nLab
n-fold complete Segal space

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 $\mathrm{Cat}(\mathrm{Cat}(\cdots \mathrm{Cat}(\mathrm{\infty }\mathrm{Grpd})))$. 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 $(\mathrm{\infty },1)$-category.

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

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

$$X_{•}:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\times \cdots \times {\Delta }^{\mathrm{op}}\to \mathrm{Top}$$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

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

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

Definition

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Related concepts

• Theta-spaces, Segal n-categories

• higher Segal space

• semi-Segal space

References

The definition originates in the thesis

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

which however remains unpublished.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

• Clark Barwick, Chris Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories (arXiv:1112.0040, slides)

The basic idea was being popularized and put to use in

• Jacob Lurie, On the Classification of Topological Field Theories

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

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

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