Re: Some books on category and topos theory
On 07 Nov 2008, at 15:57, Mirek Dobsicek wrote: > > Bruno Marchal in an older post wrote: >>> Also, >>> can you elaborate a bit more on the motivation behind category >>> theory? >>> Why >>> was it invented, and what problems does it solve? What's the >>> relationship >>> between category theory and the idea that all possible universes >>> exists? >> >> >> Tim makes a very genuine remark (but he writes so much I fear that >> has >> been unnoticed!). He said: read Tegmark (Everything paper), then >> learn >> category, then read again Tegmark. Indeed I would say category >> theory has > > Bruno, which of the Tegmark's 'Everything papers' did you have in > your mind? I guess it is this one: http://space.mit.edu/home/tegmark/index.html But it looks the paper is alive and evolves. I was thinking of its diagram of mathematical structures. Category theory put "natural" order in mathematical theories. But recursion theory is a sort of obstacle. category theory works well for sort of first person recursion theory (like with realizability, typed lambda calculus/comobinators, etc.) Then category is a must for knots and geometry ... Well come back, Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Some books on category and topos theory
Bruno Marchal in an older post wrote: >> Also, >> can you elaborate a bit more on the motivation behind category theory? >> Why >> was it invented, and what problems does it solve? What's the relationship >> between category theory and the idea that all possible universes exists? > > > Tim makes a very genuine remark (but he writes so much I fear that has > been unnoticed!). He said: read Tegmark (Everything paper), then learn > category, then read again Tegmark. Indeed I would say category theory has Bruno, which of the Tegmark's 'Everything papers' did you have in your mind? > emerged from the realisation that mathematical structures are themselves > mathematically structured. Categorist applies the every-structure principle > for each structure. Take all groups, and all morphism between groups: you > get the category of groups. It is one mathematical structure, a category > (with objects = groups and arrows = homomorphism) which, in some sense > capture the essence of group. Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Some books on category and topos theory
Title: Re: Some books on category and topos theory Tim May wrote: Whether knots are the key to physics, I can't say. [...] Knots are the key to (quantum) entanglement. s.
Re: Some books on category and topos theory
Title: Re: Some books on category and topos theory At 12:24 -0700 9/07/2002, Tim May wrote: Whether knots are the key to physics, I can't say. Certainly there are suggestive notions that particles might be some kind of knots in spacetime (of some dimensionality)... Interesting! Chromosomes make knots, also. But my reading of Louis Kauffman, a great enthusiast and a great pedagogue, makes me believe that knots could be much more: -Knots could be "type" of multiverse! (multi-histories "skeleton"). Much more: -Invariant of knots could be quantum statistical abstract physical theories! As you know perhaps I get a proof that with the comp hyp, the mind-body problem is partially reduced into a derivation of physics from machine's psychology: poetically: physical realities is a web of infinite machine dreams(*). Machine's psychology can be approximated (at least) by the many possible intensionnal (modal) variant of the Godel Lob logic of Self-reference (restricted to the \Sigma_1 arithmetical sentences (= genuily accessible by the Universal Dovetailer when true)). This gives the Z logics. Normaly the Z logics should give ... a skeleton of physical theories. Knots will perhaps provide the missing link. Knots would give a shortcut between comp psychology and quantum physics (or quantum psychology?). Toposes and monoidal/braided categories are most probably important part of the big drama in the middle term run. Aaaah... knots are so cute, also :-) (I'm afraid I am not vaccinated against the mathematical sirens, perhaps) -Bruno (*) IN case you are interested, here are relevant links to a step by step version of the basic uda proof (conversation with Joel Dobrzelewski): UDA step 1 http://www.escribe.com/science/theory/m2971.html UDA step 2-6 http://www.escribe.com/science/theory/m2978.html UDA step 7 8 http://www.escribe.com/science/theory/m2992.html UDA step 9 10 http://www.escribe.com/science/theory/m2998.html UDA last question http://www.escribe.com/science/theory/m3005.html Joel 1-2-3 http://www.escribe.com/science/theory/m3013.html Re: UDA... http://www.escribe.com/science/theory/m3019.html George'sigh http://www.escribe.com/science/theory/m3026.html Re:UDA... http://www.escribe.com/science/theory/m3035.html Joel's nagging question http://www.escribe.com/science/theory/m3038.html Re:UDA... http://www.escribe.com/science/theory/m3042.html
