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higher Segal space

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Higher category theory

higher category theory

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

Contents

Idea

An nn-fold Segal space is an nn-fold pre-category object in ∞Grpd. If this happens to be an actual nn-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)(n+1)-Segal space is a complete Segal space object in an (∞,1)-category of complete nn-Segal spaces.

This is discussed at n-fold complete Segal space.

Dyckerhoff-Kapranov

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 XX be a simplicial topological space or bisimplicial set or generally a simplicial object in a suitable simplicial model category.

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

Say that XX is a 2-Segal object if for all nn and all TT as above, the induced morphisms

X n:=[Δ n,X]X T:=[Δ 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 12, section 3.6).

Under some conditions DW 2-Segal spaces X X_\bullet induce Hall algebra structures on X 1X_1 (Dyckerhoff-Kapranov 12, section 8).

References

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

  • Jacob Lurie, (,2)(\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, Higher Segal spaces, talk at Steklov Mathematical Institute (2011) (video)

Revised on May 22, 2014 06:04:44 by Stupido? (87.165.99.68)