[Haskell-cafe] Re: monads and groups -- instead of loops
Arie, Thanks for your thoughtful reply. Comments in-lined. Best wishes, --greg Date: Thu, 2 Aug 2007 03:06:51 +0200 (CEST) From: Arie Peterson [EMAIL PROTECTED] Subject: Re: [Haskell-cafe] Re: monads and groups -- instead of loops To: haskell-cafe@haskell.org Message-ID: [EMAIL PROTECTED] Content-Type: text/plain;charset=iso-8859-1 Math alert: mild category theory. Greg Meredith wrote: But, along these lines i have been wondering for a while... the monad laws present an alternative categorification of monoid. At least it's alternative to monoidoid. I wouldn't call monads categorifications of monoids, strictly speaking. A monad is a monoid object in a category of endofunctors (which is a monoidal category under composition). Sorry, i was being as fast and loose with the term as the rest of the communities concerned with 'categorification' seem to be. What do you mean by a 'monoidoid'? I only know it as a perverse synonym of 'category' :-). Indeed. In the spirit of this thought, does anyone know of an expansion of the monad axioms to include an inverse action? Here, i am following an analogy monoidoid : monad :: groupoid : ??? First of all, I don't actually know the answer. The canonical option would be a group object in the endofunctor category (let's call the latter C). This does not make sense, however: in order to formulate the axiom for the inverse, we would need the monoidal structure of C (composition of functors) to behave more like a categorical product (to wit, it should have diagonal morphisms diag :: m a - m (m a) ). It seems to me that there are two basic possibilities, here. One is that the ambient categories over which one formulates computational monads are almost always some type of Linear-Cartesian situation. So, you can possibly exploit the additional structure there. That's certainly been the general flavor of the situation that motivates me. Otherwise, you can go the route of trying to excavate structure that might give meaningful interpretations. This has appeal in that it is more general and might actually uncover something, but as you observe it's not immediate. i haven't wrestled with the idea in anger, yet, because i thought it such an obvious thing to try that someone would have already done the work and was hoping just to get a reference. Your note suggests that it might be worth digging a little. i wonder... does a Hopf algebra-like structure do the job? Maybe there is a way to get it to work, though. After all, what we (in FP) call a commutative monad, is not commutative in the usual mathematical sense (again, C does not have enough structure to even talk about commutativity). My intuition tells me this could be quite generally useful to computing in situation where boxing and updating have natural (or yet to be discovered) candidates for undo operations. i'm given to understand reversible computing might be a good thing to be thinking about if QC ever gets real... ;-) If this structure is to be grouplike, the inverse of an action should be not only a post-inverse, but also a pre-inverse. Is that would you have in mind? (If I'm not making sense, please shout (or ignore ;-) ).) Greetings, Arie -- L.G. Meredith Managing Partner Biosimilarity LLC 505 N 72nd St Seattle, WA 98103 +1 206.650.3740 http://biosimilarity.blogspot.com ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: monads and groups -- instead of loops
On 1 Aug 2007, at 21:23, Greg Meredith wrote: But, along these lines i have been wondering for a while... the monad laws present an alternative categorification of monoid. At least it's alternative to monoidoid. In the spirit of this thought, does anyone know of an expansion of the monad axioms to include an inverse action? Here, i am following an analogy monoidoid : monad :: groupoid : ??? I'm not sure that's the right question. A monoid is a category with one object. A group is a category with one object, where every arrow is an iso. A groupoid is a category (with potentially more than one object), where every arrow is an iso. A monad is monoid in the category of endofunctors. Your groupad, i.e. a group in the category of endofunctors, would boil down to having a monad m that has an unreturn :: m a - a. In the light of things like unsafePerformIO, this is maybe not what you want. More interesting, however, is generalizing a monad to a monoid in functor categories in general, as opposed to just endofunctors. You could call this a monadoid. I've heard Tarmo Uustalu talk about this once (Kan-extensions, coends, *mumble mumble mumble*). Mike Spivey's MSFP paper (http://spivey.oriel.ox.ac.uk/mike/msfp.pdf) mentions A-monads, which is a step in that direction. Anyhow, just a thought. Wouter ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: monads and groups -- instead of loops
Haskellians, But, along these lines i have been wondering for a while... the monad laws present an alternative categorification of monoid. At least it's alternative to monoidoid. In the spirit of this thought, does anyone know of an expansion of the monad axioms to include an inverse action? Here, i am following an analogy monoidoid : monad :: groupoid : ??? i did a search of the literature, but was probably using the wrong terminology to try to find references. i would be very grateful for anyone who might point me in the right direction. My intuition tells me this could be quite generally useful to computing in situation where boxing and updating have natural (or yet to be discovered) candidates for undo operations. i'm given to understand reversible computing might be a good thing to be thinking about if QC ever gets real... ;-) Best wishes, --greg On 8/1/07, Greg Meredith [EMAIL PROTECTED] wrote: Haskellians, Though the actual metaphor in the monads-via-loops doesn't seem to fly with this audience, i like the spirit of the communication and the implicit challenge: find a pithy slogan that -- for a particular audience, like imperative programmers -- serves to uncover the essence of the notion. i can't really address that audience as my first real exposure to programming was scheme and i moved into concurrency and reflection after that and only ever used imperative languages as means to an end. That said, i think i found another metaphor that summarizes the notion for me. In the same way that the group axioms organize notions of symmetry, including addition, multiplication, reflections, translations, rotations, ... the monad(ic axioms) organize(s) notions of snapshot (return) and update (bind), including state, i/o, control, In short group : symmetry :: monad : update Best wishes, --greg -- L.G. Meredith Managing Partner Biosimilarity LLC 505 N 72nd St Seattle, WA 98103 +1 206.650.3740 http://biosimilarity.blogspot.com -- L.G. Meredith Managing Partner Biosimilarity LLC 505 N 72nd St Seattle, WA 98103 +1 206.650.3740 http://biosimilarity.blogspot.com ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: monads and groups -- instead of loops
Math alert: mild category theory. Greg Meredith wrote: But, along these lines i have been wondering for a while... the monad laws present an alternative categorification of monoid. At least it's alternative to monoidoid. I wouldn't call monads categorifications of monoids, strictly speaking. A monad is a monoid object in a category of endofunctors (which is a monoidal category under composition). What do you mean by a 'monoidoid'? I only know it as a perverse synonym of 'category' :-). In the spirit of this thought, does anyone know of an expansion of the monad axioms to include an inverse action? Here, i am following an analogy monoidoid : monad :: groupoid : ??? First of all, I don't actually know the answer. The canonical option would be a group object in the endofunctor category (let's call the latter C). This does not make sense, however: in order to formulate the axiom for the inverse, we would need the monoidal structure of C (composition of functors) to behave more like a categorical product (to wit, it should have diagonal morphisms diag :: m a - m (m a) ). Maybe there is a way to get it to work, though. After all, what we (in FP) call a commutative monad, is not commutative in the usual mathematical sense (again, C does not have enough structure to even talk about commutativity). My intuition tells me this could be quite generally useful to computing in situation where boxing and updating have natural (or yet to be discovered) candidates for undo operations. i'm given to understand reversible computing might be a good thing to be thinking about if QC ever gets real... ;-) If this structure is to be grouplike, the inverse of an action should be not only a post-inverse, but also a pre-inverse. Is that would you have in mind? (If I'm not making sense, please shout (or ignore ;-) ).) Greetings, Arie ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
