# 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 $Cat(Cat(\cdots Cat(\infty Grpd)))$. 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, $X^\bullet : \Delta^{op} \to Top$ satisfying some conditions and thought of as a model for an $(\infty,1)$-category.

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

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

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

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

$\array{ X_{m+n,\bullet} &\to& X_{n,\bullet} \\ \downarrow && \downarrow \\ X_{m,\bullet} &\to& X_{0,\bullet} }$

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

In analogy of how it works for complete Segal spaces, the completness condition on an $n$-fold complete Segal space demands that the $(n-1)$-fold complete Segal space in degree zero is (under suitable identifications) the infinity-groupoid which is the core of the (infinity,n)-category which is being presented. Since the embedding of $\infty$-groupoids into ($n-1$)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an $n$-fold complete Segal space involves lots of degeneracy conditions in degree 0.

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## References

The definition originates in the thesis

• Clark Barwick, $(\infty,n)$-$Cat$ 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which however remains unpublished.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

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

Revised on February 20, 2014 04:20:58 by Urs Schreiber (77.80.20.34)