The observation that the theory of strict n-groupoids is not enough to give a good model for homotopy n-types, led Grothendieck to ask for a theory of n-categories with weakly associative composition. This will be the main subject of our book, in particular we use the terminology n-category to mean some kind of object in a possible theory with weak associativity, or even a composition which is only defined up to homotopy, or perhaps some other type of weakening (as will be briefly discussed in Chapter 4).

There are a certain number of basic elements expected of any theory of n-categories, and which can be explained without refering to a full definition. It will be useful to start by considering these. Our discussion follows Tamsamani's paper [250], but really sums up the general expectations for a theory of n-categories which were developed over many years starting with Bénabou and continuing through the theory of strict n-categories and Grothendieck's manuscript.

For this chapter, we will use the terminology “n-category” to mean any object in a generic theory of n-categories. We will sometimes use the idea that our generic theory should admit cartesian products and disjoint sums.

A globular theory

We saw that a strict n-category has, in particular, an underlying globular set. This basic structure should be conserved, in some form, in any weak theory.

(OB)–An n-category A should have an underlying set of objects denoted Ob(A).