## December 28, 2014

### 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.

Posted at December 28, 2014 9:37 PM UTC

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### 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!

Posted by: Greg on December 28, 2014 9:54 PM | Permalink | Reply to this

### Re: A Call for Examples

Whether all of the examples that I am tempted to use are reasonable is another question $\ldots$ :)

For instance, I am very taken with Tom’s characterization of the universal property of the Banach space of Lebesgue integrable functions on the unit interval. I certainly don’t expect that everyone will be familiar with Banach spaces, but there is a new, relatively popular, Analysis II class covering measure and integration.

I do intend to mention the Freyd-Leinster universal property of the unit interval. I’ll admit, I don’t feel like I understand this result in a very deep way, but it’s very cool.

Posted by: Emily Riehl on December 28, 2014 10:08 PM | Permalink | Reply to this

### Re: A Call for Examples

In some representation theory courses, induction of representations is sometimes described via an explicit and unenlightening formula. Actually, for representations of finite groups, it can be described as either a left or a right adjoint to the forgetful functor given by restriction of representations; this is one of the simplest examples of an ambidextrous adjunction.

A relation between two sets naturally induces an adjunction between the posets of subsets of those sets which further restricts to a contravariant equivalence between two subposets of closed subsets. This abstracts at least three important Galois correspondences in undergraduate-ish mathematics: the correspondence between subextensions of a Galois extension and subgroups of a Galois group, the correspondence between ideals of a commutative ring and subvarieties of its spectrum, and the correspondence between collections of sentences in a first-order theory and collections of models of those sentences. There is some generalization involving enriched profunctors but I don’t know familiar examples of this more general construction off the top of my head.

In representation theory, one reason to think of representations of a group as modules over its group algebra is that elements of the group algebra provide additional operations on representations, and in particular elements of the center of the group algebra are where the idempotents projecting to each isotypic component of the representation live. Even if it had never occurred to you to think about group algebras, you could in fact recover them from the category of representations by trying to figure out what an operation on a representation is and then trying to classify them. Eventually it will occur to you to look at natural endomorphisms of the forgetful functor. The forgetful functor is represented by the group algebra as a representation of the group, so by the Yoneda lemma, its natural endomorphisms can be identified with the group algebra as an algebra. This tells you not only that you should look at the group algebra for operations but also that all (unary) operations live in the group algebra. Similarly for representations of Lie algebras and the universal enveloping algebra.

Posted by: Qiaochu Yuan on December 28, 2014 11:26 PM | Permalink | Reply to this

### Re: A Call for Examples

Cool, Qiaochu, thank you. I was aware separately of the three Galois correspondences but wouldn’t have thought to unify them as contravariant equivalences arising from a relation between the appropriate pairs of sets.

Regarding the explicit and unenlightening formula for (co)induced representations, is there another way to see that for finite groups the left and right adjoints to restriction coincide? I usually argue by unpacking the formulas for (enriched) left and right Kan extension in this special case and observing that the finite sum in the former case is isomorphic to the finite product in the latter. Can you argue from the universal property that the adjunction is ambidextrous?

Posted by: Emily Riehl on December 29, 2014 2:25 AM | Permalink | Reply to this

### Re: A Call for Examples

Can you argue from the universal property that the adjunction is ambidextrous?

Well, you have to use somewhere that the group is finite…

Posted by: Mike Shulman on December 29, 2014 5:31 AM | Permalink | Reply to this

### Re: A Call for Examples

While talking about cool examples of Galois correspondences, it’s probably worth mentioning that when you see one of these (or an adjunction more generally), there’s often an interesting theorem telling us how to make this an adjunction an equivalence: witness the Fundamental Theorem of Galois Theory and the Nullstellensatz (although these answer slightly different questions: in both cases we have a whole class of adjunctions; in the former case we identify which of these adjunctions are equivalence, while in the latter case we restrict each adjunction to a subcategory to get an equivalence).

Another example in this vein, I guess not so familiar to undergraduates, is the correspondence between locales and topological spaces, and the ensuing characterization of sober spaces. But at least soberfication explains the non-closed points in a Zariski topology, which may be more familiar.

