Hi all,
I'm trying to learn Haskell and have come across Monads. I kind of understand monads now, but I would really like to understand where they come from. So I got a copy of Barr and Well's Category Theory for Computing Science Third Edition, but the book has really left me dumbfounded. It's a good book. But I'm just having trouble with the proofs in Chapter 1--let alone reading the rest of the text. Are there any references to things like "Hom Sets" and "Hom Functions" in the literature somewhere and how to use them? The only book I know that uses them is this one. Has anyone else found it frustratingly difficult to find details on easy-to-diget material on Category theory. The Chapter that I'm stuck on is actually labelled Preliminaries and so I reason that if I can't do this, then there's not much hope for me understanding the rest of the book... Maybe there are books on Discrete maths or Algebra or Set Theory that deal more with Hom Sets and Hom Functions? Thanks, Mark Spezzano. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
I should probably add that I am trying various proofs that involve injective and surjective properties of Hom Sets and Hom functions.
Does anyone know what Hom stands for? I need a text for a newbie. Mark On 02/02/2010, at 9:56 PM, Mark Spezzano wrote: > Hi all, > > I'm trying to learn Haskell and have come across Monads. I kind of understand monads now, but I would really like to understand where they come from. So I got a copy of Barr and Well's Category Theory for Computing Science Third Edition, but the book has really left me dumbfounded. It's a good book. But I'm just having trouble with the proofs in Chapter 1--let alone reading the rest of the text. > > Are there any references to things like "Hom Sets" and "Hom Functions" in the literature somewhere and how to use them? The only book I know that uses them is this one. > > Has anyone else found it frustratingly difficult to find details on easy-to-diget material on Category theory. The Chapter that I'm stuck on is actually labelled Preliminaries and so I reason that if I can't do this, then there's not much hope for me understanding the rest of the book... > > Maybe there are books on Discrete maths or Algebra or Set Theory that deal more with Hom Sets and Hom Functions? > > Thanks, > > Mark Spezzano. > > _______________________________________________ > Haskell-Cafe mailing list > [hidden email] > http://www.haskell.org/mailman/listinfo/haskell-cafe > > _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Hom(A, B) is just a set of morphisms from A to B.
Mark Spezzano wrote: > I should probably add that I am trying various proofs that involve injective and surjective properties of Hom Sets and Hom functions. > > Does anyone know what Hom stands for? > > I need a text for a newbie. > > Mark > > On 02/02/2010, at 9:56 PM, Mark Spezzano wrote: > >> Hi all, >> >> I'm trying to learn Haskell and have come across Monads. I kind of understand monads now, but I would really like to understand where they come from. So I got a copy of Barr and Well's Category Theory for Computing Science Third Edition, but the book has really left me dumbfounded. It's a good book. But I'm just having trouble with the proofs in Chapter 1--let alone reading the rest of the text. >> >> Are there any references to things like "Hom Sets" and "Hom Functions" in the literature somewhere and how to use them? The only book I know that uses them is this one. >> >> Has anyone else found it frustratingly difficult to find details on easy-to-diget material on Category theory. The Chapter that I'm stuck on is actually labelled Preliminaries and so I reason that if I can't do this, then there's not much hope for me understanding the rest of the book... >> >> Maybe there are books on Discrete maths or Algebra or Set Theory that deal more with Hom Sets and Hom Functions? >> >> Thanks, >> >> Mark Spezzano. >> >> _______________________________________________ >> Haskell-Cafe mailing list >> [hidden email] >> http://www.haskell.org/mailman/listinfo/haskell-cafe >> >> > > _______________________________________________ > Haskell-Cafe mailing list > [hidden email] > http://www.haskell.org/mailman/listinfo/haskell-cafe > Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Mark Spezzano-3
You may try Pierce's "Basic Category Theory for Computer Scientists" or Awodey's "Category Theory", whose style is rather introductory. Both of them (I think) have a chapter about functors where they explain the Hom functor and related topics.
