From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8443 Path: news.gmane.org!not-for-mail From: Emily Riehl Newsgroups: gmane.science.mathematics.categories Subject: a call for examples Date: Sun, 28 Dec 2014 16:52:55 -0500 Message-ID: Reply-To: Emily Riehl NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 8.1 \(1993\)) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1419863397 16065 80.91.229.3 (29 Dec 2014 14:29:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 29 Dec 2014 14:29:57 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Dec 29 15:29:50 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Y5bKP-0001TI-6Q for gsmc-categories@m.gmane.org; Mon, 29 Dec 2014 15:29:49 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:56844) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Y5bJM-0005O0-2V; Mon, 29 Dec 2014 10:28:44 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Y5bJL-00020Z-V5 for categories-list@mlist.mta.ca; Mon, 29 Dec 2014 10:28:43 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8443 Archived-At: Hi all, I am writing in hopes that I might pick the collective brain of the = categories list. This spring, I will be teaching an undergraduate-level = category theory course, entitled =E2=80=9CCategory theory in context=E2=80= =9D: http://www.math.harvard.edu/~eriehl/161 It has two aims:=20 (i) To provide a thorough =E2=80=9CCambridge-style=E2=80=9D introduction = to the basic concepts of category theory: representability, (co)limits, = adjunctions, and monads. (ii) To revisit as many topics as possible from the typical = undergraduate curriculum, using category theory as a guide to deeper = understanding. For example, when I was an undergraduate, I could never remember whether = the axioms for a group action required the elements of the group to act = via *automorphisms*. But after learning what might be called the first = lemma in category theory -- that functors preserve isomorphisms -- I = never worried about this point again. Over the past few months I have been collecting examples that I might = use in the course, with the focus on topics that are the most = =E2=80=9Csociologically important=E2=80=9D (to quote Tom Leinster=E2=80=99= s talk at CT2014) and also the most illustrative of the categorical = concept in question. (After all, aim (i) is to help my students = internalize the categorical way of thinking!) Here are a few of my favorites: * The Brouwer fixed point theorem, proving that any continuous = endomorphism of the disk admits a fixed point, admits a slick proof = using the functoriality of the fundamental group functor pi_1 : Top_* = =E2=80=94> Gp. Assuming the contrapositive, you can define a continuous = retraction of the inclusion S^1 =E2=80=94> D^2. Applying pi_1 leads to = the contradiction 1=3D0. * The inverse image of a function f : A =E2=80=94> B, regarded as a = functor f* : P(B) =E2=80=94> P(A) between the posets of subsets of its = codomain and domain, admits both adjoints and thus preserves both = intersections and unions. By contrast, the direct image, a left adjoint, = preserves only unions. * Any discrete group G can be regarded as a one-object groupoid in which = case a covariant Set-valued functor is just a G-set. The unique = represented functor is the G-set G, with its translation (left = multiplication) action. By contrast, a *representable* functor X, not = yet equipped with the natural ($G$-equivariant) isomorphism $G \cong X$ = defining the representation, is a $G$-torsor. I learned this from John = Baez=E2=80=99s this week=E2=80=99s finds: http://math.ucr.edu/home/baez/torsors.html My favorite example is still the one that John uses: n-dimensional = affine space is most naturally a R^n-torsor. * The universal property defining the tensor product V @ W as the = initial vector space receiving a bilinear map=20 @ : V x W =E2=80=94> V @ W can be used to extract its construction. The projection to the quotient = V @ W =E2=80=94> V @ W/ by the vector space spanned by the image = of the bilinear map @ must restrict along @ to the zero bilinear map, as = of course does the zero map. Thus V @ W must be isomorphic to the span = of the vectors v @ w, modulo the bilinearity relations. * By the existence of discrete and indiscrete spaces, all of the limits = and colimits one meets in point-set topology -- products, gluings, = quotients, subspaces -- are given by topologizing the (co)limits of the = underlying sets. Of course this contradicts our experience with the = constructions of colimits in algebra. * On that topic, the construction of the tensor product of commutative = rings or the free product of groups can be understood as special cases = of the general construction of coproducts in an EM-category admitting = coequalizers. I would be very grateful to hear about other favorite examples which = illustrate or are clarified by the categorical way of thinking. My view = of what might be accessible to undergraduates is relatively expansive, = particularly in the less-obviously-categorical areas of mathematics such = as analysis. I have also posted this query to the n-Category Cafe and am hoping to = collect examples there as well: https://golem.ph.utexas.edu/category/2014/12/a_call_for_examples.html Best wishes to all for a happy and productive new year. Emily Riehl -- Benjamin Peirce & NSF Postdoctoral Fellow Department of Mathematics, Harvard University www.math.harvard.edu/~eriehl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]