The edgewise subdivision criterion for 2-Segal objects

Abstract

We show that the edgewise subdivision of a 2-Segal object is always a Segal object, and furthermore that this property characterizes 2-Segal objects.

Decomposition-space slices are toposes

“…Remark 2.2. The object part of lemma 2.1 has been observed also by Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer [1], who furthermore establish a partial converse: they show (in the more general setting of simplicial objects in a combinatorial model category) that sd( ) is Segal if and only if is 2-Segal [4] (i.e. a "non-unital decomposition space").…”

Quasi-2-Segal sets

“…In Section 4, we prove for quasi-2-Segal sets a version of the path space criterion for 2-Segal spaces. In Section 5, we prove some technical results about pushout-products and pushout-joins of 2-Segal horns which yield the characterization of quasi-2-Segal sets in terms of having contractible fillers of horn of the form Λ 0,2 [3] and Λ 1,3 [3]. In Section 6, we show that quasi-2-Segal sets satisfy a version of the special horn lifting property from quasi-categories, and we show that there is a model structure for quasi-2-Segal sets.…”

“…We can think of a 2-Segal set as being like a weak version of a category, where we view the 0-simplices as objects, the 1-simplices as morphisms, and the 2-simplices as witnessing composition of morphisms, except that the composite of two morphisms x → y → z need not be unique or even defined. Having unique lifts of 2-Segal spine extensions implies that this weak notion of composition is still associative in a certain sense, given by the correspondence between the two 2-Segal spines of ∆ [3]:…”

“…The two broken subsets of {0, 1, 2, 3} are {0, 2} and {1, 3}. The corresponding 2-Segal horns Λ 0,2 [3] and Λ 1,3 [3] are precisely the two triangulations of the square, our two 2-Segal spines from Subsection 2.9.…”

“…In particular, we see that for S set • (Ab), the choices of fillers of horns of the form Λ 0,2 [3] and Λ 1,3 [3] are indeed unique up to homotopy in the sense we would hope for.…”