Alvaro. 2010/2/2 Mark Spezzano <[hidden email]> I should probably add that I am trying various proofs that involve injective and surjective properties of Hom Sets and Hom functions. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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Mark Spezzano wrote:
> I need a text for a newbie. While the other books suggested are excellent, I think that they would be hard going if you find Barr & Wells difficult. The simplest introduction to the ideas of category theory that I know is "Conceptual Mathematics" by F W Lawvere & S H Schanuel. There are a great many online resources including many good books on category theory. But Barr & Wells is one of the best for application to Computing. ael _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Álvaro García Pérez-2
2010/2/2 Álvaro García Pérez <[hidden email]>
You may try Pierce's "Basic Category Theory for Computer Scientists" or Awodey's "Category Theory", whose style is rather introductory. Both of them (I think) have a chapter about functors where they explain the Hom functor and related topics. I think Awodey's book is pretty fantastic, actually, but I'd avoid Pierce. Unlike "Types and Programming Languages", I think "Basic Category Theory..." is a bit eccentric in its presentation and doesn't help the reader build intuition.
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Mark Spezzano <mark.spezzano <at> chariot.net.au> writes:
> > Maybe there are books on Discrete maths or Algebra or Set Theory that deal more with Hom Sets and Hom Functions? > Googling "haskell category theory" I got: http://en.wikibooks.org/wiki/Haskell/Category_theory http://www.haskell.org/haskellwiki/Category_theory and many others. The latter has a list of books. Perhaps people could update with books they are familiar with and add comments? Dominic. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Creighton Hogg-4
On Tue, 02 Feb 2010 09:16:03 -0800, Creighton Hogg wrote:
> 2010/2/2 Álvaro García Pérez <[hidden email]> > >> You may try Pierce's "Basic Category Theory for Computer Scientists" or >> Awodey's "Category Theory", whose style is rather introductory. Both of them >> (I think) have a chapter about functors where they explain the Hom functor >> and related topics. >> > > I think Awodey's book is pretty fantastic, actually, but I'd avoid Pierce. > Unlike "Types and Programming Languages", I think "Basic Category > Theory..." is a bit eccentric in its presentation and doesn't help the > reader build intuition. I have written an overview of various category theory books, which you may find useful, at the following site: Learning Haskell through Category Theory, and Adventuring in Category Land: Like Flatterland, Only About Categories http://dekudekuplex.wordpress.com/2009/01/16/learning-haskell-through-category-theory-and-adventuring-in-category-land-like-flatterland-only-about-categories/ Hope this helps. -- Benjamin L. Russell -- Benjamin L. Russell / DekuDekuplex at Yahoo dot com http://dekudekuplex.wordpress.com/ Translator/Interpreter / Mobile: +011 81 80-3603-6725 "Furuike ya, kawazu tobikomu mizu no oto." -- Matsuo Basho^ _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Sun, 07 Feb 2010 01:38:08 +0900
"Benjamin L. Russell" <[hidden email]> wrote: > On Tue, 02 Feb 2010 09:16:03 -0800, Creighton Hogg wrote: > > > 2010/2/2 Álvaro García Pérez <[hidden email]> > > > >> You may try Pierce's "Basic Category Theory for Computer > >> Scientists" or Awodey's "Category Theory", whose style is rather > >> introductory. Both of them (I think) have a chapter about functors > >> where they explain the Hom functor and related topics. > >> > > > > I think Awodey's book is pretty fantastic, actually, but I'd avoid > > Pierce. Unlike "Types and Programming Languages", I think "Basic > > Category Theory..." is a bit eccentric in its presentation and > > doesn't help the reader build intuition. > > I have written an overview of various category theory books, which > you may find useful, at the following site: > > Learning Haskell through Category Theory, and Adventuring in Category > Land: Like Flatterland, Only About Categories > http://dekudekuplex.wordpress.com/2009/01/16/learning-haskell-through-category-theory-and-adventuring-in-category-land-like-flatterland-only-about-categories/ > > Hope this helps. It does. Does anybody have any opinions on Pitt, "Category Theory and Computer Science" ? Brian _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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Mark Spezzano <mark.spezzano <at> chariot.net.au> writes:
