August 2009 - Categories - categories.org.au

Research positions
by Maria Manuel Clementino 26 Feb '14

26 Feb '14

5-YEAR RESEARCH POSITIONS The Centre for Mathematics of the University of Coimbra (CMUC) invites applications for five 5-year research positions in all areas of Mathematics. Applicants should have a PhD in Mathematics and at least three years of post-doctoral experience, with an international-level publication record and demonstrated ability to perform independent research. Preference will be given to candidates working in one of the areas in which the members of CMUC are currently active. A very good command of English is required. The appointments will be made for three years, renewable for two more. Successful applicants are expected to develop research within the areas of interest of CMUC. They should also contribute to graduate studies, supervising students from the Department of Mathematics. The basic salary is equivalent to the index 195 of the Portuguese research career (in the order of 42 500 euros a year), in accordance with the Ciência 2008 guidelines http://alfa.fct.mctes.pt/apoios/contratacaodoutorados/edital2008 Benefits include private health insurance and social security. Applicants should send • cover sheet (available from www.mat.uc.pt/cmuc/coversheet.pdf) including names and contact information of three persons who can provide a letter of recommendation • curriculum vitae (publication list included) • a 5-year research proposal (2 pages max), emphasizing their own research goals as pdf files to cmuc(a)mat.uc.pt. The subject of the e-mail message must include the job reference C2008 FCTUC CMUC v15-v19. The deadline for applications is *September 10, 2008.* ------------------------------ More information can be found at www.mat.uc.pt

1 1

separable locale
by Thomas Streicher 17 May '13

17 May '13

Recently rereading Fourman's "Continuous Truth" I came across the term "separable locale" but could nowhere find an explanation. Does it mean a cHa A for which there exists a countable subset B such that ever a in A is the supremum of those b in B with b leq a. This would be the point free account of "second countable", i.e. having a countable basis. Of course, second countable T_) spaces are separable, i.e. have a countable dense set. Is this reading the "usual" one? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

5 4

Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

2 1

typed lambda calculus:cartesian closed categories::sorted pi calculus:?
by Mike Stay 05 Sep '09

05 Sep '09

As far as I can tell, the major difference between lambda calculus and pi calculus is that the rewrite rules in pi calculus aren't confluent, so it doesn't make a lot of sense to consider rewrite-equivalence classes of terms. What kind of category (or bicategory) does sorted pi calculus give? -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2 3

Wikibook
by Bodo Pareigis 30 Aug '09

30 Aug '09

Hi John, Moss Sweedler and I once tried to collect some of the facts on generators and cogenerators for Hopf algebras and the like in On Generators and Cogenerators. (with M. Sweedler) Manuscr. Math. 2 (49-66) 1970. Best wishes (from Oceanside) Bodo -- Prof. emer. Dr. Bodo Pareigis Dept. of Mathematics University of Munich pareigis(a)math.lmu.de [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

Novemberfest
by Steve Awodey 29 Aug '09

29 Aug '09

************************************************************************ ***** Category Theory NovemberFest Carnegie Mellon University, Pittsburgh Saturday - Sunday, November 14-15, 2009 ************************************************************************ ***** We invite you to join us in Pittsburgh this Fall for a weekend meeting in Category Theory, a translation of the traditional OctoberFest. A more formal announcement will be sent in early September; at this time we just want to allow you to arrange your schedule to include a visit to Pittsburgh if you wish to participate. The meeting will be held at Carnegie Mellon University. Steve Awodey Dana Scott ************************************************************************ ****** awodey(a)cmu.edu [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

The Wikibook on Category Theory
by Charles Wells 24 Aug '09

24 Aug '09

The Category Theory book on Wikibooks<http://en.wikibooks.org/wiki/Category_Theory>has been languishing for three years, except for some good work done by Physis <http://en.wikibooks.org/wiki/User:Physis>. I have spent the last two days reorganizing it and cleaning it up. I have broken the book into pages and introduced some stubs. I have rewritten the Introduction and most of the chapter on Categories. The chapter on Natural Transformations covers the topic but I recommend it be rewritten in a more labeled style with bulleted lists, subsections, etc (like the Categories chapter) The other chapters need to be completely rewritten to fix the TeX, get rid of the page references (to someone's lecture notes, presumably), and so on. This is a good project for retired category theorists. Younger mathematicians get no academic rewards at all for contributing to Wikis. Note: If you are interested in something special, just write it up! For example you could write up typed lambda-calculus as an example, or a section on sheaves, or an introduction to 2-categories. If you need to refer to something not yet written, just add a stub. *Wikibooks do not have to be done in order. * -- professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://sixwingedseraph.wordpress.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm astounding math stories: http://www.abstractmath.org/MM//MMAstoundingMath.htm personal website: http://www.abstractmath.org/Personal/index.html sixwingedseraph.facebook.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

