* Re: Haskell Arrows and internal categories [not found] <AANLkTinJEsOLaF4iaMcgJoypZktx4q-AFCpTtmw+5oz=@mail.gmail.com> @ 2010-09-27 23:45 ` Mike Stay 2010-09-29 12:06 ` Bas Spitters 2010-09-30 13:39 ` Mike Stay [not found] ` <AANLkTik1xN6-cDm+0DezF3xfRUhKRrR95ZBiC-zrYj-R@mail.gmail.com> 1 sibling, 2 replies; 5+ messages in thread From: Mike Stay @ 2010-09-27 23:45 UTC (permalink / raw) To: categories On Mon, Sep 27, 2010 at 3:50 PM, Mike Stay <metaweta@gmail.com> wrote: > I'm trying to understand Arrows in Haskell, > http://www.haskell.org/arrows/index.html > but since I haven't become literate yet, I'm not sure I'm getting > everything right. It looks to me like an Arrow is a monoidal closed > category object in Hask. Is that all there is to it? Hmm. After reading "Freyd is Kleisli, for Arrows", it now looks to me like an Arrow is an enrichment. It consists of a V-profunctor - A:C^op x C -> V, where V is a monoidal category, together with a natural transformation - arr:Hom => A and dinatural transformations - compose:A(b,c) x A(a,b) => A(a,c) - first:A(a,b) => A(a tensor c, b tensor c) satisfying various coherence laws. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Haskell Arrows and internal categories 2010-09-27 23:45 ` Haskell Arrows and internal categories Mike Stay @ 2010-09-29 12:06 ` Bas Spitters 2010-09-30 10:24 ` Jeff Egger 2010-09-30 13:39 ` Mike Stay 1 sibling, 1 reply; 5+ messages in thread From: Bas Spitters @ 2010-09-29 12:06 UTC (permalink / raw) To: Mike Stay; +Cc: categories You may be interested in the work of my colleagues: "Categorical semantics for Arrows" (2009) Bart Jacobs, Ichiro Hasuo and Chris Heunen Journal of Functional Programming, 19(3-4):403-438, 2009 http://www.comlab.ox.ac.uk/people/chris.heunen/publications/2008/arrows/arrows.pdf "Arrows, like Monads, are Monoids" (2006) Bart Jacobs and Chris Heunen (ENTCS 158:219-236) in the proceedings of MFPS 22 http://www.comlab.ox.ac.uk/people/chris.heunen/publications/2006/arrows/arrows.pdf Bas On Tue, Sep 28, 2010 at 1:45 AM, Mike Stay <metaweta@gmail.com> wrote: > On Mon, Sep 27, 2010 at 3:50 PM, Mike Stay <metaweta@gmail.com> wrote: >> I'm trying to understand Arrows in Haskell, >> http://www.haskell.org/arrows/index.html >> but since I haven't become literate yet, I'm not sure I'm getting >> everything right. It looks to me like an Arrow is a monoidal closed >> category object in Hask. Is that all there is to it? > > Hmm. After reading "Freyd is Kleisli, for Arrows", it now looks to me > like an Arrow is an enrichment. > > It consists of a V-profunctor > - A:C^op x C -> V, > where V is a monoidal category, together with a natural transformation > - arr:Hom => A > and dinatural transformations > - compose:A(b,c) x A(a,b) => A(a,c) > - first:A(a,b) => A(a tensor c, b tensor c) > satisfying various coherence laws. > -- > Mike Stay - metaweta@gmail.com > http://www.cs.auckland.ac.nz/~mike > http://reperiendi.wordpress.com > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Haskell Arrows and internal categories 2010-09-29 12:06 ` Bas Spitters @ 2010-09-30 10:24 ` Jeff Egger 0 siblings, 0 replies; 5+ messages in thread From: Jeff Egger @ 2010-09-30 10:24 UTC (permalink / raw) To: Bas Spitters, Mike Stay; +Cc: categories I should also point out recent work of my colleagues: "What is a Categorical Model of Arrows?" Robert Atkey. Mathematically Structured Functional Programming 2008. http://personal.cis.strath.ac.uk/~raa/arrows.pdf http://personal.cis.strath.ac.uk/~raa/msfp-slides-arrows.pdf "Idioms are oblivious, arrows are meticulous, monads are promiscuous." Sam Lindley, Philip Wadler, and Jeremy Yallop. In the proceedings of MSFP 2008. ENTCS. http://homepages.inf.ed.ac.uk/slindley/papers/idioms-arrows-monads.pdf "The arrow calculus." Sam Lindley, Philip Wadler and Jeremy Yallop. In the Journal of Functional Programming 20:1, pages 51-69, January 2010. Cambridge University Press. http://homepages.inf.ed.ac.uk/slindley/papers/arrow-calculus.pdf Cheers, Jeff. ----- Original Message ---- > From: Bas Spitters <spitters@cs.ru.nl> > To: Mike Stay <metaweta@gmail.com> > Cc: categories <categories@mta.ca> > Sent: Wed, September 29, 2010 1:06:16 PM > Subject: categories: Re: Haskell Arrows and internal categories > > You may be interested in the work of my colleagues: > "Categorical semantics for Arrows" (2009) Bart Jacobs, Ichiro Hasuo > and Chris Heunen > Journal of Functional Programming, 19(3-4):403-438, 2009 >http://www.comlab.ox.ac.uk/people/chris.heunen/publications/2008/arrows/arrows.pdf > >f > > "Arrows, like Monads, are Monoids" (2006) Bart Jacobs and Chris Heunen > (ENTCS 158:219-236) in the proceedings of MFPS 22 >http://www.comlab.ox.ac.uk/people/chris.heunen/publications/2006/arrows/arrows.pdf > >f > > Bas > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Haskell Arrows and internal categories 2010-09-27 23:45 ` Haskell Arrows and internal categories Mike Stay 2010-09-29 12:06 ` Bas Spitters @ 2010-09-30 13:39 ` Mike Stay 1 sibling, 0 replies; 5+ messages in thread From: Mike Stay @ 2010-09-30 13:39 UTC (permalink / raw) To: categories Thanks everyone for your responses, both on- and off-list! -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
[parent not found: <AANLkTik1xN6-cDm+0DezF3xfRUhKRrR95ZBiC-zrYj-R@mail.gmail.com>]
* Re: Haskell Arrows and internal categories [not found] ` <AANLkTik1xN6-cDm+0DezF3xfRUhKRrR95ZBiC-zrYj-R@mail.gmail.com> @ 2010-09-28 17:42 ` Mike Stay 0 siblings, 0 replies; 5+ messages in thread From: Mike Stay @ 2010-09-28 17:42 UTC (permalink / raw) To: categories On Mon, Sep 27, 2010 at 4:45 PM, Mike Stay <metaweta@gmail.com> wrote: > Hmm. After reading "Freyd is Kleisli, for Arrows", it now looks to me > like an Arrow is an enrichment. > > It consists of a V-profunctor > - A:C^op x C -> V, > where V is a monoidal category, together with a natural transformation > - arr:Hom => A It was pointed out to me that this doesn't make sense unless C is already V-enriched (which was the case in the two examples I saw, Set and Hask), but if we have a monoidal functor i:Set -> V, then we can define - arr:i Hom => A. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
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2010-09-27 23:45 ` Haskell Arrows and internal categories Mike Stay
2010-09-29 12:06 ` Bas Spitters
2010-09-30 10:24 ` Jeff Egger
2010-09-30 13:39 ` Mike Stay
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