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Stream: learning: questions

Topic: When are kappa-accessible monads monadic?

James Deikun (Nov 04 2025 at 17:06):

This seems to be the case for Set and any regular $\kappa$? But what about when the base category is, say, some other locally presentable category? Or a presentable $(\infty,1)$-category?

Nathanael Arkor (Nov 04 2025 at 17:51):

Monadic over what?

James Deikun (Nov 04 2025 at 17:52):

$\kappa$-accessible endofunctors is the obvious choice, but something else nontrivial could also be interesting.

Nathanael Arkor (Nov 04 2025 at 18:37):

It's true for any locally $\kappa$-presentable category by Lack's On the monadicity of finitary monads. I don't see any reason this would fail for $(\infty, 1)$-categories.

Mike Shulman (Nov 06 2025 at 22:48):

Have you looked at Lack's proof and it seems like it should work for $(\infty,1)$-categories? Or do you just mean there's no obvious reason the statement would fail to be true there?

Nathanael Arkor (Nov 07 2025 at 07:48):

For monadicity over $\kappa$-accessible endofunctors, the full force of Lack's result is not necessary: it suffices to know that categories of monoids in nice monoidal categories are monadic over their base categories. I would suppose that this result has already been established for $(\infty, 1)$-categories, but if not it presumably will follow from the work Simon Henry spoke about at CT this year.

James Deikun (Nov 07 2025 at 14:43):

Has anybody actually established any monadicity results for the category of algebras of an $\infty$-operad in an $\infty$-category other than $\mathcal S$? It could very well be in Higher Algebra or Kerodon for all I have a handle on those books, but I don't think I've seen it cited.

James Deikun (Nov 07 2025 at 14:47):

I guess there's some stuff in HA section 3.1.3 ... and 4.7.3. After that Lurie moves on to other stuff. He never seems to establish anything like "algebras of operads in presentable $\infty$-categories are algebras of a monad" though.