Posted by: Tim Campion on December 29, 2014 9:53 PM | Permalink | Reply to this

### Re: A Call for Examples

Here’s something that’s incredibly basic, which undergrad math majors really need to hear: we get the set of natural numbers by decategorifying (= forming the set of isomorphism classes) of the category of finite sets. Thus, addition, multiplication and exponentiation of natural numbers arise from coproducts, products and exponentials in the category of finite sets!

Seeing how this work in some detail is a nice way to learn a lot of basic category-theoretic concepts. For example, you can show students that an equation like this:

$a^{b + c} = a^b \times a^c$

comes from an isomorphism — and even if they don’t right away learn the most general reason behind this isomorphism, they’ll never look at equations like this in the same way again.

I like to emphasize how remarkable it is that these insights about elementary arithmetic were only discovered in the second half of the 20th century! It shows that there’s probably a lot of very simple stuff left to be discovered. When you’re a student, I think it’s encouraging to know this.

I also like to emphasize that decategorifying FinSet is the reason natural numbers and arithmetic operations on them were invented in the first place! We meet finite sets of things in the world; the natural numbers are an abstraction based on this.

If one studies categorification one soon discovers an amazing fact: many deep-sounding results in mathematics are just categorifications of facts we learned in high school! There is a good reason for this. All along, we have been unwittingly ‘decategorifying’ mathematics by pretending that categories are just sets. We ‘decategorify’ a category by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a mere set: the set of isomorphism classes of objects.

To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and ‘count’ it, setting up an isomorphism between it and some set of ‘numbers’, which were nonsense words like ‘one, two, three,…’ specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented.

According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification.

Posted by: John Baez on December 29, 2014 6:25 AM | Permalink | Reply to this

### Re: A Call for Examples

It’s a funny coincidence that Paul Taylor tells virtually the same parable involving a shepherdess in his book (I think his is named Bo Peep), if I recall correctly.

Posted by: Todd Trimble on December 29, 2014 9:11 AM | Permalink | Reply to this

### Re: A Call for Examples

Dear Todd,

Do you happen to remember the name of the book you are talking about? I am curious about Paul Taylor’s exposition of this.

Cheers.

Posted by: Dan Frumin on January 3, 2015 12:42 PM | Permalink | Reply to this

### Re: A Call for Examples

The title is Practical Foundations of Mathematics. A very interesting and unique book; in my case definitely an acquired taste (I found it difficult to read and follow in the beginning), but I would advise anyone to persist because it’s packed with wisdom and good examples.

Posted by: Todd Trimble on January 3, 2015 1:22 PM | Permalink | Reply to this

### Re: A Call for Examples

When undergrads (possible future: engineers) complain about math being “too abstract”, I ask them which part of mathematics is “not too abstract”. They usually tell me a couple of examples, one of them is always “natural numbers and counting”.

Then, I ask them to explain to me what means “four”.

Then the debate always goes like: “This is four” –> “No. These are some of your fingers.” etc. etc. Gradually, they realize how infinitely abstract concepts “counting” and “numbers” actually are.

Posted by: Gejza Jenča on December 29, 2014 9:27 AM | Permalink | Reply to this

### Re: A Call for Examples

Thanks John. I like this very much.

An interesting expository question is when to introduce exponentials.

By the time I get around to defining adjunctions, it’s not such a big step to parametrized adjunctions, as in Mac Lane, with our favorite closed monoidal categories being the principal motivating examples. But perhaps that’s too long to wait, especially given that one of Eilenberg and Mac Lane’s first examples of a (extraordinary) natural isomorphism was tensor-hom for a triple of abelian groups. (For time reasons, extraordinary naturality might end up being a topic for the homework rather than for lecture.)

I suppose I could introduce the exponential as the thing that represents the functor $hom(A x - ,B)$, but this doesn’t feel very natural to me.

Slightly better would be to introduce the evaluation map $A \times B^A \to B$ as a universal arrow in the comma category $(A \times -) \downarrow B$. I think this might be how Steve Awodey does it in his book, but I don’t have it to hand at the moment and can’t remember for sure.

