higher Segal space
Higher category theory
higher category theory
Extra properties and structure
An -fold Segal space is an -fold pre-category object in ∞Grpd. If this happens to be an actual -fold category object it is an n-fold complete Segal space.
In (Lurie, section 1.3) a recursive definition is given: a (complete) -Segal space is a complete Segal space object in an (∞,1)-category of complete -Segal spaces.
This is discussed at n-fold complete Segal space.
In (DyckerhoffKapranov 12) a 2-Segal space is defined to be a simplicial space with a higher analog of the weak composition operation known from Segal spaces.
Let be a simplicial topological space or bisimplicial set or generally a simplicial object in a suitable simplicial model category.
For let be the -polygon. For any triangulation of let be the corresponding simplicial set. Regarding as the cellular boundary of that polygon provides a morphism of simplicial sets .
Say that is a 2-Segal object if for all and all as above, the induced morphisms
are weak equivalences.
Warning. A Dyckerhoff-Kapranov “2-Segal spaces” is not itself a model for an (∞,2)-category. Instead, it is a model for an (∞,1)-operad (Dyckerhoff-Kapranov 12, section 3.6).
Under some conditions DW 2-Segal spaces induce Hall algebra structures on (Dyckerhoff-Kapranov 12, section 8).
The notion of higher Segal space as a model for (∞,n)-categories is discussed in
- Jacob Lurie, -categories and the Goodwillie calculus I (pdf)
For more references along these lines see at n-fold complete Segal space
The Dyckerhoff-Kapranov “higher Segal spaces” above are discussed in
- Tobias Dyckerhoff, Higher Segal spaces, talk at Steklov Mathematical Institute (2011) (video)
Revised on March 12, 2016 05:02:02
by Tim Porter