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.

Definition

Definition

A Segal category is

• a simplicial simplicial set (bisimplicial set) $X$

• such that $X_0$ (the set of points) is a discrete (= constant) simplicial set

• the morphisms

  $${\varphi }_k:X_k\to X_1{\times }_{X_0}\cdots {\times }_{X_0}{X}_1(k\mathrm{factors})$$\phi_k: X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1 \;\;(k factors)

  induced by the face maps $X_{\{0,\cdots ,k\}}\to X_{\{i,i+1\}}$ for $0\le i\le k-1$ are weak equivalences of simplicial sets for $k\ge 2$.

Remark

The object $X_1{\times }_{X_0}{X}_1$ has the interpretation as the space of composable 1-morphisms in $X$. The weak equivalence $X_2\to X_1{\times }_{X_0}{X}_1$ given by the above definition together with the remaining face map ${\delta }^1:X_2\to X_1$ provide an ∞-anafunctor

$$\begin{array}{ccc}{X}_2& \stackrel{\delta 1}{\to }& X_1\\ {}^{\simeq }\downarrow \\ X_1{\times }_{X_0}{X}_1\end{array}$$\array{ X_2 &\stackrel{\delta1}{\to}& X_1 \\ {}^{\mathllap{\simeq}}\downarrow \\ X_1 \times_{X_0} X_1 }

that encodes the composition operation in the Segal category $X$.

References

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.

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)