November 2000 - Categories - categories.org.au

Re: Categories ridiculously abstract
by John Duskin 06 Dec '00

06 Dec '00

Readers of the cat list may be interested in the one meaningful post to Slate's "The Fray" in reply to Holt's MSRI "Diary" article. It was made by David Yetter: category theory David Yetter 28 Nov 2000 20:29 It is sad more than a decade on since the proof of the remarkable categorical coherence theorem of Shum that mathematicians can continue to view category theory as a mere linguistic convention or useless abstraction. Shum's theorem shows that axioms completely natural from the internal dynamic of category theory completely characterize framed tangles, relative versions of the framed knots and links which are central to smooth topology in 3 and 4 dimensions (notice the dimensionality of space and of space-time: hardly divorced from meaning.) Other categories satisfying the same axioms include the categories of representations of quantum groups, physically motivated algebraic structures which have become central objects of study for mathematicians from many old branches of mathematics. Indeed, Shum's theorem, a theorem of category theory, is the only really satisfying explanation for the intimate connection between quantum groups and low-dimensional topology.

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Categories - too abstract?
by baez＠newmath.UCR.EDU 30 Nov '00

30 Nov '00

Michael Barr write: > I don't think one should blame the guy whose remarks Peter quoted. He is > not a mathematician and presumably knows nothing more than some college > level mathematics. He has picked up that attitude from the high-powered > mathematicians that inhabit places like MSRI (and the CRM, Fields Inst., > and PIMS in Canada). Ignoring the fact that category theory was fathered > by two of the most eminent mathematicians of the last century and > god-fathered by arguably the very greatest, they still go around saying > that it is without content and nothing but meaningless abstraction. When I am confronted with this attitude, I sometimes use a couple of arguments in addition to the usual one of listing things that category theory is good for. I present them here in a rather blunt form; they can be toned down as politeness dictates. 1) "You say category theory is `too abstract'. But if you don't like abstraction, why in the world are you doing mathematics? Maybe you should be in finance, where the numbers all have dollar signs in front of them. Complaining that a piece of mathematics is `too abstract' is a bit like saying that the ocean is `too wet'." 2) "You say category theory is `too abstract'. Good! Don't learn it! That way, I'll make progress in my research faster than you." > And here is a question: are categories more abstract or less abstract > than sets? Indeed, what we often take as "greater abstraction" is really unfamiliarity. The real answer to the problem of people thinking categories are "too abstract" is to keep explaining category theory and how it's useful in a wide variety of problems... until people get used to it. Physicists used to think groups were too abstract! John Baez

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category of fraction and set-theoretic problem
by Philippe Gaucher 30 Nov '00

30 Nov '00

Bonjour, I have a general question about localizations. I know that for any category C, if S is a set of morphisms, then C[S^-1] exists. And moreover if C is small, then C[S^{-1}] is small as well (as in the Borceux's book Handbook of categorical algebra I) If S is not small, and if we suppose that all sets are in some universe U, then the previous construction gives a solution as a V-small category for some universe V with U \in V (the objects are the same but the homsets need not to be U-small). So it does not work if one wants to get U-small homsets. Another way is to have a calculus of fractions (left or right) and if S is locally small as defined in Weibel's book "Introduction to homological algebra". But in my case, the Ore condition is not satisfied. Hence the question : is there other constructions for C[S^{-1}] ? Thanks in advance. pg.

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Monads without unit?
by Jason C Reed 30 Nov '00

30 Nov '00

I had been looking at the standard way the topological closure A |-> CA can be recovered from the derived set operation A |-> A' (which takes A to the set of the accumulation points of A) by setting CA := A' u A, and observed that it can be generalized to categorical language. If you take thing made up of a coproduct-preserving functor D and a natural transformation t : D^2 -> D, such that t is `associative' in the usual sense of asserting that t o t_D = t o Dt, and also p_{X,Y} o t_{X + Y} = (t_X + t_Y) o p_{DX,DY} o Dp_{X,Y} where p is the canonical natural transformation D(X + Y) -> DX + DY , then if one defines TX := DX + X eta_X := inr_{DX + X} mu_X := ([1_{DX},1_{DX}] + X) a_{DX,DX,X} (([t_X,1_{DX}] o p_{DX,X}) + TX) (where [ , ] is copairing and a_{X,Y,Z} is the associativity isomorphism X + (Y + Z) -> (X + Y) + Z) then turns out that (T, eta, mu) is a monad. Has anyone already studied these? I'd be particularly interested in a type-theoretic or logical (along the lines of the correspondence between monads and modal operators) interpretation of such a structure, if any. ---Jason