> Does anyone know what Hom stands for? 'Hom' stands for 'homomorphism' --a way of changing (morphism) between two structures while keeping some information the same (homo-). Any algebra text will define morphisms aplenty --homomorphisms, epimorphisms, monomorphisms, and the like. These are maps on groups that preserve group operations (or on rings that preserve ring operations, etc.) In a topology text, you will find information on what are called continuous functions; they're morphisms too, in disguise. You can find a thinner disguise when you look at continuously invertible continuous functions, which are called homeomorphisms. If you proceed to differential geometry, you'll see smooth maps --they're morphisms too, and the invertible ones are called diffeomorphisms. This-morphisms, that-morphisms --if you're trying to come up with a general theory that describes all of them, it's natural just to call them 'morphisms'; but, as with the word 'colonel', the word and the symbol come to us via different routes, so that 'Hom(omorphism)' survives instead as the abbreviation. The crucial point in learning category theory is the realisation that, despite all the fancy terminology, it is at heart nothing but a way of talking about groups, rings, topological spaces, partial orders, etc. --all at once, so no wonder it seems abstract! _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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On Tue, Feb 2, 2010 at 5:26 AM, Mark Spezzano <[hidden email]> wrote:
Hi all, I've looked through at least a dozen. For neophytes, the best of the bunch BY FAR is Goldblatt, Topoi: the categorial analysis of logic . Don't be put off by the title. He not only explains the stuff, but he explains the problems that motivated the invention of the stuff. He doesn't cover monads, but he covers all the basics very clearly, so once you've got that down you can move to another author for monads. -gregg _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Feb 16, 2010, at 9:43 AM, Gregg Reynolds wrote: I've looked through at least a dozen. For neophytes, the best of the bunch BY FAR is Goldblatt, Topoi: the categorial analysis of logic . Don't be put off by the title. He not only explains the stuff, but he explains the problems that motivated the invention of the stuff. He doesn't cover monads, but he covers all the basics very clearly, so once you've got that down you can move to another author for monads. He does cover monads, briefly. They're called "triples" in this context, and the chapter on interpretations of the intuitionistic logic depend on functorial/monadic techniques. If I remember correctly, he uses the techniques and abstracts from them. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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I haven't seen anybody mentioning «Joy of Cats» by Adámek, Herrlich &
Strecker: It is available online, and is very well-equipped with thorough explanations, examples, exercises & funny illustrations, I would say best of university lecture style: http://katmat.math.uni-bremen.de/acc/. (Actually, the name of the book is a joke on the set theorists' book «Joy of Set», which again is a joke on «Joy of Sex», which I once found in my parents' bookshelf... ;-)) Another alternative: Personally, I had difficulties with the somewhat arbitrary terminology, at times a hindrance to intuitive understanding - and found intuitive access by programming examples, and the book was «Computational Category Theory» by Rydeheart & Burstall, also now available online at http://www.cs.man.ac.uk/~david/categories/book/, done with the functional language ML. Later I translated parts of it to Haskell which was great fun, and the books content is more beginner level than any other book I've seen yet. The is also a programming language project dedicated to category theory, «Charity», at the university of Calgary: http://pll.cpsc.ucalgary.ca/charity1/www/home.html. Any volunteers in doing a RENAME REFACTORING of category theory together with me?? ;-)) Cheers, Nick Mark Spezzano wrote: > Hi all, > > I'm trying to learn Haskell and have come across Monads. I kind of understand monads now, but I would really like to understand where they come from. So I got a copy of Barr and Well's Category Theory for Computing Science Third Edition, but the book has really left me dumbfounded. It's a good book. But I'm just having trouble with the proofs in Chapter 1--let alone reading the rest of the text. > > Are there any references to things like "Hom Sets" and "Hom Functions" in the literature somewhere and how to use them? The only book I know that uses them is this one. > > Has anyone else found it frustratingly difficult to find details on easy-to-diget material on Category theory. The Chapter that I'm stuck on is actually labelled Preliminaries and so I reason that if I can't do this, then there's not much hope for me understanding the rest of the book... > > Maybe there are books on Discrete maths or Algebra or Set Theory that deal more with Hom Sets and Hom Functions? > > Thanks, > > Mark Spezzano. > > _______________________________________________ > Haskell-Cafe mailing list > [hidden email] > http://www.haskell.org/mailman/listinfo/haskell-cafe > > _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Thu, Feb 18, 2010 at 04:27, Nick Rudnick wrote:
