### A Call for Examples

#### Posted by Emily Riehl

This spring, I will be teaching an undergraduate-level category theory course: Category theory in context. 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 $\pi_1 \colon \mathbf{Top}_* \to \mathbf{Gp}$. Assuming the contrapositive, you can define a continuous retraction of the inclusion $S^1 \hookrightarrow D^2$. Applying $\pi_1$ leads to the contradiction $1=0$ in $\mathbb{Z}$.

The inverse image of a function $f \colon A \to B$, regarded as a functor $f^* \colon P(B) \to P(A)$ 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 $\mathbf{Set}$-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 and my favorite example is still the one that John uses: $n$-dimensional affine space is most naturally a $\mathbb{R}^n$-torsor.The universal property defining the tensor product $V \otimes W$ as the initial vector space receiving a bilinear map $\otimes \colon V \times W \to V \otimes W$ can be used to extract its construction. The projection to the quotient $V \otimes W \to V \otimes W/\langle v \otimes w\rangle$ by the vector space spanned by the image of $\otimes$ must restrict along $\otimes$ to the zero bilinear map, as of course does the zero map. Thus $V \otimes W$ must be isomorphic to the span of the vectors $v \otimes w$, 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.

## Re: A Call for Examples

I’m not sure I have any useful examples to suggest, but I did want to take a moment to marvel at the diversity of what might be considered a “typical undergraduate curriculum”. I wish such things were typical for our undergraduates!