nLab higher Segal space

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

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

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

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

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

Idea

There are several ways to think about $d$-Segal spaces:

Higher associativity parameterized by polyhedra

A $1$-Segal space $C$ 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 $I$ into subintervals $I_1,\dots,I_n$, and for any choice of labelings of the endpoints of these intervals by objects $c_0,\dots,c_n$, and any choice of labelings $\gamma_1 \in C(c_0,c_1),\dots,\gamma_n \in C(c_{n-1},c_n)$ of the intervals $I_1, \dots, I_n$, the Segal condition provides us with a “composite” labeling $\gamma_n \circ \dots \circ \gamma_1$ of the whole interval $I$, in a coherent way. “Coherence” here means that the composition is continuous in the $\gamma_i$‘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: $\gamma_3 \circ (\gamma_2 \circ \gamma_1) = \gamma_3 \circ \gamma_2 \circ \gamma_1$.

A $2$-Segal space is, like a $1$-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 $d$-Segal spaces are simplicial spaces with higher associativity data parameterized by triangulations of $d$-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 $a \to b \to c$ 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 $X$ 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

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

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 $X_\bullet$ induce Hall algebra structures on $X_1$ (Dyckerhoff-Kapranov 12, section 8).

Examples

A central motivating example comes from $K$-theory. If $C$ is a Quillen-exact category, then $S_\bullet C$ is a 2-Segal space. Here $S_\bullet$ is the Waldhausen S-construction. There is one object of $S_\bullet C$, denoted $0$. There is a morphism $0 \to 0$ for each object of $C$. A composite of in $S_\bullet C$ of two objects $c,c' \in C$ is an object $c'' \in C$ equipped with a short exact sequence $0 \to c \to c'' \to c' \to 0$. 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 $\mathcal{R}_\bullet$-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).