This chapter introduces the main object of study, the notion of M-precategory. The terminology “precategory” has been used in several different ways, notably by Janelidze [152]. The idea of the word is to invoke a structure coming prior to the full structure of a category. For us, a categorical structure means a category weakly enriched over M following Segal's method. Then a “precategory” will be a kind of simplicial object without imposing the Segal conditions. The passage from a precategory to a weakly enriched category consists of enforcing the Segal conditions using the small object argument. The basic philosophy behind this construction is that the precategory contains the necessary information for defining the category, by generators and relations. The small object argument then corresponds to the calculus whereby the generators and relations determine a category, this operation being generically denoted Seg. Splitting up things this way is motivated by the fact that simplicial objects satisfying the Segal condition are not in any obvious way closed under colimits. When we take colimits we get to arbitrary simplicial objects or precategories, which then have to generate a category by the Seg operation.

The calculus of generators and relations is themain subject of several upcoming chapters. In the present chapter, intended as a reference, we introduce the definition of precategory appropriate to our situation, and indicate the construction of some important examples which will be used later.