Segal categories can be seen as an intermediate model between simplicial categories and complete Segal spaces. Like complete Segal spaces, they are simplicial spaces with a notion of up-to-homotopy composition encoded by a Segal condition, but rather than imposing the completeness condition, we simply ask that the space in degree zero be discrete. Thus, Segal categories more closely resemble simplicial nerves of simplicial categories. Because this discreteness assumption is simpler than completeness, as objects Segal categories are more straightforward to define than complete Segal spaces. However, asking that a given simplicial set be discrete (rather than homotopy discrete) is awkward from a homotopical point of view, so doing homotopy theory with these structures is substantially more difficult. In particular, neither of the two model structures for Segal categories can be obtained as a localization of a model category with levelwise weak equivalences.

Segal categories first appeared in a paper of Dwyer, Kan, and Smith [58], although not under that name. They were developed extensively by Hirschowitz and Simpson [70], and their homotopy theory was investigated by Pellissier [97]. For the purposes of establishing Quillen equivalences, we consider two different model structures on the category of Segal precategories; SeCatc resembles an injective model structure, that its cofibrations are precisely the monomorphisms, and is more easily compared to CSS, whereas SeCat f more closely resembles a projective model structure and is more easily compared to SC. The notation is further justified by the fact that the fibrant objects in these model categories are Segal categories which are Reedy fibrant and projective fibrant, respectively, as simplicial spaces. However, these two model structures have the same weak equivalences, and they are Quillen equivalent to one another.

Basic Definitions and Constructions

As in the case of complete Segal spaces, the category of Segal categories alone does not have the necessary categorical properties to have a model structure. Thus, we need to work in a larger category to get a model structure for Segal categories. However, here we work in a more restricted category than that of all simplicial spaces.