All Segal objects are generalised monads in spans
David Kern
TL;DR
The paper develops a general framework that extends the double ∞-category of spans to a wide class of algebraic patterns $\mathfrak{P}$, constructing a $\mathfrak{P}$-monoidal ∞-category of $\mathfrak{P}$-shaped spans in a suitable base, and proving that $\mathfrak{P}$-monads in this span category are equivalent to Segal $\mathfrak{P}$-objects in the base. In particular, for the cell pattern $Θ^{\mathrm{op}}$ this yields a homotopical reformulation of weak $\omega$-categories, and the framework recovers and unifies generalized multicategories of Burroni, Hermida, Leinster and Cruttwell–Shulman. The core result ties monadic and Segal perspectives in a broad, fibrational setting, with concrete instances including internal categories, commutative monoids, and higher/iterated spans, illustrating how various flavours of multicategories arise as special cases. The fibrational viewpoint and univalence considerations further situate the construction within modern higher topos theory, suggesting a unifying approach to Segal objects across dimensions.
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 $Θ^{\mathrm{op}}$, this recovers a homotopical reformulation of Batanin's original definition of weak $ω$-categories, and in general can be seen as a variant of the generalised multicategories of Burroni, Hermida, Leinster and Cruttwell-Shulman.