All Segal objects are generalised monads in spans

Abstract

We extend Barwick's and Haugseng's construction of the double $\infty$-category of spans in a pullback-complete $\infty$-category $\mathfrak{C}$ to more general shapes: for a large class of algebraic patterns $\mathfrak{P}$, we define a $\mathfrak{P}$-monoidal $\infty$-category of $\mathfrak{P}$-shaped spans in $\mathfrak{C}$, and we identify $\mathfrak{P}$-monads in it with Segal $\mathfrak{P}$-objects in $\mathfrak{C}$. For the cell pattern $\Theta^{\mathrm{op}}$, this recovers a homotopical reformulation of Batanin's original definition of weak $\omega$-categories, and in general can be seen as a variant of the generalised multicategories of Burroni, Hermida, Leinster and Cruttwell-Shulman.