nLab
higher Segal space

Idea

An $n$-fold Segal space is an $n$-fold pre-category object in ∞Grpd. If this happens to be an actual $n$-fold category object it is an n-fold complete Segal space.

Definition

Lurie

In (Lurie, section 1.3) a recursive definition is given: a (complete) $(n+1)$-Segal space is a complete Segal space object in an (∞,1)-category of complete $n$-Segal spaces.

This is discussed at n-fold complete Segal space.

Dyckerhoff-Kapranov

In (DyckerhoffKapranov) 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 $X$ be a simplicial topological space or bisimplicial set or generally a simplicial object in a suitable simplicial model category.

For $n\in \mathbb{N}$ let $P_n$ be the $n$-polygon. For any triangulation $T$ of $P_n$ let ${\Delta }^T$ be the corresponding simplicial set. Regarding ${\Delta }^n$ as the cellular boundary of that polygon provides a morphism of simplicial sets ${\Delta }^T\to {\Delta }^n$.

Say that $X$ is a 2-Segal object if for all $n$ and all $T$ as above, the induced morphisms

$$X_n:=[{\Delta }^n,X]\to X_T:=[{\Delta }^T,X]$$X_n := [\Delta^n, X] \to X_T := [\Delta^T,X]

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)-operads (Dyckerhoff-Kapranov, section 3.6).

Under some conditions DW 2-Segal spaces $X_{•}$ induce Hall algebra structures on $X_1$ (Dyckerhoff-Kapranov, section 8).

References

The notion of higher Segal space as a modl for (∞,n)-categories is discussed in

• Jacob Lurie, $(\mathrm{\infty },2)$-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, Mikhail Kapranov, Higher Segal spaces I, (arxiv:1212.3563)

• Tobias Dyckerhoff, Higher Segal spaces, talk at Steklov Mathematical Institute (2011) (video)

• Mikhail Kapranov, Higher Segal spaces, talk at IHES (2012) (video)