$2$-Segal spaces are simplicial spaces satisfying a $2$-dimensional generalisation of the Segal condition. It has been shown by Dyckerhoff--Kapranov and G\'alvez-Carrillo--Kock--Tonks that the space of $1$-simplices $S_1$ of a reduced, unital $2$-Segal space $S_\bullet$ carries an associative, multi-valued product. This notion is formalised by saying that $S_1$ is an algebra object in the category of correspondences of spaces.

In this talk I will show that the multi-valued product on $S_1$ is part of a lax, multi-valued bialgebraic structure. Namely, $S_1$ is a so-called lax bialgebra object in the $2$-category of iterated correspondences of spaces. The construction of this structure proceeds combinatorially by leveraging the fact that every simplicial space defines a contravariant realisation functor from simplicial sets to spaces. An upshot of this construction is that it elucidates the exact role played by the $2$-Segal condition.

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