# nLab n-fold complete Segal space

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

An $n$-fold complete Segal space is a homotopy theory-version of an n-fold category: an $n$-fold category object internal to ∞Grpd hence an n-category object in an (∞,1)-category, hence an object in $\mathrm{Cat}\left(\mathrm{Cat}\left(\cdots \mathrm{Cat}\left(\infty \mathrm{Grpd}\right)\right)\right)$. This is a model for an (∞,n)-category.

A complete Segal space is to be thought of as the nerve of a category which is homotopically enriched over Top: it is a simplicial object in Top, ${X}^{•}:{\Delta }^{\mathrm{op}}\to \mathrm{Top}$ satisfying some conditions and thought of as a model for an $\left(\infty ,1\right)$-category.

An $\left(\infty ,n\right)$-category is in its essence the $\left(n-1\right)$-fold iteration of this process: recursively, it is a category which is homotopically enriched over $\left(\infty ,n-1\right)$-categories.

This implies then in particular that an $\left(\infty ,n\right)$-category in this sense is an $n$-fold simplicial topological space

${X}_{•}:{\Delta }^{\mathrm{op}}×{\Delta }^{\mathrm{op}}×\cdots ×{\Delta }^{\mathrm{op}}\to \mathrm{Top}$X_\bullet : \Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op} \to Top

which satisfies the condition of Segal spaces (characterizing nerves of categories, recall) in each variable, in that all the squares

$\begin{array}{ccc}{X}_{m+n,•}& \to & {X}_{n,•}\\ ↓& & ↓\\ {X}_{m,•}& \to & {X}_{0,•}\end{array}$\array{ X_{m+n,\bullet} &\to& X_{n,\bullet} \\ \downarrow && \downarrow \\ X_{m,\bullet} &\to& X_{0,\bullet} }

are homotopy pullbacks of $\left(n-1\right)$-fold Segal spaces.

(…)

## References

The definition originates in the thesis

• Clark Barwick, $\left(\infty ,n\right)$-$\mathrm{Cat}$ as a closed model category PhD (2005)

which however remains unpublished. It appears in print in section 12 of

The basic idea was being popularized and put to use in

A detailed discussion in the general context of internal categories in an (∞,1)-category is in section 1 of

• Jacob Lurie, $\left(\infty ,2\right)$-Categories and the Goodwillie Calculus I (arXiv:0905.0462)

For related references see at (∞,n)-category .

Revised on November 23, 2012 13:00:14 by Urs Schreiber (82.169.65.155)