nLab
Segal category

Idea

The notion of Segal category is one of the models for that of (∞,1)-category. It can be understood as modelling the notion of an SSet-enrichment up to coherent homotopy , i.e. a weak enrichment.

As such it is very similar to the notion of complete Segal space.

A Segal category is a weak form of S-categories?, in which composition is only defined up to a coherent system of equivalences.

Segal categories were defined in 1974 (implicitly) by Graeme Segal. They were named Segal categories first by William Dwyer–Daniel Kan–Jeff Smith in 1989. In their famous book Homotopy invariant algebraic structures on topological spaces, John Boardman and Rainer Vogt used quasi-categories; a quasi-category is a simplicial set satisfying the weak Kan condition, so quasi-categories are also called weak Kan complexes. All of these are Quillen equivalent models of $(\mathrm{\infty }\mathrm{,1})$-categories.

Definition

A Segal category is a simplicial space? (i.e. a simplicial object in an appropriate category of spaces, such as topological spaces or Kan complexes) $X$ such that $X_0$ (the set of points) is a discrete simplicial set and the Segal map?

$${\varphi }_k:X_k\to X_1{\times }_{X_0}\cdots {\times }_{X_0}{X}_1$$\phi_k: X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1

(induced by $X({\alpha }_i):X_k\to X_1$) assigned to $X$ is a weak equivalence of simplicial sets for $k\ge 2$.

References

An overview is on pages 164 to 169 of

• Andre Joyal, The theory of Quasi-Categories and its Applications notes from a lecture at Simplicial Methods in Higher Categories

A discussion with emphasis on the comparison of the various model category structures is in

• Julia Bergner, A survey of $(\mathrm{\infty },1)$-categories (arXiv)

• see also Segal space