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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJun 24th 2019

A stub.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2019

Do I have this generalisation right for graded monads in any 2-category?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2019

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJul 4th 2019

Looks good.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJul 5th 2019

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 $\mathbf{B} M \to 1$?

• CommentRowNumber6.
• CommentAuthorPaoloPerrone
• CommentTimeJul 5th 2019

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.)

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJul 5th 2019
• (edited Jul 5th 2019)

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 $1$?

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJul 6th 2019

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJul 6th 2019

Looking at the entry, currently it doesn’t make clear why one should care about the concept. What use have people put this to?

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2019

I added a general way to produce graded monads. Perhaps that begins to answer Urs’s question. But it would be good to include some concrete examples.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2019

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.

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2019

Is that a good bet even when thinking of monads as lax functors?

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeJul 7th 2019

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.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2019

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.

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2019

• CommentRowNumber16.
• CommentAuthorTobias Fritz
• CommentTimeJul 9th 2019

Expanded on the utility of graded monads, and modified the grading of the graded list monad to include zero (since the list monad should contain the empty list).

• CommentRowNumber17.
• CommentAuthorTobias Fritz
• CommentTimeJul 9th 2019

corrected & clarified previous edits

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeJul 9th 2019

Great, thanks!

I added annotation so we know the authors of theorems within the text.

Strangely here we have to separate letters in math mode, so $m n$ rather than $mn$.

• CommentRowNumber19.
• CommentAuthorTobias Fritz
• CommentTimeJul 9th 2019

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.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 1st 2020