem for $\omega$-categories. A primary example of a globular monoidal category is the globular category of $n$-spans $Span$. The $0$-spans are just the sets. The $1$-spans are the spans in $Set$ in the usual sense. In some informal sense, an $n$-span is a relation between two $(n-1)$-spans. This globular monoidal category plays the same role for higher-order category theory as the category of sets does for ordinary category theory. These results allow me to formulate the notion of higher order operad. An $\omega$-operad will associate an $n$-span to every $n$-cell in $Tr$ for every $n\ge 0$.} There are also the units and multiplications and some axioms for these operations. The category of non-symmetric operads (in the category of sets) is just a one-dimensional subcategory of the category of $\omega$-operads. Finally, Iintroduce a notion of a contractible $\omega$-operad, So the main definition is: A weak $\omega$-category is a globular set together with the structure of algebra over a universal contractible $\omega$-operad. I construct also a fundamental $n$-groupoid functor from topological spaces to the category of weak $n$categories for all $n$ including $\omega$ and consider another examples of weak $n$-categories, hifger operads and their algebras.