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Re: RFN (Request for Notation)
by Paul Levy 29 Nov '00

29 Nov '00

> > (3) I badly need a good name for the sets > Nat_k = {n \in Nat | n < k } > These are widely used and I am surprised that > there is no satisfactory name in wide-spread use. > These are NOT the sets Z_k = Z mod k, > although the Nat_k form a system of distinct, > canonical representatives for the Z_k. > These are the set of array indices in computer > languages such as C and SML. In this use, the > Nat_k have nothing whatsoever to do with Z_k > and I certainly do not want to confuse the students! > I agree that there is a need for a standard name, and that Nat_k would be a confusing name. I have been calling this set $k for some time but I am happy to change my macro if there is some other accepted name or if $ has some other mathematical meaning. More generally, if k is an ordinal, one writes $k for the set of ordinals less than k, the canonical well-ordered set of order-type k. (The traditional ZF definition of ordinal makes k equal to $k, but that is just an implementation.) A similarly useful terminology for arrays and the like is "obaz", which indicates the use of the "ordinals begin at zero" convention. Thus you can refer to the cell with index 7 in your array as the obaz 7th cell instead of as the 8th cell. You can't call it the 7th cell, without qualification, because in English the obao ("ordinals begin at one") convention is the established default, unfortunately. As an example, today is the obaz 28th day of the obaz 10th month of the year obaz 1999, which is the final year of the obaz 19th century and not the obaz zeroth year of the obaz 20th century as many obaoists mistakenly believe. Though I'm hardly the zeroth person to point this out (obaz). Warning: this usage may alienate the less tolerant of your friends. Regards Paul -- Paul Blain Levy Computer Science Department, Boston University http://www.dcs.qmw.ac.uk/~pbl/ If language were arbitrary, it wouldn't be interesting.

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Re: David Benson's questions on terminology
by Robert J. MacG. Dawson 29 Nov '00

29 Nov '00

Paul Taylor wrote: > > (1) I would say (rather strongly) that it is ill-conceived to > try to generalise the successor relation from the natural numbers > to arbitrary partial orders. The successor relation is an aspect > of the inductive/recursive/well founded structure on N, and it > is wrong to confuse well founded relations (which are necessarily > IRreflexive) with partial arders (which are Reflexive). > > See Sections 2.7, 3.1 and elsewhere in "Practical Foundations". I don't think David was trying to generalize the successor relation in the sense of finding a "moral equivalent" in a poset for the natural numbers' successor _function_. All he wants - I think - is a notation for "a > b and there is no a>c>b". I would suggest using an indefinite article with a noun formation: " a is _a_ successor of b" or a prepositional formation that does not connote uniqueness or necessary existence: "a is immediately above b" Bob Pare and I used "<!" for this in our 1993 paper on tileorders. It may be - is this what you're getting at, Paul? - that if one finds a successor relation is natural or useful for what one's looking at, then one should wonder hard about whether it would be better thought of as a well-founded structure rather than as a poset, so as to avoid the repetition of "and not equal to". But there are certainly cases where after the wondering one would conclude "no it isn't." -Robert Dawson

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More about names and notation
by David B. Benson 28 Nov '00

28 Nov '00

Dear Colleagues, Thank you for the responses! (1) Several pointed out that my <<followers>> relations are said to be <<covering>> relations in combinatorics. These relations are defined as relations which are 1. acyclic 2. without interpolants, as Robert Dawson mentioned. A followers relation for discrete partial order R is the least relation whose reflexive and transitive closure is R. This agrees entirely with the definition of a successor relation being the least relation whose transitive closure is a discrete strict total order and whose reflexive and transitive closure is a discrete total order. In the case of successor, there is a distinct next element, as in the succession of the Kings (and Queens) of England. In the case of followers, there is in general a set of next elements, such as the followers of Cromwell. (2) While several alternatives were offered, I will try Paul Taylor's <<instance>> of a relation to describe an ordered pair (x,y) \in R. (3) The problem of a good name for the sets Nat_k remains. Suggestions included finite ordinals numerals order ideals and by far the most appropriate for my purposes, index sets [This problem of a good name is worthy of some further attention. Computer science second year students will think of <<finite ordinals>> as a number, not a set, of <<numerals>> as representations of the digits in a number system, and have not be exposed to order ideals.] Cheers, David -- Professor David B. Benson (509) 335-2706 School of EE and Computer Science (EME 102A) (509) 335-3818 fax PO Box 642752, Washington State University dbenson(a)eecs.wsu.edu Pullman WA 99164-2752 U.S.A.

