On 3/27/07, Dave at haskell.org <Dave at haskell.org> wrote:
Given the amount of material posted at haskell.org and elsewhere explaining IO, monads and functors, has anyone considered publishing a comprehensive book explaining those subjects? (I am trying to read all the material online, but books are easier to read and don't require sitting in front of a computer to do so. Plus I can write in books :-). )
I've thought about writing extended tutorials on the relationship between Haskell programming and category theory but there's a problem I run into. It's tempting to make the identifications: types<->objects, Haskell function<->arrows, suitably polymorphic functions<->natural transformations, and so on. But the fact is, this doesn't really work in the obvious way even though it seems like it should at first (eg. Haskell functions aren't always total functions in the mathematical sense and if you allow partial functions you can do weird stuff). So either:
(1) we need some technical work to patch up the differences (and unfortunately I don't know what the patch-up is), (2) we restrict ourselves to certain types of Haskell function for which the theory works or (3) we deliberately leave things a little vague.
I usually tend to go for option (3), but that wouldn't be satisfactory for an extended treatment.
Has anyone else given this subject much thought? -- Dan Dan,
It's a long time since I looked at this so this might be a complete red herring but have you thought about using the category of \omega-CPOs. Types are then objects (\omega-cpos) and functions are then arrows (continuous functions in the \omega-cpo sense). These papers may be of use: 1. Meijer http://research.microsoft.com/~emeijer/Papers/CWIReport.pdf 2. Mitchell Wand: Fixed-Point Constructions in Order-Enriched Categories 3. Freyd: Recursive Types Reduced to Inductive Types. Dominic.