Re: Some books on category and topos theory
On Tuesday, July 9, 2002, at 11:08 AM, Bruno Marchal wrote: > > Me too. Now, I feel almost like you about ... knot theory. > And this fit well with your cat-enthusiasm, for knot theory is > a reservoir of beautiful and TOE-relevant categories > (the monoidal one). I've just > ordered Yetter's book: functorial(*) knot theory. It is the number 24 > of Kauffman series on Knots and Everything (sic) at World > Scient. Publ Co. A series which could be a royal series for this > list ... > May I recommand the n° 1, by Louis Kauffman himself: knots and physics? > A must for the (quantum) toes, and (I speculate now) the comp toe too! I've looked at some of the knot series books, but have put them off for now. A good book to prepare for these books is Colin Adams, "The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots," 1994. Whether knots are the key to physics, I can't say. Certainly there are suggestive notions that particles might be some kind of knots in spacetime (of some dimensionality)...a lot of people have played with knots, loops, kinks, and braids for the past century. One thing that Tegmark got right, I think, is the notion that a lot of branches of mathematics and a lot of mathematical structures probably go into making up the nature of reality. This is at first glance dramatically apposite the ideas of Fredkin, Toffoli, Wheeler1970, and Wolfram on the generation of reality from simple, local rules. Wolfram has made a claim in interviews, and perhaps somewhere in his new book, that he thinks the Universe may be generated by a 6-line Mathematica program! However, while I am deeply skeptical that a 6-line Mathematica program underlies all of reality, enormous complexity, including conceptual complexity, can emerge from very simple rules. A very simple example of this is the game of Go. From extremely simple rules played with two types of stones on a 19 x 19 grid we get "emergent concepts" which exist in a very real sense. For example, a cluster of stones may have "strength" or "influence." Groups of stones develop properties which individual stones don't have. Abstraction hierarchies abound. The Japanese have hundreds of names for these emergent, higher-order structures and concepts. All out of what is essentially a cellular automata. So even if our universe is a program running as a screen saver on some weird alien's PC, all sorts of complexity can emerge. Getting down to earth, most of this complexity is best seen as mathematics, I think. I expect to take a closer look at knots after I get more math under my belt. > Now I know the z logics really should have "tensorial semantics", > sort of many related (glued) von neumann type of logics (which are > themselves atlases of boolean logics). > But where (in Zs logic) those damned tensorial categories come from??? > Knots gives hints!!! This would explain the geometrical appearance > of realities. > > Bruno > > (*) For the other: "functorial" really means categorial. Functors are > the morphisms between categories. The first chapter of Yetter's book > is an intro to category theory, the second one, on Knot theory, ... Exciting stuff. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Some books on category and topos theory
At 9:24 -0700 9/07/2002, Tim May wrote: >Reading styles differ, but I have come to favor the "hawk spiral." I >see hawks spiralling in the thermals near my house, and this is how >I like to learn. I read something from one book, think, read from >another, think, try to compare what the authors are saying, read >from another, go back to the earlier book and read more, and so on. >A helix, covering the same material many times. I have always read books in this way (except courses and novels). In each field I'm interested in I have master books I read and reread and satellites I consult and reconsult. Sometimes I am not sure if I exist only but as a slave of dormant ideas in books manipulating me for spreading in some ways ... My books are like butterflies jumping from tables to sofas, following me everywhere trying to exchange ideas, linking notions, etc. I am only an humble servitor ... :) >And much of what Tegmark outlines in his large chart can be >dramatically simplified and abstracted. And we need to do that! because we are distributed in the tegmark big structure in such a way that from our local views, the global accessible view is mathematically more rich than the big structure! (well more on that later). >(Aside: I believe this is a big part of what thinking is about: >applying thoughts/concepts/morphisms/etc. from one area to another. >Perhaps category theory will push AI in new ways. Part of