Dear Colin, The Templeton foundation http://en.wikipedia.org/wiki/John_Templeton_Foundation is presently supporting a 2 years research program in set theory called THE INFINITY PROJECT at the CRM in Barcelona: http://www.crm.cat/InfinityProject/ There seem to be an endless number of projects with the same name: http://www.infinity-project.org/ http://fusionanomaly.net/tip.html We some luck, we may be able to convince the Templeton Foundation to support a research project in higher category theory and homotopy theory: http://ncatlab.org/nlab/show/infinity-category http://ncatlab.org/nlab/show/A-infinity-algebra http://ncatlab.org/nlab/show/E-infinity-ring http://ncatlab.org/nlab/show/L-infinity-algebra http://ncatlab.org/nlab/show/%28infinity%2C1%29-operad On the serious side, I think that we should make an effort to find a better terminology in higher category theory. I confess that I do not particularly cherish the name "quasi-category", although I am responsible for introducing it. It seems better than "weak Kan complex" because the theory of these objects behaves very much like category theory. The name "infinity-category" is no better than "quasi-category". infinity=endless Jacob Lurie has expressed the same concern in a private discussion with me. best, Andre -------- Message d'origine-------- De: categories@mta.ca de la part de Colin McLarty Date: jeu. 12/11/2009 20:29 À: categories@mta.ca Objet : categories: Re: categorical foundations 2009/11/12 <Andre.Rodin@ens.fr>: writes
ETCS is the formal basis of CCAF.
This is simply false. On some versions ETCS is a part of CCAF but even then it is in no sense prior to other parts.
ETCS relies on a pre-formal notion of set or collection just like ZF or any other axiomatic theory built with Hilbert-Tarski axiomatic method.
Do you mean that every formalized axiom system uses arithmetical notions such as "finite string of symbols." This is why that formal axioms cannot be the real basis of our knowledge of math, but it has no more bearing on categorical axioms than any others. Or do you think that pre-formal notions of "set" or "collection" are all based on iterated membership and Zermelo's form of the axiom of extensionality, so that CCAF is less basic than ZFC? That is a common belief among logicians who have not read Zermelo's critique of Cantor (where Zermelo points out that Cantor did not hold these beliefs) and who know a great deal more of ZFC than of other mathematics. In fact, long before mathematicians could analyze the continuum into a discrete set of points plus a topology, they were well aware of collections like the collection of rigid motions of the plane -- and that "collection" is a category. It is not just a ZFC set of motions but comes with composition of motions and with an object that the motions act on.
Elements of a new properly categorical method of theory-building are present in the "basic theory" (BC) that follows ETC. (I mean, in particular, the "redefinition" of functor in BC as 2-->A, etc. The standard definition of functor given earlier in ETC never reappears in BC.)
The "standard" definition of functor appears as the definition of a small category in the category of sets.
However in CCAF these new features are not yet developed into an autonomous axiomatic method - or into a new way of formalisation of pre-formal concepts, if you like.
Well, yes, they are developed into one. That was Bill's achievement with CCAF. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]