I haven't seen anybody mentioning «Joy of Cats» by Adámek, Herrlich & Strecker: This book reads quite nicely! I love the illustrations that pervade the technical description, providing comedic relief. I might have to go back a re-learn CT... again. Excellent recommendation! For those looking for resources on category theory, here are my collected references: http://www.citeulike.org/user/spl/tag/category-theory Sean _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In this case, I would regard it as desirable to -- in best refactoring manner -- to identify a wording in this language instead of the abuse of terminology quite common in maths, e.g. * the definition of open/closed sets in topology with the boundary elements of a closed set to considerable extent regardable as facing to an «outside» (so that reversing these terms could even appear more intuitive, or «bordered» instead of closed and «unbordered» instead of open), or * the abuse of abandoning imaginary notions in favour person's last names in tribute to successful mathematicians... Actually, that pupils get to know a certain lemma as «Zorn's lemma» does not raise public conciousness of Mr. Zorn (even among mathematicians, I am afraid) very much, does it? * 'folkloristic' dropping of terminology -- even in Germany, where the term «ring» seems to originate from, since at least a century nowbody has the least idea it once had an alternative meaning «gang,band,group», which still seems unsatisfactory... Here computing science has explored ways to do much better than this, and it might be time category theory is claimed by computer scientists in this regard. Once such a project has succeeded, I bet, mathematicians will pick up themselves these work to get into category theory... ;-) As an example, let's play a little: Arrows: Arrows are more fundamental than objects, in fact, categories may be defined with arrows only. Although I like the term arrow (more than 'morphism'), I intuitively would find the term «reference» less contradictive with the actual intention, as this term * is very general, * reflects well dual asymmetry, * does harmoniously transcend the atomary/structured object perspective -- a an object may be in reference to another *by* substructure (in the beginning, I was quite confused lack of explicit explicatation in this regard, as «arrow/morphism» at least to me impled objekt mapping XOR collection mapping). Categories: In every day's language, a category is a completely different thing, without the least association with a reference system that has a composition which is reflective and associative. To identify a more intuitive term, we can ponder its properties, * reflexivity: This I would interpret as «the references of a category may be regarded as a certain generalization of id», saying that references inside a category represent some kind of similarity (which in the most restrictive cases is equality). * associativity: This I would interpret as «you can *fold* it», i.e. the behaviour is invariant to the order of composing references to composite references -- leading to «the behaviour is completely determined by the lower level reference structure» and therefore «derivations from lower level are possible» Here, finding an appropriate term seems more delicate; maybe a neologism would do good work. Here one proposal: * consequence/?consequentiality? : Pro: Reflects well reflexivity, associativity and duality; describing categories as «structures of (inner) consequence» seems to fit exceptionally well. The pictorial meaning of a «con-sequence» may well reflect the graphical structure. Gives a fine picture of the «intermediating forces» in observation and the «psychologism» becoming possible (-> cf. CCCs, Toposes). Con: Personalized meaning has an association with somewhat unfriendly behaviour. Anybody to drop a comment on this? Cheers, Nick Sean Leather wrote: On Thu, Feb 18, 2010 at 04:27, Nick Rudnick wrote: _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On 18 Feb 2010, at 14:48, Nick Rudnick wrote:
> * the definition of open/closed sets in topology with the boundary > elements of a closed set to considerable extent regardable as facing > to an «outside» (so that reversing these terms could even appear > more intuitive, or «bordered» instead of closed and «unbordered» > instead of open), I take "closed" as coming from being closed under limit operations - the origin from analysis. A closure operation c is defined by the property c(c(x)) = c(x). If one takes c(X) = the set of limit points of X, then it is the smallest closed set under this operation. The closed sets X are those that satisfy c(X) = X. Naming the complements of the closed sets open might have been introduced as an opposite of closed. Hans _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Doralicio
Am Donnerstag 18 Februar 2010 14:48:08 schrieb Nick Rudnick:
> even in Germany, where the > term «ring» seems to originate from, since at least a century nowbody > has the least idea it once had an alternative meaning «gang,band,group», Wrong. The term "Ring" is still in use with that meaning in composites like Schmugglerring, Autoschieberring, ... _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Hi Daniel,
