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Stream: deprecated: id my structure

Topic: horizontal monads on a fibration?

James Deikun (Aug 04 2023 at 14:38):

So a monad in the bicategory $\bold{Fib}$ consists of a fibration $p: E \to B$, a monad $\overline{\mathbb{T}} = (\overline{T}, \overline{\eta}, \overline{\mu})$ on $E$, and a monad $\underline{\mathbb{T}} = (\underline{T}, \underline{\eta}, \underline{\mu})$ on $B$, such that $T = (\overline{T},\underline{T}) : p \to p$ is a map of fibrations, $\overline{\eta}, \overline{\mu}$ are componentwise lifts of their underlined equivalents, and $p : \overline{\mathbb{T}} \to \underline{\mathbb{T}}$ is a strict map of monads.

Call such a monad horizontal if $\overline{\eta}, \overline{\mu}$ are componentwise Cartesian. Given such a horizontal monad, call $\overline{\mathbb{T}}$ a $p$-Cartesian structure on $\underline{\mathbb{T}}$. Then, it seems, a $\mathrm{cod}$-Cartesian structure uniquely exists iff $\underline{\mathbb{T}}$ is a [[Cartesian monad]] in the usual sense, and a $\mathrm{dom}$-Cartesian structure corresponds to a "trivialization": a section of the forgetful functor $U^{\underline{\mathbb{T}}} : B^{\underline{\mathbb{T}}} \to B$.

Has anyone run into these before or know a reference? All the references to "fibred" or "fibered" monads seem to only talk about the "vertical" kind, where $\underline{\mathbb{T}}$ is the identity monad.