Greetings categorists! A PDF version of my thesis entitled "Symmetric Operads for Globular Sets" is now available at http://www.maths.mq.edu.au/~mweber/ -------------------------------------------------------------------------------------------------- ABSTRACT This thesis considers various issues related to the adaptation of Batanin's higher operads to the problem of describing $k$-fold monoidal weak $n$-categories and related structures. These higher operads have been found to be successful in providing descriptions of the monads on $\Glob$, the category of globular sets, whose algebras are structures like strict and weak $\omega$-categories. One of the features of the endofunctors and monads on $\Glob$ that arise in this theory, is that they bear a formal resemblance to power series. The first contribution of this thesis is to view this phenomenon as an instance of a general notion of power series. This notion is based on a presheaf category together with a chosen full subcategory, whose objects turn out to be the exponents of the resulting power series. Other examples of this notion include the usual notions of power series which correspond to certain endofunctors of $\Set$. By definition, $k$-fold monoidal weak $n$-categories are weak $(n+k)$-categories which have exactly one $r$-cell for $r < k$. The second contribution of this thesis is the description and analysis of a simple construction called \emph{suspension}, which takes a finitary monad $T$ on $\Glob$, and produces another finitary monad whose algebras are one-$0$-cell $T$-algebras. Applying this construction $k$ times to $T$ produces the monad whose algebras are ``$k$-fold monoidal $T$-algebras''. Batanin's higher operads are describable as cartesian monad morphisms into the monad $\TREE$, whose algebras are strict $\omega$-categories. Such a description is possible since $\TREE$ satisfies a certain property: it is a club in the sense of Kelly. However, applying suspension at least twice to Batanin operads gives rise naturally to a monad morphism into a monad which is not a club. This situation has more in common with \emph{symmetric} operads in $\Set$. The third contribution of this thesis is to provide a common generalisation of symmetric operads in $\Set$ and Batanin's operads. This new operad notion also includes the clubs originally considered by Kelly in the early 1970's for the study of categories with structure and their coherence. -------------------------------------------------------------------------------------------------- Regards, Mark Weber