homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
category object in an (∞,1)-category, groupoid object
-fold complete Segal spaces are a model for (∞,n)-categories, i.e. the homotopical version of n-categories.
We can view strict n-categories as n-fold categories where part of the structure is trivial; for example, strict 2-categories can be described as double categories where the only vertical morphisms are identities. -fold Segal spaces similarly result from viewing -categories as a special class of -fold internal -categories in ∞-groupoids.
If is an -category, then -categories internal to can be defined as certain simplicial objects in (namely those satisfying the “Segal condition”). Thus -fold internal -categories in -groupoids correspond to a class of -simplicial -groupoids, and -fold Segal spaces are defined by additionally specifying certain constancy conditions.
To describe the correct homotopy theory of -categories we also want to regard the fully faithful and essentially surjective morphisms between -fold Segal spaces as equivalences. It turns out that, just as in the case of Segal spaces, the localization at these maps can be accomplished by restricting to a full subcategory of complete objects.
If is an -category with pullbacks, we say that a simplicial object satisfies the Segal condition if the squares
are all pullbacks. Such Segal objects give the -categorical version of internal categories as algebraic structures. (I.e. we have not inverted a class of fully faithful and essentially surjective morphisms.)
If denotes the full subcategory of spanned by the Segal objects, then this is again an -category with pullbacks, so we can iterated the definition to obtain a full subcategory of of Segal -objects in .
We can now inductively define -fold Segal objects by imposing constancy conditions: An -fold Segal object in is a Segal -object such that
When is the -category of spaces (or -groupoids) we refer to -fold Segal objects as -fold Segal spaces.
We now define fully faithful and essentially surjective morphisms between -fold Segal inductively in terms of the corresponding notions for Segal spaces:
A morphism between -fold Segal spaces is fully faithful and essentially surjective if:
An -fold Segal space is complete if:
There are several equivalent ways to reformulate these inductive definitions. For example, a morphism is fully faithful and essentially surjective if and only if:
The complete -fold Segal spaces are precisely the -fold Segal spaces that are local with respect to the fully faithful and essentially surjective morphisms. Thus the localization of the -category of -fold Segal spaces at this class of morphisms is equivalent to the full subcategory of complete -fold Segal spaces.
This was first proved in Barwick’s thesis, generalizing Rezk’s proof in the case . Later, Lurie extended the notion of complete Segal objects to more general contexts than spaces, which allows an inductive definition of complete -fold Segal spaces as complete Segal objects in complete -fold Segal spaces. The theorem for -fold Segal spaces then follows by inductively applying the generalization of Rezk’s theorem (for the case ) to this setting.
(…)
The definition originates in the thesis
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
A Quillen adjunction relating -complicial sets to -fold complete Segal spaces:
For related references see at (∞,n)-category .
Last revised on May 2, 2023 at 05:48:46. See the history of this page for a list of all contributions to it.