I have completed the publication data for this item:
(In particular I have added the DOi link, so that it’s possible to actually see the article with a click or two.)
As usual, I am also copying this item into the author’s pages “Selected writings”-lists.
]]>I have touched the Idea-section here, rewording at various places in an attempt to make it flow more nicely.
Where the lax -action is mentioned, I have hyperlinked with action of a monoidal category.
]]>Note that graded monads are enriched relative monads.
]]>Added
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.
]]>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 rather than .
]]>corrected & clarified previous edits
]]>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).
]]>Added another example.
]]>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.
]]>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 for instance.
If I were trying to answer #9, I would probably start by looking at the references.
]]>Is that a good bet even when thinking of monads as lax functors?
]]>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.
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.
]]>Looking at the entry, currently it doesn’t make clear why one should care about the concept. What use have people put this to?
]]>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.
]]>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 ?
]]>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 (the grading) into an endofunctor category . If instead we view the graded monad as a lax 2-functor into a bicategory , I don’t know how to obtain a monad (lax 2-functor ) via a (2-dimensional?) Kan extension. Naively I would say that if we tried to do that, then the functor and its Kan extension (along ) could have different image objects in , only related by a universal 1-cell which is not necessarily (pseudo)invertible. If , 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.)
]]>I see Paolo Perrone was speaking here of a colimit of graded monads. With the formulation via lax 2-functors, , is there a slick way of speaking of some 2-categorical Kan extension along ?
]]>Looks good.
]]>Added an example.
]]>Do I have this generalisation right for graded monads in any 2-category?
]]>A stub.
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