category object in an (∞,1)-category, groupoid object
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.
There are several ways to think about $d$-Segal spaces:
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.
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.
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
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).
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.
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
The notion of unital 2-Segal space is also discovered independently under the name of a decomposition space in
There are many sequels including
Last revised on November 11, 2022 at 08:59:42. See the history of this page for a list of all contributions to it.