tag:blogger.com,1999:blog-11295132.post4054854230903521571..comments2017-01-21T08:58:46.751-08:00Comments on A Neighborhood of Infinity: Some thoughts on reasoning and monadsDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-11295132.post-11188426493387743352008-11-29T14:43:00.000-08:002008-11-29T14:43:00.000-08:00When I learned category theory, a subobject of d w...When I learned category theory, a subobject of d was defined as an equivalence class of monics with codomain d. This definition via equivalence classes is independent of the choice of representative - is this what's going on with your 'abuse of notation'?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-45538394252662488102008-11-18T20:47:00.000-08:002008-11-18T20:47:00.000-08:00The meet-join example doesn't make sense as wr...The meet-join example doesn't make sense as written. Did you mean something like the following? "Given f :: A -> B and g :: C -> D we can define h :: (A,C) -> Either B D as<BR/><BR/>h w = let x = fst w<BR/> y = f x<BR/> z = Left y<BR/> in z<BR/>"<BR/><BR/>i.e., h = Left . f . fst<BR/>or alternately h = Right . g . snd.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-31199631121111496342008-11-17T01:34:00.000-08:002008-11-17T01:34:00.000-08:00What would a category with commutative monads look...What would a category with <I>commutative</I> monads look like? I don't know how to introduce this particular commutativity into the lattice of subsets of a topological space.davenoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-73354099466667760592008-11-16T07:27:00.000-08:002008-11-16T07:27:00.000-08:00Andrej,Even without any category theory it's easy ...Andrej,<BR/><BR/>Even without any category theory it's easy to assign sets and functions for the example of posets. But of course the elements you get aren't necessarily the ones you're imagining as you write x :: a.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-57084178845797511342008-11-15T16:58:00.000-08:002008-11-15T16:58:00.000-08:00Dan, I just wanted to mention that arrows in any (...Dan, I just wanted to mention that arrows in any (small enough) category can be seen as actual functions, or families of functions. The Grothendieck construction embeds (non-fully) any small category into Sets, which turns morphisms into functions, while (the more reasonable) Yoneda embedding embeds (fully) any small category into presheaves, which turns morphisms into families of functions (natural transformations). I am sure you could write a nice Haskell post about this, if you haven't yet :-)Andrej Bauerhttp://andrej.com/noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-90386866517264866642008-11-15T16:03:00.001-08:002008-11-15T16:03:00.001-08:00Derek,Should have known you'd know a relevant pape...Derek,<BR/><BR/>Should have known you'd know a relevant paper! I'll check it out when I get a chance.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-57668968668969912382008-11-15T16:03:00.000-08:002008-11-15T16:03:00.000-08:00Dylan, that's exactly why I like to use the vector...Dylan, that's exactly why I like to use the <A HREF="http://sigfpe.blogspot.com/2008/08/hopf-algebra-group-monad.html" REL="nofollow">vector space monad</A>! I guess "Sweedler notation" is an example of what I'm talking about too. But what motivated me to write this post were the examples where the arrows aren't functions.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-44852251089895056912008-11-15T15:38:00.000-08:002008-11-15T15:38:00.000-08:00One case where a pointful notation is really more ...One case where a pointful notation is really more useful is when dealing with maps to and from tensor products. If you have a map D from V to V tensor V, would you rather write<BR/><BR/>(D tensor Id) (D (v))<BR/><BR/>or<BR/><BR/>let (x,y) = D v; (a,b) = D x in (a,b,y)<BR/><BR/>?<BR/><BR/>(Actually, what I really find most helpful is neither of these two, but rather a graphical notation with graphs.)Dylan Thurstonhttp://math.columbia.edu/~dptnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-71282805710807385392008-11-15T15:37:00.000-08:002008-11-15T15:37:00.000-08:00In response to your last paragraph. do-notation is...In response to your last paragraph. do-notation is a notational variant of (one of*) Moggi's monadic metalanguage(s) which is indeed, precisely the internal language of a category with monads. And monads are related to lax logic. See <A HREF="http://www.cs.cmu.edu/~fp/papers/mscs00.pdf" REL="nofollow">A Judgemental Reconstruction of Modal Logic</A> which also refers to (a fuller version of) Moggi's monadic metalanguage.<BR/><BR/>* The "Simple Metalanguage" in <A HREF="http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf" REL="nofollow">Notions of computation and monads</A>.Derek Elkinsnoreply@blogger.com