Arrows, like Monads, are Monoids

By Chris Heunen and Bart Jacobs

At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fairly civilised, by showing that they correspond to monoids in suitable subcategories of bifunctors Cop × C → C. This shows that, at a suitable level of abstraction, arrows are like monads …

Comment viewing options

Select your preferred way to display the comments and click "Save settings" to activate your changes.

Extremely cool paper

As are a lot of Bart Jacob's papers. Personally I have a soft spot for Distributive laws for the Coinductive Solution of Recursive Equations (previous discussion) and Coalgebras and monads in the semantics of Java (apparently not discussed).

Now that we've seen a lot of abstractions in programming languages that are monoid-like, I really wonder when will semirings will become just as popular? Some people, like Dexter Kozen, have been making them more mainstream (often under the name of Kleene Algebra). They have been showing up in quite a few other areas as well (polynomial functors form a semiring [and thus containers], ``features'' also can be modeled by them, and apparently constraints too).

Jacobs as model

As a previous postdoc with Bart, I'd like to suggest that he is a model theoretical computer scientist, moving from categories to smart cards to voting machines with ease.

More theoreticians should get their hands dirty like Bart; and more dirty programmers should learn more about the various theories underlying computation.

I, and Bart I am sure, are excited to see a forum like this one where the two worlds meet.

What is a Categorical Model of Arrows?

In What is a Categorical Model of Arrows? Robert Atkey criticises the Heunen & Jacobs paper for ignoring the additional structure that Arrows have over Freyd categories, and denies the claimed equivalence holds.

Kudos to Sam Lindley.