This chapter continues the study of weakly enriched categories using Segal's method. We use the model category for algebraic theories, developed in the previous chapter, to get model structures for Segal precategories on a fixed set of objects. This structure will be studied in detail later, to deal with the passage from a Segal precategory to the Segal category it generates.

Then we consider the full category of Segal precategories, with movable sets of objects, giving various definitions and notations. Constructing a model structure in that case is the main subject of the subsequent chapters.

The reader will note that this division of the global argument into two pieces, was present already in Dwyer–Kan's treatment of the model category for simplicial categories. They discussed the model category for simplicial categories on a fixed set of objects in a series of papers (Dwyer and Kan [101, 102, 103]); but it wasn't until some time later in their unpublished manuscript with Hirschhorn [99], which subsequently became Dwyer et al. [100], and then Bergner's paper [39] that the global case was treated.

For the theory of weak enrichment following Segal's method, the corresponding division and introduction of the notion of left Bousfield localization for the first part was suggested in Barwick's thesis [16].

Assume throughout that M is a tractable left proper cartesian model category. See Section 7.7 for an explanation and first consequences of the cartesian condition.