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Stream: theory: category theory

Topic: p.r.a. monads from adjunctions

Brendan Murphy (Mar 24 2024 at 16:44):

Recall that a functor $T : \mathcal{C} \to \mathcal{D}$ on a category $\mathcal{C}$ with a terminal object $1$ is called p.r.a. if the induced functor $T_1 : \mathcal{C} \to \mathcal{D}/T(1)$ is a right adjoint, and a monad is called p.r.a. if its underlying functor is p.r.a. and its unit/multiplication are cartesian (have cartesian naturality squares). What conditions on an adjunction $F \negmedspace: \mathcal{C} \rightleftarrows \mathcal{D}:\negmedspace G$ tell us that the induced monad on $\mathcal{C}$ is p.r.a.?

Brendan Murphy (Mar 24 2024 at 16:47):

If $\mathcal{C}, \mathcal{D}$ have all pullbacks, $F, G$ preserve pullbacks, and the adjunction unit & counit are cartesian then we can interpret this as an adjunction in the $2$-category of such things and get that the induced monad is again cartesian (which is a necessary condition)

Brendan Murphy (Mar 24 2024 at 16:49):

That the unit/multiplication of the induced monad are cartesian natural transformations means the unit of the adjunction must be cartesian and the "conjugate whiskering" $G\varepsilon F$ of the counit must be cartesian

Kevin Carlson (aka Arlin) (Mar 26 2024 at 18:26):

If $\mathcal C$ is cototal then $T_1$ will be a right adjoint if and only if $T$ preserves connected limits, so the main thing you want is that $F$ preserves connected limits. I'm not sure what to say about the cartesian natural transformations beyond what you've already said.