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Stream: learning: questions

Topic: Beck monadicity in Fib/A

fosco (Nov 07 2023 at 11:24):

Does anyone know a precise statement, and a reference, for a Beck monadicity theorem for a functor $U : X \to Y$ fibered over $A$? By this I mean, at your choice:

• $U$ is isomorphic to the forgetful functor $Alg(T) \to Y$ from the category of algebras of a monad on $Y$, fibered over $A$; or slightly more formally,
• $U$ is a 1-cell of $Fib/A$, the 2-cat of fibrations over $A$, cartesian functors over $A$ and vertical 2-cells, with the property that it has a left adjoint $F$, and $X$ is (equivalent to) the Eilenberg-Moore object (in Fib) of the monad (in Fib) $UF$.

From Street's formal theory I can't understand whether a 1-cell $u : x\to y$ of a 2-category with enough EM-objects is monadic (in the obvious sense) iff the functor $u_* : K(a,x) \to K(a,y)$ is monadic for every $a$. Is it a consequence of the fact that Yoneda reflects limits, or something subtle is going on?

Nathanael Arkor (Nov 07 2023 at 17:35):

fosco said:

From Street's formal theory I can't understand whether a 1-cell $u : x\to y$ of a 2-category with enough EM-objects is monadic (in the obvious sense) iff the functor $u_* : K(a,x) \to K(a,y)$ is monadic for every $a$.

This sounds very similar to Corollary 8.1 of The formal theory of monads. Are you asking if your statement follows from Corollary 8.1, or asking why 8.1 is true?

Nathanael Arkor (Nov 07 2023 at 17:38):

If you'd rather have a monadicity theorem of the form "a functor is monadic iff it is right adjoint and creates certain colimits", rather than one in terms of the 2-Yoneda embedding, it should be possible to derive one from the formal monadicity theorem in Relative monadicity.

fosco (Nov 08 2023 at 07:08):

I was asking if my statement follows from 8.1; and yes, I'd like to have something like the second statement...

Nathanael Arkor (Nov 08 2023 at 13:21):

A pair of antiparallel 1-cells $f : y \to x$ and $u \colon x \to y$ are adjoint iff for all $a \in \mathcal K$, $\mathcal K(a, f) \dashv \mathcal K(a, u)$. Also, a 1-cell $u \colon x \to y$ is left adjoint iff for all $a \in \mathcal K$, $K(a, u)$ is left adjoint 2-natural in $a$. (See here for instance.) So you can use either of these facts to deduce something similar to your statement from Corollary 8.1. However, your exact statement doesn't hold: you either need to fix the left adjoint, or you need to assume 2-naturality in $a$.

Nathanael Arkor (Nov 08 2023 at 13:22):

(Although note that fixing a left adjoint results exactly in Corollary 8.1.)

Nathanael Arkor (Nov 08 2023 at 13:24):

To apply the results from Relative monadicity, you need a (virtual) equipment of categories fibred over $A$, and you need to prove that the category of algebras satisfies the universal property of an algebra object. But I imagine these are straightforward modifications of the ordinary setting.