* 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
[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
* 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
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2010-09-27 23:45 ` Haskell Arrows and internal categories Mike Stay
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2010-09-30 10:24 ` Jeff Egger
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