Simplicial spaces, lax algebras and the 2-Segal condition

Abstract

Dyckerhoff--Kapranov and Gálvez-Carrillo--Kock--Tonks independently introduced the notion of a $2$-Segal space, that is, a simplicial space satisfying $2$-dimensional analogues of the Segal conditions, as a unifying framework for understanding the numerous Hall algebra-like constructions appearing in algebraic geometry, representation theory and combinatorics. In particular, they showed that every $2$-Segal object defines an algebra object in the $\infty$-category of spans. In this paper we show that this algebra structure is inherited from the initial simplicial object $\Delta [\bullet ]$. Namely, we show that the standard $1$-simplex $\Delta [1]$ carries a lax algebra structure. As a formal consequence the space of $1$-simplices of a simplicial space is also a lax algebra. We further show that the $2$-Segal conditions are equivalent to the associativity of this lax algebra.

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Cited by in corpus (5)

• The incidence comodule bialgebra of the Baez-Dolan construction
• Frobenius objects in the category of spans
• The universal Hall bialgebra of a double 2-Segal space
• Higher Segal spaces and Lax ${\mathbb{A}}_{\infty }$-algebras
• Hall categories and KLR categorification