higher Segal space in nLab

Contents

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 d≥2d \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 1∈C(c 0,c 1),…,γ n∈C(c n−1,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 n∘…∘γ 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 2∘γ 1)=γ 3∘γ 2∘γ 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 a→b→ca \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 n∈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 T→Δ 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 0→00 \to 0 for each object of CC. A composite of in S •CS_\bullet C of two objects c,c′∈Cc,c' \in C is an object c″∈Cc'' \in C equipped with a short exact sequence 0→c→c″→c′→00 \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 do not see at n-fold complete Segal space – that is a different concept.

The Dyckerhoff-Kapranov “higher Segal spaces” above are discussed in

• Tobias Dyckerhoff, Mikhail Kapranov, Higher Segal spaces I, arxiv:1212.3563; now part of the book T. Dyckerhoff, M. Kapranov, Higher Segal spaces, Springer LNM 2244 (2019) doi

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

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

• Tashi Walde, On the theory of higher Segal spaces, thesis, Brexen 2020 pdf

• Matthew B. Young, Relative 2-Segal spaces, Algebraic & Geometric Topology 18 (2018) 975-1039 [doi:10.2140/agt.2018.18.975]

  We introduce a relative version of the 2–Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2–Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the ℛ •\mathcal{R}_\bullet-construction from Grothendieck–Witt theory. We show that a relative 2–Segal space defines a categorical representation of the Hall algebra associated to the base 2–Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2–Segal spaces.

The notion of unital 2-Segal space is also discovered independently under the name of a decomposition space in

• Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks, Decomposition spaces, incidence algebras and Möbius inversion, arXiv:1404.3202

There are many sequels including

• Joachim Kock, David I. Spivak, Decomposition-space slices are toposes, arXiv:1807.06000
• Louis Carlier, Joachim Kock, Antipodes of monoidal decomposition spaces, arXiv:1807.11858
• Matthew Feller, Richard Garner, Joachim Kock, May U. Proulx, Mark Weber, Every 2-Segal space is unital, arXiv:1905.09580
• Nicolas Behr, Joachim Kock, Tracelet Hopf algebras and decomposition spaces, arXiv:2105.06186
• Philip Hackney, Joachim Kock, Culf maps and edgewise subdivision, arXiv:2210.11191; Free decomposition spaces, arXiv:2210.11192

See also

• Jonte Gödicke. An ∞\infty-Category of 2-Segal Spaces (2024). (arXiv:2407.13357).