  Dyckerhoff--Kapranov and G\'alvez-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.
