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Looks good.
I see Paolo Perrone was speaking here of a colimit of graded monads. With the formulation via lax 2-functors, $\mathbf{B} M \to K$, is there a slick way of speaking of some 2-categorical Kan extension along $\mathbf{B} M \to 1$?
If you refer to a way of obtaining a non-graded monad from a graded monad via a Kan extension in the 2-category of monoidal categories and lax monoidal functors, the answer is yes - we do it for example in the paper presented in that blog post (link here), Theorems 4.14 and B.1. The idea of a “monoidal Kan extension” is not due to us – we review some of the approaches in the literature here.
However, those are Kan extensions in the 2-category of monoidal categories. In that approach, a graded monad is a lax monoidal functor from a monoidal category $M$ (the grading) into an endofunctor category $[C,C]$. If instead we view the graded monad as a lax 2-functor $\mathbf{B}M\to K$ into a bicategory $K$, I don’t know how to obtain a monad (lax 2-functor $1\to K$) via a (2-dimensional?) Kan extension. Naively I would say that if we tried to do that, then the functor $\mathbf{B}M\to K$ and its Kan extension $1\to K$ (along $\mathbf{B}M\to 1$) could have different image objects in $K$, only related by a universal 1-cell which is not necessarily (pseudo)invertible. If $K=\mathbf{Cat}$, that would mean that the graded monad and the resulting monad would be potentially on different categories.
(I hope I’m understanding your question correctly though.)
Ok, thanks. I wonder what the benefits of the second (lax 2-functor) approach are. A few months ago I was wondering about factorization of monads in that setting. It seems you can’t generate the Kleisli and Eilenberg-Moore adjunctions as you can through the inclusions of the walking monad into the walking adjoint as here.
What does one gain in thinking of monads as lax 2-functors from $1$?
One thing is that lax and colax monad morphisms (between monads with varying base categories) are lax and colax natural transformations between such lax 2-functors.
Looking at the entry, currently it doesn’t make clear why one should care about the concept. What use have people put this to?
Should we be heeding Mike’s advice?
My personal conclusion is that whenever we start seeing lax functors appearing, it’s a good bet that our 2-categories are really double categories.
Is that a good bet even when thinking of monads as lax functors?
I don’t think so. There I was talking about the question of lax functors forming the morphisms in a category, i.e. composing them with each other and so on. Here we’re looking instead at lax functors mainly as “diagrams”, i.e. as being the objects of a category rather than the morphisms in one. And of course we’re interested in monads in plenty of 2-categories that aren’t double categories in any nontrivial way, like $Cat$ for instance.
If I were trying to answer #9, I would probably start by looking at the references.
It may be more of a computer science thing, such as modalities in bounded linear logic and the graded state monad.
I’m looking into this a little since philosophers have devised graded modalities for degrees of necessity, obligation, and so on.
Thanks! I’ll keep these things in mind for future edits.
By the way, one reason for restricting to monoidal categories as opposed to general 2-categories is that monoidal categories with lax monoidal functors can easily be treated as a 2-category, while arguably the collection of all 2-categories ought to be considered a 3-category. So it’s a technical simplification to consider merely deloopings.
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