Posted by: Emily Riehl on December 29, 2014 7:00 PM | Permalink | Reply to this

### Re: A Call for Examples

A related historical note:

In chapter IV of their “General theory of natural equivalences,” Eilenberg and Mac Lane regard the categories of sets and well-ordered sets as full subcategories of the category of preorders (which they call “quasi-ordered sets”) and order-preserving functions. They then mention a related categorification of arithmetic:

Specifically, the general theory of arithmetic of partially ordered sets, as developed recently by Birkhoff, can be viewed as the construction of a large number of functors (cardinal power, ordinal power, and so on) defined on suitable subcategories of $\mathfrak{Q}$, together with a collection of natural equivalences and transformations between these functors.

The reference is to:

Garrett Birkhoff, Generalized arithmetic, Duke Math. J. vol. 9 (1942) pp. 283-302.

The quoted passage ends with a rather curious footnote:

Note, however, that the ordinary cardinal sum of two sets $A$ and $B$ does not give rise to a functor, because the definition applies only when the sets $A$ and $B$ are disjoint.

Posted by: Emily Riehl on December 29, 2014 7:19 PM | Permalink | Reply to this

### Re: A Call for Examples

I like to emphasize how remarkable it is that these insights about elementary arithmetic were only discovered in the second half of the 20th century! … I also like to emphasize that decategorifying FinSet is the reason natural numbers and arithmetic operations on them were invented in the first place!

These statements seem a bit contradictory to me. (-: I would have thought that people knew long before the second half of the 20th century that $a^{b+c} = a^b \times a^c$ comes from a bijection of finite sets; by “these insights” do you mean specifically the fact that such isomorphisms have abstract categorical proofs?

Posted by: Mike Shulman on December 30, 2014 12:29 AM | Permalink | Reply to this

### Re: A Call for Examples

Mike wrote:

by “these insights” do you mean specifically the fact that such isomorphisms have abstract categorical proofs?

Sorry, my remark wasn’t very clear. While the natural numbers were invented to count finite sets, and people knew a bunch of properties of the natural numbers are connected to those of finite sets, there are a lot of insights that can only arise — or can only become theorems — after you have some category-theoretic concepts in hand.

For example: in the category of finite sets, addition, multiplication and exponentiation can be defined by universal properties. I find it particularly wonderful that addition and multiplication are dual to each other: coproduct and product. Before 1950 or so, if you’d told someone you could take the definition of addition and get the definition of multiplication simply by ‘turning around the arrows’, they’d have thought you were nuts! (In fact, most people still would.)

Posted by: John Baez on December 30, 2014 1:35 AM | Permalink | Reply to this

### Re: A Call for Examples

Ah, I see. Yes, you’re absolutely right.

Posted by: Mike Shulman on December 30, 2014 3:45 AM | Permalink | Reply to this

### Re: A Call for Examples

The situation is thoroughly confused by the fact that people are generally taught to think of multiplication as repeated addition and exponentiation as repeated multiplication. The fact that these definitions work to define multiplication and exponentiation on the natural numbers $\mathbb{N}$ is a fact about $\mathbb{N}$, not a fact about multiplication and exponentiation in more general contexts (e.g. the real or complex numbers, where these definitions fail miserably).

Posted by: Qiaochu Yuan on January 2, 2015 10:36 AM | Permalink | Reply to this

### Re: A Call for Examples

What exactly is special about the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$ that allows definitions as repetitions, and does this hold elsewhere? In a Boolean algebra you cannot define $\vee$ as repeated $\wedge$ or $\to$ as repeated $\vee$.

Posted by: RodMcGuire on January 3, 2015 7:54 PM | Permalink | Reply to this

### Re: A Call for Examples

$a^{b+c} = a^b \times a^c$

I like to emphasize how remarkable it is that these insights about elementary arithmetic were only discovered in the second half of the 20th century!

I like the example, but telling undergraduates the above is intellectually irresponsible. You cannot honestly think that these “insights” are recent. I’m no great mathematical talent and I noticed such things in high school; yet somehow Newton, Euler, and Gauss missed them?

In the Disquisitiones Gauss gives two proofs of Fermat’s Little, one of which is by showing how to divide an explicitly defined set of $x^p$ elements into groups of $p$ elements each, plus $x$ elements left over.

In the combinatorics literature, dating back to the 19th century, such “categorified proofs” of binomial identities are utterly run-of-the-mill.

