Subject: categories: generalized computads

From: [email-masked] (Michael Batanin)
Date: Mon, 6 Jul 1998 16:49:05 +1000

Dear collegues,

the following  preprint

"Computads for finitary monads on globular sets"

is available at

http://www-math.mpce.mq.edu.au/~mbatanin/papers.html

>From  Introduction.

This work arose as a reflection on the foundation of higher
dimensional category theory. One of the main ingredients of any
proposed definition of weak $n$-category is the shape of diagrams
(pasting scheme) we
accept to be composable. In a globular approach \cite{Bat} each
$k$-cell
has a source and target  $(k-1)$-cell. In the opetopic approach of
Baez and Dolan \cite{BD} and the multitopic approach of Hermida,
Makkai
and Power \cite{HMP} each $k$-cell has a unique $(k-1)$-cell as
target
and a whole $(k-1)$-dimensional pasting diagram  as source.
In the theory of strict $n$-categories both source and target may be a
general pasting diagram \cite{J,StH, StP}.

 The globular approach
being the simplest one
seems too restrictive to describe the combinatorics of higher
dimensional compositions. Yet, we argue  that this is a false
impression. Moreover, we prove that this approach is a basic
one from which the other type of composable diagrams may be derived.
One  theorem proved here asserts that the category of algebras of
a finitary monad on the category of $n$-globular sets is {\bf
equivalent} to the category of algebras of an appropriate monad on
the
special category (of computads) constructed from the data of the
original monad. In the case of the monad derived from the universal
contractible operad \cite{Bat} this result may be interpreted as the
equivalence of the definitions of weak $n$-categories (in the sense
of \cite{Bat}) based on the
`globular' and general pasting diagrams. It may be also considered
 as the first step toward the proof of equivalence of the different
 definitions of weak $n$-category.

We also develop a general theory of computads and investigate some
properties of the category of generalized computads. It turned out,
that in a good situation this category is a topos (and even a presheaf
topos under some not very restrictive conditions, the property firstly
observed by S.Schanuel and reproved by A,Carboni and P.Johnstone for
$2$-computads in the sense of Street).

                      /\
                     /  \
                M --/ Co \--> MQ
                   / A C T\
                  /________\
      Centre of Australian Category Theory
  Mathematics Department, Macquarie University
        New South Wales 2109, AUSTRALIA