8 9

pushouts in REL
by soloviev＠irit.fr 23 Aug '09

23 Aug '09

To the list: I am actually travelling (in Russia) and I need more or less urgently a reference concerning pullbacks and pushouts in the category of sets and relations - do they always exist etc - I would not adress it to the list if it would be not urgent and I would not have some difficulty with search from here - Best to all - Sergei Soloviev [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

8 7

Re: [ALGTOP-L] history of ``twisted cohomology" and nonabelian cohomology
by Ronnie Brown 23 Aug '09

23 Aug '09

Dear Jim and all, This is reply to Jim Stasheff's question below. I hope it is helpful to relate my own experiences from the 1960s and later wrt nonabelian cohomology. In writing my book on topology, I got offended by having to make a detour to get the fundamental group of the circle, and then was attracted by Olum's paper {O}lum, P. \newblock \enquote{Non-abelian cohomology and {van} {Kampen}'s theorem}. {Ann. Math.} \textbf{68} (1958) 658--667. I extended his result to a Mayer-Vietoris (MV) type exact sequence in ``On a method of P. Olum'', {\em J. London Math. Soc.} 40 (1965), 303-304. and this enabled one to compute the fundamental group of, for example, a wedge of circles. ( I use an MV sequence in `Topology and groupoids' in connection with pullbacks of covering spaces.) So I decided to use this account for the book, thus giving students the advantage of an introduction to cohomology ideas. The problem was that the account when written in detail came to 30 pages (or maybe 40) and when looked at in the cold light of day seemed incredibly boring (a full account is different from Olum's research account). I was at the time looking for exercises and came across Philip Higgins paper on presentations of groupoids, which used free products with amalgamation of groupoids. So I decided to give an exercise on the fundamental groupoid of a union. Then I felt I ought to write out a solution. When I had done this, it seemed streets ahead in exposition of all that nonabelian cohomology stuff and moreover, when souped up to the *fundamental groupoid on a set of base points, *gave results not reachable by the MV sequence; for example you could not with the MV sequence deduce the _precise calculation_ of the fundamental group of a union of two open sets whose intersection had say 150 path components. (This anomaly is also significant, in illustrating the limitations of exact sequences.) So I decided to switch to an exposition of groupoids in 1-dimensional homotopy theory (also spurred by a meeting with George Mackey in 1967 where he told me of his work on ergodic groupoids). It occurred to me that if one could come to the groupoid idea from two distinct directions, then there was likely to be more in this than met the eye. At the same time, an examination of the proof of the van Kampen theorem for groupoids, suggested that the theorem should have an extension to all dimensions, if one could define homotopy gadgets with the right properties. Another stimulus was the proof (used in the book) by Frank (circulated in handwritten lecture notes) of the cellular approximation theorem, which had analogies to parts of the van Kampen proof, but failed to get algebraic results because, apparently, of the lack of an appropriate algebraic gadget in dimension n>1. It took 9 years to find such a gadget in dimension 2, and another 3 to get them it all dimensions, in work with Philip Higgins. It seemed to me unfortunate that this work aroused the opposition, for reasons never explained to me, of Frank Adams, who told people the whole programme was `ridiculous'. His opinion became the opposite only when I told him (1985?) of the extension to the non simply connected case of the Blakers-Massey description of \pi_3 of a triad, using the nonabelian tensor product (work with Loday). I did make some attempt to get the groupoid van Kampen result from nonabelian cohomology, see 40. (with P.R. HEATH and H. KAMPS), ``Groupoids and the Mayer-Vietoris sequence'', {\em J. Pure Appl. Alg.} 30 (1983) 109-129. and perhaps the earlier 11. ``Groupoids as coefficients'', {\em Proc. London Math Soc.} (3) 25 (1972) 413-426. But the higher order van Kampen theorems, and the often nonabelian calculations which result, have not been obtained by cohomological methods, but only by working directly with structures appropriate to the geometry of higher homotopies, i.e. forms of strict multiple groupoids. This confirms the comment of Philip Hall, Philip Higgins' supervisor, that one should not try to force the geometry into a given algebraic mode, but search for the algebra which models the geometry. So it seems to me that algebraic topology has been mainly restricted to, or not got out of, the single base point and `group', not `groupoid', mode, nor appreciated the possibilities of colimit type theorems in algebraic (and geometric?) topology (no algebraic or geometric topology text mentions the work with Philip Higgins). You can also see this restriction in the contrast between the unsymmetrical, choice laden, definition of the second relative homotopy group, with its compositions in one direction (recall the limitations of `Lineland' described in `Flatland') and the definition of the fundamental double groupoid of a pointed pair of spaces \rho_2(X,A), with its compositions in 2 directions: see 25. (with P.J. HIGGINS), ``On the connection between the second relative homotopy groups of some related spaces'', {\em Proc. London Math. Soc.