From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3964 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Help! Date: Sun, 07 Oct 2007 16:20:42 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019631 11116 80.91.229.2 (29 Apr 2009 15:40:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:31 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Mon Oct 8 10:13:57 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:57 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesOJ-0003xO-Tr for categories-list@mta.ca; Mon, 08 Oct 2007 10:11:24 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 21 Original-Lines: 102 Xref: news.gmane.org gmane.science.mathematics.categories:3964 Archived-At: I would think the best topics would be those that can be described with a minimum of jargon. The problem with category theory is that it is so steeped in its own jargon as to make it quite an effort to strip it out. Here are some topics where I would expect that effort to be minimal, arranged in roughly increasing order of intricacy of definition. This should more than fill a one-hour lecture, especially if there are questions. 1. Thinking of each object T of a category C as both a type and a dual type, characterize a product AxB in C as an object consisting of all pairs of T-elements of A and B over all types T, and A+B as dually consisting of all pairs of T-functionals of A and B over all dual types T in ob(C). Section 6 needs pullbacks, they could be done either here or there, it's neither here nor there. 2. The category FinBip of finite bipointed sets as the theory of cubical sets. The models are arbitrary functors M: Bip --> Set. You could look at \Delta for simplicial sets as well or instead, I'm partial to Bip perhaps because in kindergarten we tended to work more with cubical than simplicial sets (Western Australian kindergartens had strong PTOs reflecting epic entanglements). You could then continue with FinSet^op as the algebraic theory of Boolean algebras, but that would entail giving up one of the other segments. 3. Enriched categories as generalized metric spaces. People who have a hard time with abstract objects mixed in with concrete homsets (I certainly did) will be relieved to know that making the homsets just as abstract as the objects turns the definition of category into a familiar object not normally considered part of the categorical basement. 4. Presheaves on J as colimits of diagrams in J. If you use Yoneda to hide the concept of colimit the idea becomes almost trivial. In the case J = 1, as C starts from 1 and grows towards Set^1 each new set X as a new object of C is installed along with an arbitrary choice of C(1,X). The composites at X are defined by first installing C(X,Y) for all existing Y in C, defining fx: 1 --> X --> Y for each x in X and new f in C(X,Y), and taking C(X,Y) to be maximal subject to Ax[fx=gx] ==> f=g. These composites then uniquely determine the remaining composites gf: X --> Y --> Z and fg: W --> X --> Y for W other than 1. The completion is complete when every new set is necessarily isomorphic to one already present. (Does this have anything to do with Yoneda structures? Trying to read about those I discovered I no longer talked Strine.) For J the ordinals 1 and 2 as respectively the primitive vertex and the primitive edge, namely the two reflexive graphs priming the pump for the rest, there are now two types of element, vertices and edges, with Ax interpreted as quantifying over all elements of both types; otherwise everything is as for J = 1. Point out that whereas all sets are free, the free graphs are just those with trivial incidences. If you do section 6 (triples for matrix multiplication), also point out at some point that whereas Set^1 is tripleable on Set (the identity), Set^J in general is tripleable only on Set^{|J|}, important when talking Czech. 5. Toposes, but *not* the way it is explained on You-tube, which is completely unmotivated and incomprehensible for anyone who hasn't already understood them. The Explanation section http://en.wikipedia.org/wiki/Topos_theory#Explanation in the Wikipedia article on elementary toposes touches on the two points that should be in any explanation of the concept, namely (i) "subobject" predates "topos," witness CTWM which defines it in second-order language, and (ii) monics m: X' --> X are in bijection with pullbacks of the element (hence monic) t: 1 --> \Omega along morphisms f: X --> \Omega, allowing one to speak of *the* characteristic function of a monic, thereby classifying the monics by their characteristic functions, a first-order notion (whence the "elementary" in "elementary topos") that is in full agreement with the second-order notion in (i) when applied to a topos. 6. Matrix multiplication in terms of the Kleisli construction for the triple for Vct_k. I just sent out a crib sheet for that which focused on a difficulty with non-finitary (square summable) linear combinations, but that difficulty is impossible to absorb in the available time, better to stick to the finitary operations where matrix multiplication is tripleable. You could mention the Haskell programming language and how they blended the second component of the triple and Kleisli into a single operator Bind: T(X) --> (X --> T(Y)) --> T(Y), which might get any programmers in the club interested in Haskell; also point out the possibility of replacing (X --> T(Y)) by T(Y*X) and its implications for matrix algebra including Hilbert space. Stick to finite X in T(X) = k^X to save the extra step of defining finitary k^X for infinite X (but if you do decide to do that step it should suffice to point out that 6 of the 16 binary Boolean operations as 2x2 truth tables have constant rows or columns or both and then generalize to infinity). In case You-tube ever has a video on triples you should probably mention any synonyms for "triple" so the students can find the video. Vaughan Michael Barr wrote: > What would you say to an undergraduate math club about categories? I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk about > dual spaces of finite-dimensional vector spaces? Maybe, but then what? > > Michael > > >