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