nLab
complete Segal space

Idea

Complete Segal spaces are one model for $(\mathrm{\infty }\mathrm{,1})$-categories.

The rough idea is that a complete Segal space is the nerve of a category enriched weakly over Top: it is not a simplicial set, but a simplicial topological space which satisfies the homotopy theoretic analog of the condition that otherwise implies that a simplicial set is the nerve of a category.

A bit more precisely: to determine if a simplicial set $X_{•}$ arises from a category by passing to its nerve one has to check if for all natural numbers $m,n$ the square

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

is a pullback square (where the maps $p_{\cdots }^{*}$ project out the indicated parts of the objects of the simplex category $\Delta$ regarded as posets). $X^{•}$ is the nerve of a category precisely if this is the case for all $n,m$.

This condition is internalized homotopically in the category of spaces to get the definition of a Segal space. One can interpret “spaces” here as meaning either (sufficiently nice) topological spaces or simplicial sets; in the latter case a Segal space is a particular sort of bisimplicial set.

Definition

A Segal space $X_{•}$ is a simplicial space $X_{•}:{\Delta }^{\mathrm{op}}\to \mathrm{Top}$ 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 \\ {}^{p^*_{m, \cdots, m+n}}\downarrow && \downarrow^{p^*_m} \\ X_n &\stackrel{p^*_0}{\to}& X_0 }

is a homotopy pullback square.

Next, the idea is that, roughly, a Segal space $X_{•}$ is a complete Segal space if the the fundamental ∞-groupoid of $X_0$ is the maximal topological groupoid contained in the topologically enriched category of which $X_{•}$ is like the nerve of.

More precisely…

Remarks

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

References

Complete Segal spaces were originally defined by Charles Rezk.

Information is for instance on pages 29 to 31 of

• Jacob Lurie, On the classification of TFTs (pdf)

or section 4 of

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