;-)) agreed, but is the word «Ring» itself in use? The same about the English language... de.wikipedia says: « Die Namensgebung Ring bezieht sich nicht auf etwas anschaulich Ringförmiges, sondern auf einen organisierten Zusammenschluss von Elementen zu einem Ganzen. Diese Wortbedeutung ist in der deutschen Sprache ansonsten weitgehend verloren gegangen. Einige ältereVereinsbezeichnungen (wie z. B. Deutscher Ring, Weißer Ring) oder Ausdrücke wie „Verbrecherring“ weisen noch auf diese Bedeutung hin. Das Konzept des Ringes geht auf Richard Dedekind zurück; die Bezeichnung Ring wurde allerdings von David Hilbert eingeführt.» (http://de.wikipedia.org/wiki/Ringtheorie) How many students are wondering confused about what is «the hollow» in a ring every year worlwide, since Hilbert made this unreflected wording, by just picking another term around «collection»? Although not a mathematician, I've visited several maths lectures, for interest, having the same problem. Then I began asking everybody I could reach -- and even maths professors could not tell me why this thing is called a «ring». Thanks for your examples: A «gang» {of smugglers|car thieves} shows even the original meaning -- once knowed -- does not reflect the characteristics of the mathematical structure. Cheers, Nick Daniel Fischer wrote: Am Donnerstag 18 Februar 2010 14:48:08 schrieb Nick Rudnick:even in Germany, where the term «ring» seems to originate from, since at least a century nowbody has the least idea it once had an alternative meaning «gang,band,group»,Wrong. The term "Ring" is still in use with that meaning in composites like Schmugglerring, Autoschieberring, ... _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Am Donnerstag 18 Februar 2010 17:10:08 schrieb Nick Rudnick:
> Hi Daniel, > > ;-)) agreed, but is the word «Ring» itself in use? Of course, many people wear rings on their fingers. Oh - you meant "in the sense of gang/group"? It still appears as part of the name of some groups as a word of its own, otherwise, I can at the moment only recall its use in compounds. > The same about the > English language... de.wikipedia says: > > « Die Namensgebung /Ring/ bezieht sich nicht auf etwas anschaulich > Ringförmiges, sondern auf einen organisierten Zusammenschluss von > Elementen zu einem Ganzen. I don't know whether that's correct. It may be, but then the french "anneau" is a horrible mistranslation. > Diese Wortbedeutung ist in der deutschen > Sprache ansonsten weitgehend verloren gegangen. Einige > ältereVereinsbezeichnungen </wiki/Verein> (wie z. B. Deutscher Ring > </wiki/Deutscher_Ring>, Weißer Ring </wiki/Wei%C3%9Fer_Ring_e._V.>) oder > Ausdrücke wie „Verbrecherring“ weisen noch auf diese Bedeutung hin. Das > Konzept des Ringes geht auf Richard Dedekind > </wiki/Richard_Dedekind> zurück; die Bezeichnung /Ring/ wurde allerdings > von David Hilbert </wiki/David_Hilbert> eingeführt.» > (http://de.wikipedia.org/wiki/Ringtheorie) > > How many students are wondering confused about what is «the hollow» in a > ring every year worlwide, since Hilbert made this unreflected wording, You know, a "field" is a "Körper" in german, ("corps" in french), a "Ring" is a "Körper" with a hole in it (no division in general). > by just picking another term around «collection»? Although not a > mathematician, I've visited several maths lectures, for interest, having > the same problem. Then I began asking everybody I could reach -- and > even maths professors could not tell me why this thing is called a > «ring». That's often a problem with things that were named by Germans in the nineteenth or early twentieth century. They had pretty undecipherable ways of choosing metaphors and coming up with weird associations. > > Thanks for your examples: A «gang» {of smugglers|car thieves} shows even > the original meaning -- once knowed -- does not reflect the > characteristics of the mathematical structure. > > Cheers, > > Nick > > Daniel Fischer wrote: > > Am Donnerstag 18 Februar 2010 14:48:08 schrieb Nick Rudnick: > >> even in Germany, where the > >> term «ring» seems to originate from, since at least a century nowbody > >> has the least idea it once had an alternative meaning > >> «gang,band,group», > > > > Wrong. The term "Ring" is still in use with that meaning in composites > > like Schmugglerring, Autoschieberring, ... _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Doralicio
On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick <[hidden email]> wrote:
Goldblatt works for me.
Both have a border, just in different places.
Arrows don't refer.
Not necesssarily (for Kantians, Aristoteleans?) If memory serves, MacLane says somewhere that he and Eilenberg picked the term "category" as an explicit play on the same term in philosophy. In general I find mathematical terminology well-chosen and revealing, if one takes the trouble to do a little digging. If you want to know what terminological chaos really looks like try linguistics. -g _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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