it, sure. Most "explanations" are morphism in known structures. >Everything seemed directly related to problems which had fascinated >me for decades. Some of these issues I hope I have hinted at here. > >It was almost as if category and topos theory had been invented just >for mean exaggeration, but it captures my sense of wonder. I >haven't been this excited about a new area in more than a decade. I >expect I'll be doing something in this area for at least the _next_ >decade. > >My apologies if this explanation of enthusiasm is too personal for >you the reader, but I think enthusiasm is a good thing. Me too. Now, I feel almost like you about ... knot theory. And this fit well with your cat-enthusiasm, for knot theory is a reservoir of beautiful and TOE-relevant categories (the monoidal one). I've just ordered Yetter's book: functorial(*) knot theory. It is the number 24 of Kauffman series on Knots and Everything (sic) at World Scient. Publ Co. A series which could be a royal series for this list ... May I recommand the n° 1, by Louis Kauffman himself: knots and physics? A must for the (quantum) toes, and (I speculate now) the comp toe too! I knew Yetter's work a long time ago when I read his paper on the semantics on non commutative girard linear logic. Unfortunately, later, the Z logics gave me a (weakening) of quantum logic (the von neumann one), which Yetter dismisses in that paper, so I did dismiss Yetter ... Now I know the z logics really should have "tensorial semantics", sort of many related (glued) von neumann type of logics (which are themselves atlases of boolean logics). But where (in Zs logic) those damned tensorial categories come from??? Knots gives hints!!! This would explain the geometrical appearance of realities. Bruno (*) For the other: "functorial" really means categorial. Functors are the morphisms between categories. The first chapter of Yetter's book is an intro to category theory, the second one, on Knot theory, ...
Re: Some books on category and topos theory
On Tuesday, July 9, 2002, at 07:41 AM, Bruno Marchal wrote: > > Tim makes a very genuine remark (but he writes so much I fear that has > been unnoticed!). True enough...I write a lot! (The old joke applies: "I don't have enough time to write a short letter.") > He said: read Tegmark (Everything paper), then learn > category, then read again Tegmark. Well, I didn't actually say "then learn category (theory). I said spend enough time looking at category theory to get the gist of what they are. A couple of days, for example. Reading styles differ, but I have come to favor the "hawk spiral." I see hawks spiralling in the thermals near my house, and this is how I like to learn. I read something from one book, think, read from another, think, try to compare what the authors are saying, read from another, go back to the earlier book and read more, and so on. A helix, covering the same material many times. > Indeed I would say category theory has > emerged from the realisation that mathematical structures are themselves > mathematically structured. Categorist applies the every-structure > principle > for each structure. Take all groups, and all morphism between groups: > you > get the category of groups. It is one mathematical structure, a category > (with objects = groups and arrows = homomorphism) which, in some sense > capture the essence of group. Exactly. A very nice explanation. And much of what Tegmark outlines in his large chart can be dramatically simplified and abstracted. Crane, Baez, Dolan, and others call this the "categorification" process. Robert Geroch's textbook, "Mathematical Physics," uses categories and functors throughout as a unifying (and intuition-increasing) tool. Hey, let me be very clear about something: I don't know what the categorification of Tegmark's ideas are! Categories and toposes are not a magic bullet. But I know that gettting lost in the swamps of mathematical structures is a real danger, and that mathematicians have found certain unifying symmetries, structures, parallels which simplify things dramatically. Category theory is a lot like finding metaphors and parallels. (We use the term "isomorphism" almost in everyday language, so the leap to all sorts of morphisms is not great.) > > Categories arises naturally when mathematician realised that many proofs > looks alike so that it is easier to abstract a new > structure-of-structures, > then makes proofs in it, then apply the abstract proof in each structure > you want. So they define universal constructions in category (like the > "product"), which will correspond automatically to >- "and" in boolean algebra >- "and" in Heyting algebra >- group product in cat of groups >- topological product in cat of topological spaces >- Lie product in cat of Lie groups, etc. > So Category theory helps you