nLab
Segal space

Idea

A Segal space $X_{•}$ is a simplicial topological space or bisimplicial set $X_{•}:{\Delta }^{\mathrm{op}}\to \mathrm{Top}$ for which for all $m,n\in \mathbb{N}$ the square

$$\begin{array}{ccc}{X}_{m+n}& \stackrel{p_{0,\cdots ,m}^{*}}{\to }& X_m\\ {}^{p_{m,\cdots ,m+n}^{*}}\downarrow & & {\downarrow }^{p_m^{*}}\\ X_n& \stackrel{p_0^{*}}{\to }& X_0\end{array}$$\array{ X_{m+n} &\stackrel{ p^*_{0,\cdots, m} }{\to}& X_m \\ {}^{\mathllap{p^*_{m, \cdots, m+n}}}\downarrow && \downarrow^{\mathrlap{p^*_m}} \\ X_n &\stackrel{p^*_0}{\to}& X_0 }

is a homotopy pullback square.

As a model for higher categories

One definition of $(\mathrm{\infty },n)$-category is in terms of complete Segal spaces.