In this chapter, we consider one of the main examples of the theory: when M = K is the Kan–Quillen model category of simplicial sets. The notion of weakly M-enriched category then becomes the notion of Segal category, one version of the (∞, 1)-categories which are ubiquitous in applications of higher category theory. The first section is devoted to a review of the basic definitions and notations in this important case. Then, we take the opportunity to illustrate the general calculus of generators and relations as it applies to the problem of creating the loop space of a space X, together with its Segal delooping structure. This collection of structure is the Poincaré–Segal groupoid of X, a Segal category analogous to the classical Poincaré groupoid. The word “groupoid” in the terminology indicates that even the 1-morphisms are invertible up to equivalence. Keeping the higher categorical structure allows this object to reflect the full weak homotopy type of X, indeed that is what Segal's basic theorem says, as will be reviewed in the second section below. As will be pointed out in the third section, the Poincaré–Segal groupoid has a fairly simple abstract description generalizing the space of loops on X. However, explicit calculation of the infinite-dimensional space of loops is not easy, and the subsequent sections are devoted to showing how the calculus of generators and relations, developed in preceding chapters, allows us to view the loop space as something deduced algebraically from X.