# Infinity categorical analogue of 2-dimensional monad theory

I'm wondering whether there is an infinity categorical analogue to the results of Two-dimensional monad theory. For the most part, I'm interested in the relation between strict functors of infinity categories (with algebraic structure) and non-strict functors.

For example, it follows from the aforementioned paper that for every finitely complete category $$\mathcal{C}$$ there is a lex functor $$I : \mathcal{C} \rightarrow \mathcal{C}'$$ such that for every lex functor $$J : \mathcal{C} \rightarrow \mathcal{D}$$ there is a unique strict lex functor $$K : \mathcal{C}' \rightarrow \mathcal{D}$$ such that $$J = K I$$. (We assume that every finitely complete category comes with canonical choices of finite limits; strict lex functors need to preserve these canonical choices on the nose.) $$I$$ is an equivalence of categories, and $$\mathcal{C}'$$ is given by forgetting canonical limits in $$\mathcal{C}$$ and adjoining new canonical limits. The adjunction extends to a 2-adjunction, i.e. we get an isomorphism of categories $$\mathrm{Lex}(\mathcal{C}, \mathcal{D}) \cong \mathrm{sLex}(\mathcal{C}', \mathcal{D})$$.

Universal properties in the infinity categorical sense are defined via filling conditions, e.g. for inner horns in the case of composition. I suppose the obvious analogue of strict lex 1-functors would be a natural transformation of simplicial sets that preserves the canonical fillings to inner horns and the canonical fillers from the canonical limits in lex infinity categories.

I expect that the 1-categorical result I explained above also holds in the infinity categorical case. Has this (or something similar) been worked out somewhere?

• I would guess the Riehl-Verity work math.jhu.edu/~eriehl/elements.pdf might have what you want – David Roberts Feb 22 at 10:28
• In particular, chapter 9. – David Roberts Feb 22 at 10:47
• Note that all the question about whether some construction can be strictified or not are inherently model dependent, so the question only make sense if you chose a precise model for your $\infty$-categories. – Simon Henry Feb 22 at 13:01
• @DavidRoberts These are only 1-dimensional monads, or monads internal to a 2-category, though. True 2-dimensional monads, as I understand, are usually monads based at a 2-category (oftentimes Cat itself or a slice of Cat). The analogue in the Riehl-Verity theory would be, for example, a monad internal to some infinity,2-category of cosmoi. – Harry Gindi Feb 22 at 17:02
• @HarryGindi Yes and no -- I am under the impression that there are $\infty$-cosmoi whose objects are $(\infty,2)$-categories, so an internal monad therein would indeed be an $(\infty,2)$-monad. On the other hand, such a thing would probably not be strict in the way that I think the OP is imagining; for that I would be inclined to consider something like enriched monads on quasicategory-enriched categories. – Mike Shulman Feb 22 at 18:28