to make a big economy of work ... once you > invest in it, if you are using algebra. It saves your time. Exactly. Another good explanation here. And it's more than just a notational convenience. Proofs in one area, such as some branch of topology, can be transformed into proofs in other areas. (Aside: I believe this is a big part of what thinking is about: applying thoughts/concepts/morphisms/etc. from one area to another. Perhaps category theory will push AI in new ways. Perhaps the "frame problem" will be solvable with new tools.) > But, to come back to Tim remark, it hints that a giant part whole of > mathematics is naturally mathematically structured, and this should be > taken > into account. I first heard of category theory about 10 years ago. A friend of mine was working for a company in Palo Alto which was using category theory to model economic data bases (such as petroleum reserves, ports in different countries, etc. ...very probably CIA-related, now that I think about it). He didn't have the interest or insight to explain why category theory was so cool. I asked a mathematician friend of mine (Eric H., for Hal and WD) about it and he said it was about what mathematicians do when they draw diagrams on blackboards. It didn't sound very interesting. It sounded like some variant of denotational semantics. But when the light bulbs went off this spring, when I dug into the writings of Baez, Hillman, Markopoulou, and the books of Lawvere, Mac Lane (difficult), and others, I had a major epiphany, a real "Ah ha!" experience. Everything seemed directly related to problems which had fascinated me for decades. Some of these issues I hope I have hinted at here. It was almost as if category and topos theory had been invented just for mean exaggeration, but it captures my sense of wonder. I haven't been this excited about a new area in more than a decade. I expect I'll be doing something in this area for at least the _next_ decade. My apologies if this explanation of enthusiasm is too personal for you the reader, but I think enthusiasm is a good thing
Re: Some books on category and topos theory
Wei Dai asks some question to Tim May which I would like to comment taking into account some other posts. Wei Dai: >Suppose I had the time for only one book, which would you recommend? I think you (Wei) decide to look for the book by Lawvere. Good choice but you should know it is just an introduction. Now, that book is useful even for learning algebra. Some who knows algebra (i.e. groups, rings, fields, topological spaces and more importantly their morphism) could look for more advanced material perhaps. >Also, >can you elaborate a bit more on the motivation behind category theory? Why >was it invented, and what problems does it solve? What's the relationship >between category theory and the idea that all possible universes exists? Tim makes a very genuine remark (but he writes so much I fear that has been unnoticed!). He said: read Tegmark (Everything paper), then learn category, then read again Tegmark. Indeed I would say category theory has emerged from the realisation that mathematical structures are themselves mathematically structured. Categorist applies the every-structure principle for each structure. Take all groups, and all morphism between groups: you get the category of groups. It is one mathematical structure, a category (with objects = groups and arrows = homomorphism) which, in some sense capture the essence of group. Note that the category of groups is too large to be defined in Zermelo Fraenkel set theory (as almost any so called large category). The usual trick of categorist is to mention the Von Neumann Bernay Godel theory of set which has classes (collection of sets which are not sets themselves). A much modern view is to make category theory in a well chosen topos! Note that some common mathematical structure *are* categories. A group is a category (with one object: the set of group elements, and the arrows are the action of the group on itself). Partially ordered set (set with a transitive, reflexive, and antisymmetric relation on it) give other simple exemples: object of the category are the element of the partially ordered set, and the unique (here) arrow between object is the order relation). Automatically Boolean Algebra, but also Heytingian lattices, etc... are categories. Of course just a set can be made into a trivial category. Other categories lives in between groups and lattices. They have lot of objects and lot of arrows between objects. Categories arises naturally when mathematician realised that many proofs looks alike so that it is easier to abstract a new structure-of-structures, then makes proofs in it, then apply the abstract proof in each structure you want. So they define universal constructions in category (like the "product"), which will correspond automatically to - "and" in boolean algebra - "and" in Heyting algebra - group product in cat of groups - topological product in cat of topological spaces - Lie product in cat of Lie groups, etc. So Category theory helps you to make a big economy of work ... once you invest in it, if you are using algebra. It saves your time. But, to come back to Tim remark, it hints that a giant part whole of mathematics is naturally mathematically structured, and this should be taken into account. Also, as I have explained before, the whole of math cannot be entirely mathematically structured in any consistent way (that's too big). This can be shown with logic, but categories can give you a concrete feeling of the bigness and endlessness of such an enterprise. Another motivation for category is the realisation that elements of structures are not necessary for defining those structures. Objects behavior are defined (up to isomorphism) by their relationship (arrows) with other objects. That's a sort of functional or relational philosophy not so different from comp. As exercice you could try to define injection and surjection between sets without mentioning the elements! > >Does it help understand or formalize the notion of "all possible >universes"? I don't think categories can help in any direct way, although I doubt indeed we can live without categories (nor without logic-modalities) in the long run. You could try to define the category of (multi)universes. What would be a morphism between universes? Note that lawvere has try to provide math foundation through the category of all categories (with functors as arrows) but this has not succeed. He discovers toposes instead. Note that categories are difficult to marry with ... recursion theory. (Despite so-called Dominical Categories, which does the job, but that is too heavy math for me ...). >I know in logic there is the concept of a categorical theory >meaning all models of the theory are isomorphic. Does that have anything >to do with category theory? Not really. It is one of the reason it is better to use the adjective categorial in algebra and categorical in logic. But not all scientist follow this custom. Bruno
Re: Some books on category and topos theory
Hi Tim, it's really interesting to see you here. (For those who don't know, I knew Tim from the cypherpunks mailing list. Hal Finney was an active member of the list as well. See http://www.activism.net/cypherpunk/crypto-anarchy.html if you're wondering what a cypherpunk is.) Two of the most prominent cypherpunks I know are now on my Everything mailing list. I wonder what that means... Anyway, welcome! I remember your post on the cypherpunks list about category theory, but I have to admit I didn't pay it much attention since it didn't seem very relevent at the time. I guess this is my second chance to learn about category theory, so there are some questions for you. Suppose I had the time for only one book, which would you recommend? Also, can you elaborate a bit more on the motivation behind category theory? Why was it invented, and what problems does it solve? What's the relationship between category theory and the idea that all possible universes exists? Does it help understand or formalize the notion of "all possible universes"? I know in logic there is the concept of a categorical theory meaning all models of the theory are isomorphic. Does that have anything to do with category theory?
Re: Some books on category and topos theory
On Friday, July 5, 2002, at 01:16 PM, Tim May wrote: > The category and topos theory books I actually _own_ (bought through > Amazon) are: > > Oops! I left out one of the most important and accessible of the books I have and recommend: * McLarty, Colin, "Elementary Categories, Elementary Toposes," 1992. An intermediate-level, moderate-length book. Covers a lot of interesting material. Here's what Baez says: "3) John Baez, Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html and then try the books I recommended in "week68", along with this one: 4) Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, Oxford, 1992. which I only learned about later, when McLarty sent me a copy. I wish I'd known about it much sooner: it's very nice! It starts with a great tour of category theory, and then it covers a lot of topos theory, ending with a bit on various special topics like the "effective topos", which is a kind of mathematical universe where only effectively describable things exist - roughly speaking. " (end of Baez comments) By the way, the Web is a great resource for finding online books. Barr and Wells, who Bruno referred to, have put an updated version of their book "Toposes, Triples and Theories" online in PDF form. Search for it in the usual way. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