Not everything that can be described with category theory words comes from category theory, not even if it was developed after category theory. Modern enumerative combinatorics cares about explaining numerical identities via bijections (“categorifying”) but its practitioners are mostly ignorant of and uninterested in CT. The reason they care about bijections is emphatically not coming from CT.

Posted by: Jake on May 2, 2016 8:51 PM | Permalink | Reply to this

### Re: A Call for Examples

I don’t think John meant that the fact that $a^{b+c}=a^b\times a^c$ comes from a bijection was only discovered in the 20th century, but rather that the operations of coproduct, product, and exponential appearing in that bijection are categorical operations with universal properties. Newton, Euler, and Gauss couldn’t have noticed that fact because they didn’t have the language.

Posted by: Mike Shulman on May 2, 2016 10:53 PM | Permalink | Reply to this

### Re: A Call for Examples

Jake wrote:

You cannot honestly think that these “insights” are recent.

Your misinterpretation of my remark is not new either. On December 30, 2014, Mike Shulman asked:

by “these insights” do you mean specifically the fact that such isomorphisms have abstract categorical proofs?

I replied:

Sorry, my remark wasn’t very clear. While the natural numbers were invented to count finite sets, and people knew a bunch of properties of the natural numbers are connected to those of finite sets, there are a lot of insights that can only arise — or can only become theorems — after you have some category-theoretic concepts in hand.

For example: in the category of finite sets, addition, multiplication and exponentiation can be defined by universal properties. I find it particularly wonderful that addition and multiplication are dual to each other: coproduct and product. Before 1950 or so, if you’d told someone you could take the definition of addition and get the definition of multiplication simply by ‘turning around the arrows’, they’d have thought you were nuts! (In fact, most people still would.)

And, of course, the new insight about $a^{b+c} \cong a^b \times a^c$ is that it holds in any category with coproduct, product and exponential, thanks to their universal properties.

Posted by: John Baez on May 4, 2016 4:07 AM | Permalink | Reply to this

### Re: A Call for Examples

Coincidently, I just noted the other day in an article that the identities for a finite set of natural numbers, $\{b_i\}$ and $c$ a natural number

1. $c^{\sum_i b_i} = \prod_i c^{b_i}$
2. $(\prod_i b_i)^c = \prod_i (b_i)^c$

are just shadows of adjunctions involving dependent sum and dependent product in dependent type theory.

Posted by: David Corfield on May 8, 2016 7:53 AM | Permalink | Reply to this

### Re: A Call for Examples

I enjoyed this one, from Mac Lane’s book: let $\mathbb{N}$ be the additive monoid (a one-object category) of natural numbers. Let $\mathbf{Mat}$ be the category consisting of natural numbers (objects), real matrices (arrows) and multiplication of matrices (composition).

What are functors $\mathbb{N}\to\mathbf{Mat}$? What are natural transformations between them? What is an endomorphism of a functor? When are two functors isomorphic?

Then, the same questions but with a two-element chain instead of $\mathbb{N}$?

Posted by: Gejza Jenča on December 29, 2014 9:54 AM | Permalink | Reply to this

### Re: A Call for Examples

Cute. I like it. This seems like a good homework problem.

By “two-element chain” did you mean the free category with two endomorphisms of a single object? Or did you mean something else?

Posted by: Emily Riehl on December 29, 2014 6:43 PM | Permalink | Reply to this

### Re: A Call for Examples

By a two element chain I mean a poset with two comparable elements; a functor is then a not-necessarily-square matrix. The isomorphism of functors is usually called “matrix equivalence”.

The $[\mathbb{N},\mathbf{Mat}]$ thing appears to be even nicer than I assumed when I suggested this example: if we identify functors with square matrices ($F$ is the same as $F(1)$), then a natural transformation $B\to C$ is simply a matrix $A$ such that $BA=AC$.

So, to find a natural transformation $B\to C$ means to solve a homogeneous case of the Sylvester equation.

