Subject: weak \omega-categories

From: categories <[email-masked]>
Date: Mon, 28 Apr 1997 07:28:01 -0300 (ADT)

Date: Fri, 25 Apr 1997 14:56:43 +1100
From: Olga Batanin <[email-masked]>

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.