The edgewise subdivision criterion for 2-Segal objects. (English) Zbl 1427.18005

The edgewise subdivision of a simplical space is a construction leaving the geometric realization unchanged but having the effect of decomposition of the simplicial space into more simplices. It appeared first in [G. Segal, Invent. Math. 21, 213–221 (1973; Zbl 0267.55020)], while F. Waldhausen [Lect. Notes Math. 1126, 318–419 (1985; Zbl 0579.18006)] used this construction to establish the equivalence of the \(S_{\bullet}\)-construction and the \(Q\)-construction in algebraic \(K\)-theory. This paper applies this construction to the \(2\)-Segal spaces of [T. Dyckerhoff and M. Kapranov, Higher Segal spaces (to appear). Cham: Springer (2019; Zbl 1459.18001)], closely related with the decomposition spaces of [I. Gálvez-Carrillo et al., Adv. Math. 331, 952–1015 (2018; Zbl 1403.00023); ibid. 333, 1242–1292 (2018; Zbl 1403.18016); ibid. 334, 544–584 (2018; Zbl 1403.18017)]. In [J. E. Bergner et al., Topology Appl. 235, 445–484 (2018; Zbl 1422.55036); “2-Segal objects and the Waldhausen construction”, arXiv:1809.10924] the authors demonstrated that any \(2\)-Segal space abdiding by a unitality condition comes from such a construction for a suitably general input. This paper lies in the more general context of \(2\)-Segal objects in any combinatorial model category. The main result (Theorem 2.11) goes as follows:
Theorem. Let \(X\) be a simplicial object in a combinatorial model category \(\mathcal{M}\). Then \(X\) is a \(2\)-Segal object iff its edgewise subdivision \(\mathrm{esd}(X)\) is a Segal object.


18C40 Structured objects in a category (group objects, etc.)
18N50 Simplicial sets, simplicial objects
19D10 Algebraic \(K\)-theory of spaces
55U10 Simplicial sets and complexes in algebraic topology
Full Text: DOI arXiv


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