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ridiculously abstract
by Peter Freyd 28 Nov '00

28 Nov '00

If you take a look at http://slate.msn.com/diary/00-11-27/diary.asp?iMsg=1 you will find (in a column by Jim Holt) that "Category theory is a ridiculously abstract framework that takes all the meaning out of mathematics." At the end of the column is an invitation to make comments. (Please don't!) Only one such comment seems to be about category theory and it's this beauty: Subject: category theory From: John Rooney Date: 28 Nov 2000 09:49 Category theory was not so bad when the Poles invented it. It was sort of like reinventing the parts of speech or the distinction between accident and substance. Later it was like subject and predicate, noun and verb. The Dutch logicians have made it quite obscure. Here's to better understanding of category theory and a less defeatist attitude. (This almost makes sense if you substitute "categorial grammar" for "category theory". But if you do that you then have to turn Jim Lambek into a Pole.) Here's the full Holt column. I'm certain that any attempt to counter the statement about category theory -- given the mood of the piece -- can only fail. DIARY: Infiltrating the Mathematicians' Lair. Jim Holt is a member of the Mathematical Sciences Research Institute at the University of California, Berkeley, and a columnist for Lingua Franca. His book on the history of the infinitesimal, Worlds Within Worlds, will be published next fall. Posted: Monday, Nov. 27, 2000, at 4:00 p.m. PT Being at MSRI is a bit like going to heaven without all the bother and expense of dying. I don't mean the sort of heaven where you wear ermine and eat foie gras to the sound of trumpets. I mean the sort where you spend your days languidly communing with beautiful, timeless, abstract ideas: Platonic heaven. MSRI stands for the Mathematical Sciences Research Institute. It is the premier think tank in the world for pure mathematics. Even its location is heavenly: It is housed in a Corbusian glass-and-wood structure perched atop the loftiest of the hills above the University of California at Berkeley, just below the ionosphere. From my office window, I gaze down upon the skyscrapers of San Francisco, the isle of Alcatraz, the Golden Gate Bridge, the Pacific Ocean. In a few minutes I will leave my office, traverse some pristine white hallways, and join a hundred of the most eminent mathematicians from around the world in a commodious lecture room. Today's topic for contemplation: the linear p-adic group, the p-adic Galois group, and the p-affine Schur algebra. But wait. I am not a mathematician (although I have sometimes pretended to be one on NPR). I am a "trivial being," to use Paul Erdos' term for those who are not among the mathematical elect. So what am I doing in this Platonic heaven? I am here as a journalist in residence. My mandate is to convey a little of the flavor of what goes on in these ethereal precincts to my fellow trivial beings back in the material world. I also cannot help thinking of myself as an anthropologist, living among an alien tribe and observing their often strange folkways. I must be careful not to give them measles. How can I blend into this august tribe? As a longtime mathematical dilettante, I sometimes understand a little of what they are saying. I also do a good bit of faking. Luckily I have come up with a set of all-purpose trick questions that have kept my ignorance from being exposed in many a treacherous conversation. For example: "Can that result be restated in terms of category theory?" (Category theory is a ridiculously abstract framework that takes all the meaning out of mathematics.) "Isn't the constant in that equation suspiciously close to the square root of pi divided by e cubed?" "Wasn't your theorem prefigured in the work of Euler?" (Leonhard Euler, who lived in the 18th century, was the most prolific mathematician in history; nearly everything is prefigured in his work.) "But can you prove that lemma for the case of n=3?" By the shrewd use of such feints, I, a trivial being, have been able to chat as an apparent peer with many of my colleagues at MSRI. Above all, I am careful not to let conversations about things like p-adic Galois groups go on for too long. When skating over thin ice, speed is your ally.

