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higher Segal space

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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

Contents

Warning

There are several unrelated generalizations of the concept of a Segal space which might be thought of as “higher Segal spaces”. For example, one might discuss

  • n n -fold Segal spaces?, a model for (,n)(\infty,n)-categories.

  • n n -uple Segal spaces?, a model for cubical (,n)(\infty,n)-categories.

  • d d -Segal spaces? in the sense of Dyckerhoff and Kapranov, a model for something like an (,1)(\infty,1)-category, but without uniqueness of composites (for d2d \geq 2) and with higher associativity only in dimension dd and above.

This article discusses dd-Segal spaces in the sense of Dyckerhoff and Kapranov.

Idea

There are several ways to think about dd-Segal spaces:

Higher associativity parameterized by polyhedra

A 11-Segal space CC is a Segal space, i.e. a simplicial space satisfying the Segal condition. We think of the Segal condition in the following way. For every subdivision of an interval II into subintervals I 1,,I nI_1,\dots,I_n, and for any choice of labelings of the endpoints of these intervals by objects c 0,,c nc_0,\dots,c_n, and any choice of labelings γ 1C(c 0,c 1),,γ nC(c n1,c n)\gamma_1 \in C(c_0,c_1),\dots,\gamma_n \in C(c_{n-1},c_n) of the intervals I 1,,I nI_1, \dots, I_n, the Segal condition provides us with a “composite” labeling γ nγ 1\gamma_n \circ \dots \circ \gamma_1 of the whole interval II, in a coherent way. “Coherence” here means that the composition is continuous in the γ i\gamma_i‘s, but moreover that it is associative: if we compose our labelings in two steps, for example, we get the same result as if we compose our labelings in one step: γ 3(γ 2γ 1)=γ 3γ 2γ 1\gamma_3 \circ (\gamma_2 \circ \gamma_1) = \gamma_3 \circ \gamma_2 \circ \gamma_1.

A 22-Segal space is, like a 11-Segal space, a simplicial space, but it satisfies only a weakened version of the Segal condition. Instead of stipulating that labelings of triangualtions of 1-dimensional intervals may be coherently composed, we stipulate that labelings of triangulations of 2-dimensional polygons may be coherently composed.

Similarly dd-Segal spaces are simplicial spaces with higher associativity data parameterized by triangulations of dd-dimensional polyhedra.

Categories with multivalued composition

A 2-Segal space is a “category with multivalued composition”, or a category enriched in Span. A composite of two morphisms abca \to b \to c need not exist, and if it does it may not be unique. But whatever composites there are satisfy all “higher associativity conditions” one could want.

Definition

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)-operad (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).

Examples

A central motivating example comes from KK-theory. If CC is a Quillen-exact category?, then S CS_\bullet C is a 2-Segal space. Here S S_\bullet is the Waldhausen S-construction. There is one object of S CS_\bullet C, denoted 00. There is a morphism 000 \to 0 for each object of CC. A composite of in S CS_\bullet C of two objects c,cCc,c' \in C is an object cCc'' \in C equipped with a short exact sequence 0ccc00 \to c \to c'' \to c' \to 0. Thus the composite is generally not unique, but it does satisfy all the higher associativity conditions required of a 2-Segal space.

References

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)

Last revised on September 13, 2018 at 15:50:04. See the history of this page for a list of all contributions to it.