n-fold complete Segal space


Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations

Internal (,1)(\infty,1)-Categories



An nn-fold complete Segal space is a homotopy theory-version of an n-fold category: an nn-fold category object internal to ∞Grpd hence an n-category object in an (∞,1)-category, hence an object in Cat(Cat(Cat(Grpd)))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 :Δ opTopX^\bullet : \Delta^{op} \to Top satisfying some conditions and thought of as a model for an (,1)(\infty,1)-category.

An (,n)(\infty,n)-category is in its essence the (n1)(n-1)-fold iteration of this process: recursively, it is a category which is homotopically enriched over (,n1)(\infty,n-1)-categories.

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

X :Δ op×Δ op××Δ opTop X_\bullet : \Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op} \to Top

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

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

are homotopy pullbacks of (n1)(n-1)-fold Segal spaces.

In analogy of how it works for complete Segal spaces, the completness condition on an nn-fold complete Segal space demands that the (n1)(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 (n1n-1)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an nn-fold complete Segal space involves lots of degeneracy conditions in degree 0.




The definition originates in the thesis

  • Clark Barwick, (,n)(\infty,n)-CatCat 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which however remains unpublished.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 March 12, 2016 05:04:24 by Tim Porter (