} (3) 36 (1978) 193-212. This contrast gets more significant in higher dimensions. For all these reasons, my inclination is to look for the applications of the `appropriate' (whatever that is!) structures rather than cohomology with coefficients in such structures, where lots of detail is likely to get lost. These results could not have been obtained without the intuitions on multiple compositions easily allowed by a cubical approach. In any case, it is useful to analyse successes and failures, in deciding how to proceed. Hence my above account of some failures in nonabelian cohomology! What if anything have I missed? See also http://www.bangor.ac.uk/r.brown/einst.html How can one apply these (strict, for my preference!!) multiple groupoids in other areas???? (recall Grothendieck's work on the fundamental group!!) Good luck Ronnie jim stasheff wrote: > RONALD BROWN wrote: >> Jim asks: >> ____________________________________ >> >> How does >> >> `Nonabelian algebraic topology: filtered spaces, crossed complexes, >> cubical homotopy groupoids'. >> >> relate to the current rage for `nonabelian cohomology'? >> >> if it's answered in the above paper/book, more precise coordinates >> please > very helpful except, like Urs Schreiber, you answer my question only > partially > > do you *address* how yours relates to the current rage in alg geom, > math phys, cat theory or what ever? > > Ronnie, you already know but others may not Paul Olum @ 1950 already > refers to `nonabelian cohomology'!! >> >> ________________________________________________________ >> >> With regard to the book, that is dealt with in Chapter 12, >> "Nonabelian cohomology of spaces and of groups". The idea is that >> cohomology requires coefficients, and those dealt with in this book >> are `crossed complexes', which include groups, groupoids, and crossed >> modules, as well as algebraic models of spaces with homotopy groups >> in only two dimensions 1 and n > 1 . Crossed complexes form a linear >> model of homotopy types, i.e. no Whitehead product. So they are >> limited, but nonetheless linear models, as in linear algebra, have >> uses. They can be nonabelian in dimensions 1 and 2, and give complete >> information on 2-types (which is a step up from groups). >> >> The use of crossed modules as coefficients in nonabelian cohomology >> was pioneered by Dedecker, first for cohomology of groups, and >> later, in two papers in Canad. J. Math., of spaces. >> >> Crossed complexes as coefficients for cohomology appear in a >> Brown-Higgins paper on the classifying space of a crossed complex >> referred to earlier. >> >> Wrt filtered spaces: the theme of the book is that the functor >> \Pi: (filtered spaces) \to (crossed complexes) >> defined using fundamental groupoid \pi_1(X_1,X_0) and relative >> homotopy groups \pi_n(X_n,X_{n-1},x), x\in X_0, n \geq 2, can serve >> as a new foundation for algebraic topology because \Pi preserves >> certain colimits and certain tensor products, proofs of which (not so >> easy!) do not require singular homology or simplicial approximation. >> (The proofs do require the cubical higher homotopy groupoid \rho of a >> filtered space.) Many spaces, e.g. CW-complexes, classifying spaces >> of groups, or of higher categories, come with a filtration, or >> several filtrations, so this starting point is not unreasonable. If >> you are just given a space, you can get a filtered space by taking >> the (or a) singular complex and realising that as a CW-complex. >> >> The properties of BC, the classifying space of the crossed complex, >> require knowledge of the homotopy properties of simplicial or cubical >> sets. >> >> One reason \rho is useful is that it allows compositions in n >> directions in dimension n, whereas relative homotopy groups are >> limited to a 1-dimensional composition; and their definition is >> unaesthetic since it involves choices. They still form an essential >> part of the theory, and are useful for calculation rather than >> conjecture or proof, and because they are easier to relate to >> standard algebraic topology. >> >> There is a way of getting a Cech theory. >> >> I do not know the uses of crossed complexes in analogues of >> Poincar\'e duality. >> >> More general nonabelian cohomology requires more general >> coefficients, for which there are candidates, either strict models, >> e.g. crossed n-cubes of groups, or there is a lot of play with weak >> higher categorical structures. >> With the crossed complex model, you can do some calculations, >> particularly of 2-types. >> >> It seems reasonable to start by developing the linear theory, and, as >> I tried to communicate, Henry Whitehead's Combinatorial Homotopy II >> was a very powerful pointer to the structures required and the >> possible applications. There is a great string of work also by Hans >> Baues, in several books, going further in many ways, but from a >> different standpoint. >> >> Hope that helps >> >> Ronnie >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2 1

Reference on coverages
by Bas Spitters 22 Aug '09

22 Aug '09

The following is well-known: Given a posite we can construct the corresponding locale and the corresponding Grothendieck topos. Theorem: The locale is the subobject classifier of the topos. However, I fail to find a reference for this fact. Please help. Bas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

3 2