In the next two chapters we explore one possible definition of weak ω-category. Its formal shape is very simple: we take the category ε of globular sets and the free strict ω-category monad T on ε, construct a certain T operad L, and define a weak ω-category to be an L-algebra.

In order to see that this is a reasonable definition, and to get a feel for the concepts involved, we proceed at a leisurely pace. The present chapter is devoted to contemplation of the monad T (Section 8.1), of T -operads (= globular operads, Section 8.2), and of their algebras (Section 8.3). In Chapter 9 we define the particular globular operad L and look at its algebras – that is, at weak ω-categories. To keep the explanation in this chapter uncluttered, the discussion of the finite-dimensional case (n-categories) is also deferred to Chapter 9; we stick to ω-categories here.

Globular operads are an absolutely typical example of generalized operads.