Part of the Lecture Notes in Computer Science book series LNCS, volume Abstract We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between monads and relative monads. Arrows are also an instance of relative monads. This process is experimental and the keywords may be updated as the learning algorithm improves. Download to read the full conference paper text References 1.
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Thinking about a PhD? There was also the need to write down all the work done in Hawaii, and Dusko, Filippo and I just finished writing a meaty technical report, which will soon appear on the arXiv. The time that I actually did spent writing blog stuff was distributed across a number of different articles; so now I have a whole bunch of half-finished stuff. I really need to start managing this whole project a little better! At the same graduate school, Alexander Kurz was giving a course on category theory.
As part of the distributive laws of props story, you realise that categories are a special case of the concept of monad. And monoids—of course—are a special case of the concept of category. The story is full of pleasant categorical mental gymnastics.
Think of it as the categorical equivalent of a 5km fun run. Composition is associative, and composing with identities does nothing. So every arrow can be composed with every other arrow. So every monoid is a kind of category, a category with one object.
Categories, it would seem, are the more general concept. Maybe categories should actually have been called monoidoids? What matters is getting to the core of the issues, and not getting too hung up on extraneous details. In this specific case, an endofunctor is a 1-cell in the 2-category Cat of categories, functors, and natural transformations. The data defining a monad is thus one 1-cell and two 2-cells, subject to some commutative diagrams of 2-cells.
The fact that this data lives in Cat brings nothing important to the story: the definition clearly makes sense in any 2-category. The 2-category Span Set is one of my favourites. Trust me. The objects of Span Set are sets. Composition of spans involves taking pullbacks. The interesting question is what happens when we consider a monad in Span Set. Now we need to give two 2-cells, that is, morphisms of spans. So far so good. All this data has another name: a category.
So, a category is just a special case of the more general concept of monad, right? We talked about monads in the generality of 2-categories. In fact, the concept of monoid makes sense in any monoidal category. We saw before that to give a category with one object is to give a monoid. The objects are the 1-cells, and monoidal product on 1-cells is composition. So what is a monoid in [C, C], considered as a monoidal category? So monads are really special kinds of monoids.
And categories are really special kinds of monads. And ….
…, a monoid is a category, a category is a monad, a monad is a monoid, …
Monad (category theory)