Dear Emily and all, Just to put a slightly different emphasis, I like to present categories and groupoids as good examples of structures having the dual roles of (i) algebraic structures in their own right, and also (ii) of value for talking about mathematical structures. The advantage of (i) is the move to the interest in *partial *algebraic structures, and so allow an easy move to higher dimensional algebra, as the study of partial algebraic structures whose operations have domains defined by geometric conditions, and also gets people used to the transition from groups to groupoids. Thus the groupoid object {\cal I}, which has two objects 0,1 and one non identity morphism \iota: 0 \to 1, is very easy to understand, but when one identifies 0 and 1 one gets the integers! Hence the fundamental group of the circle is the integers! This process of identification is best understood in the context of saying the functor Ob: Groupoids \to Sets is a bifibration. Groupoids are an example of an algebraic structure with structure in levels 0,1 and the functor Ob forgets the top level. This idea is of course relevant to other constructions in homotopy theory, since in homotopy theory identifications in low dimensions have higher dimensional homotopical implications. Our book "Nonabelian algebraic topology: ....." has an Appendix in fibrations of categories because the notion crops up in terms of earlier results, often connected with excision. That book does not use monads: it would be interesting to know of expository improvements by using that notion. Analogously one can define the natural numbers in terms of a pushout of categories, using the category *2*, and identifying 0 and 1. The notion of fibration of groupoids is also useful algebraically, and for modelling homotopy theory, including exact sequences, particularly at the bottom end. See for example a recent article arXiv:1207.6404 <http://arxiv.org/abs/1207.6404> This notion is used in my book "Topology and groupoids" to construct operations on certain homotopy sets, generalising the usual change of base point. The point is that groupoids are used in that book as a combinatorial and calculational tool, not just for theoretical reasons. Again, covering space theory is treated there using covering morphisms of groupoids, which are equivalent to actions of groupoids on sets. While writing, I mention http://education.lms.ac.uk/2014/12/alexander-grothendieck-some-recollections... Best regards Ronnie On 28/12/2014 21:52, Emily Riehl wrote: 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 ???Category theory in context???: http://www.math.harvard.edu/~eriehl/161 It has two aims: (i) To provide a thorough ???Cambridge-style??? 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 ???sociologically important??? (to quote Tom Leinster???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_* ???> Gp. Assuming the contrapositive, you can define a continuous retraction of the inclusion S^1 ???> D^2. Applying pi_1 leads to the contradiction 1=0. * The inverse image of a function f : A ???> B, regarded as a functor f* : P(B) ???> 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???s this week???s 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 @ : V x W ???> V @ W can be used to extract its construction. The projection to the quotient V @ W ???> V @ W/<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/ ]

* 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.

Related to this is the use of what is essentially the Yoneda lemma in normalisation proofs. For example Britton's lemma (giving normal forms for words in an HNN extension) proceeds by taking the set of "normal forms" and showing that it is a torsor for the group G at issue. In other words one takes a presheaf on G and shows that it is representable, and so is isomorphic to G itself. Another argument of a similar Yoneda kind is the usual proof of the Poincare--Birkhoff--Witt theorem (giving a basis for the universal enveloping algebra of a Lie algebra). In fact the technique is very widely applicable (and very widely applied); for instance you could prove the normal form theorem for the simplicial category using it. A related (but more general) technique is what is sometimes called "normalization by evaluation" by computer scientists; Peter Dybjer has some articles on the relation to the Yoneda lemma. But that is maybe straying a bit far from undergraduate mathematics. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]