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RFN (Request for Notation)
by David B. Benson 27 Nov '00

27 Nov '00

Dear Colleagues, I am preparing notes for a sophomore (second year) course, and I am finding several places wherein I can find no standard notation. Your suggestions will be most appreciated. (1) Everybody knows that {(n,n+1) | n \in Z} is the <<successor>> relation. What to call the corresponding idea when the underlying order is only a partial order? I am currently using <<followers>> as in phrases such as ``...the followers relation for the subsets of the finite set {a, b, c}...'' in which ({a},{a,b}) is in the followers relation, but {{a},{a,b,c}) is not, anymore that (n,n+2) is in the successor relation. Is there a better word than <<followers>>? (2) I need a snappy name for an order pair in a relation R. The books I have seem to just say ``...the ordered pair (x,y) in relation R...'' The problem is that there are many uses of ordered pairs, and this is a specific use, a description of the fact that x is R-related to y by the fact that (x,y) \in R. The word ``association'' will not do as this has other meaning in computer science. I am considered <<relator>> for an order pair in a relation, but have the impression that this word has been used for other purposes in the literature. (3) I badly need a good name for the sets Nat_k = {n \in Nat | n < k } These are widely used and I am surprised that there is no satisfactory name in wide-spread use. These are NOT the sets Z_k = Z mod k, although the Nat_k form a system of distinct, canonical representatives for the Z_k. These are the set of array indices in computer languages such as C and SML. In this use, the Nat_k have nothing whatsoever to do with Z_k and I certainly do not want to confuse the students! Thank you in advance for any and all suggestions, David -- Professor David B. Benson (509) 335-2706 School of EE and Computer Science (EME 102A) (509) 335-3818 fax PO Box 642752, Washington State University dbenson(a)eecs.wsu.edu Pullman WA 99164-2752 U.S.A.

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Visit to Bangor of Eric Goubault
by Ronnie Brown 27 Nov '00

27 Nov '00

Dr E Goubault (ENS, Paris; Commission Energie Atomique) will visit the School of Informatics, University of Wales, Bangor, UK, on Dec 19, 20 2000 and give two talks, the second of which will be more mathematical: Tuesday, Dec 19, 4:00pm, Room 149 Geometric semantics for the analysis of concurrent and distributed software ABSTRACT: This talk will be focused on applications of geometrical ideas and modeling to computer scientific problems, among which: - relationship with other models for concurrency (for instance transition systems and Petri nets). - static analysis of concurrent programs (mostly Java-like threads); in particular, deadlocks, reachable states, scheduling properties (joint work with Martin Raussen and Lisbeth Fajstrup), state-space reduction techniques. - scheduling properties of distributed systems; in particular serialisability conditions for databases (after Jeremy Gunawardena and Martin Raussen), computability in fault-tolerant systems (after Maurice Herlihy and Sergio Rajsbaum). Wednesday, Dec 20th 11:15 pm, Room S5 Recent developments of "geometric concurrency theory" ABSTRACT: In this talk, I will present some of the "geometrical theories" used for modeling concurrent systems. Among these are: - the (di-)topological approaches where the flow of concurrent executions is modeled by a topological space of states plus local partial orders on it. This is very much related to the "progress graphs" of E. W. Dijkstra. I will also outline some links with similar objects introduced in domain theory. Some of the algebraic topology which is useful for describing interesting (scheduling) properties of concurrent systems will be described (many results from Martin Raussen and Lisbeth Fajstrup, Aalborg University). I will also refer to recent work by Stefan Sokolowski, Gdansk University. - the semi-cubical set approach which is in some way a combinatorial counterpart of the topological approach. This is very much related to ordinary transition systems semantics. This was described first by Vaughan Pratt and Rob van Glabbeek, Stanford University. - the omega-categorical approach (cubical and/or globular of course) initiated also by Vaughan Pratt. Most of this work has been developped by Philippe Gaucher, IRMA, Strasbourg University. I will add some recent developments (joint work with Philippe Gaucher), and also some other "applications" of these ideas (in Logics, joint work with Jean Goubault-Larrecq). All are welcome. It is hoped the above schedule will allow good time for discussion. Enquiries to Ronnie Brown r.brown(a)bangor.ac.uk

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