Simplicial spaces, lax algebras and the 2-Segal condition

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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 ∞-category of spans. In this paper we show that this algebra structure is inherited from the initial simplicial object Δ[]. Namely, we show that the standard 1-simplex Δ[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.