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?

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    $\begingroup$ I would guess the Riehl-Verity work math.jhu.edu/~eriehl/elements.pdf might have what you want $\endgroup$ – David Roberts Feb 22 at 10:28
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    $\begingroup$ In particular, chapter 9. $\endgroup$ – David Roberts Feb 22 at 10:47
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    $\begingroup$ 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. $\endgroup$ – Simon Henry Feb 22 at 13:01
  • $\begingroup$ @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. $\endgroup$ – Harry Gindi Feb 22 at 17:02
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    $\begingroup$ @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. $\endgroup$ – Mike Shulman Feb 22 at 18:28

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