n-fold complete Segal space

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/ω-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

**category object in an (∞,1)-category**, groupoid object

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.

(…)

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

- 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

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

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

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

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