Posted by: Gejza Jenča on December 31, 2014 5:37 PM | Permalink | Reply to this

### Re: A Call for Examples

The various product-like constructions on vector spaces: show that the direct sum of vector spaces is a biproduct, and also that the tensor product doesn’t admit either natural injections or natural projections. I found products of vector spaces very confusing when I was an undergraduate (there were two of them, but they didn’t seem to correspond to product and union of sets), and only when I had discovered category theory did it all start to make sense.

Posted by: Graham White on December 29, 2014 11:46 AM | Permalink | Reply to this

### Re: A Call for Examples

These are great suggestions, thanks.

This confused me momentarily:

also that the tensor product doesn’t admit either natural injections or natural projections.

Any non-zero vector $v \in V$ defines an injective natural transformation $id \Rightarrow V \otimes -$ and any non-zero functional $v^* \colon V \to k$ defines a surjective natural transformation $V \otimes - \Rightarrow \id$. But what you’re saying, I assume, is that no choices satisfy the universal properties of the coproduct or product, a very important point.

I would also like to tease apart the difference between infinite direct sums and products. It’s easy to check that the usual constructions have the appropriate universal properties and that the map whose components are identities or zero defines an injection

$\oplus_\alpha V_\alpha \to \prod_\alpha V_\alpha,$

but I’m wondering whether there is a formal way to see that this isn’t an isomorphism when the index set is infinite.

I suppose that if all the vector spaces are finite dimensional you could argue as follows: Mapping out of $k$, elements of $\prod_\alpha V_\alpha$ are tuples of vectors $(v_\alpha)_\alpha$. Since addition is coordinatewise, it’s clear that this vector space is infinite dimensional.

Similarly, we can identify the dual of $\oplus_\alpha V_\alpha$ by mapping into $k$. Using the isomorphisms $V_\alpha \cong V_\alpha^\ast$, this is $\prod_\alpha V_\alpha^\ast \cong \prod_\alpha V_\alpha$. Since the dual of the direct sum is isomorphic to the infinite-dimensional product, the direct sum must be strictly smaller.

But maybe this is silly, using a big theorem about dualizable vector spaces to prove something elementary.

Posted by: Emily Riehl on December 29, 2014 8:03 PM | Permalink | Reply to this

### Re: A Call for Examples

I’m wondering whether there is a formal way to see that this isn’t an isomorphism when the index set is infinite.

Here is an argument using a sort of “Eilenberg swindle”. If $\coprod_{a\in A} V_a \to \prod_{a\in A} V_a$ is an isomorphism in some category, then we can “add up” $A$-indexed families of parallel morphisms $\{f_a : V\to W \}$ by considering the composite $V \xrightarrow{(f_a)} \prod_a W \cong \coprod_a W \xrightarrow{fold} W$ (or any of several other similar definitions, all equivalent). It’s easy to check that these possibly-infinitary operations are associative, in an appropriate sense, and that when $A$ is finite they reproduce the usual addition on hom-sets in (say) vector spaces.

In particular, since $\mathbb{N}\cong 1+\mathbb{N}$, associativity implies that the “infinite sum” $\sum_{i\in \mathbb{N}} f$, when it exists, is equal to $f + \sum_{i\in \mathbb{N}} f$, for any map $f$. Thus, if we are in some category (such as vector spaces) where we can also subtract morphisms, we must have $f=0$ for all $f$.

By contrast, in categories where you can’t subtract, this map can be an isomorphism for infinite indexing sets. E.g. in the category of suplattices, it is an isomorphism for all (small) indexing sets.

Posted by: Mike Shulman on December 30, 2014 12:16 AM | Permalink | Reply to this

### Re: A Call for Examples

Here are one or two things I am happy I already learned early on in my mathematical eduction as well as a bunch of things I wish I would have learned earlier.

• It’s fun figuring out what (co-)products are in a totally ordered set. Doing this really drove home the point early on to me that products don’t have to look anything like the Cartesian product of sets.

• Biproducts in additive categories (or perhaps in an undergraduate course you might prefer to just talk about $R$-modules or even just vector spaces and Abelian groups): As mentioned by Graham White it can be pretty confusing that finite products and coproducts coincide in additive categories. It would have saved me a lot of trouble later on if someone had told me straight away that even more concretely the biproduct of two objects $X_1, X_2 \in \mathcal{A}$ can also be characterised as a triple $(X_1 \oplus X_2, \{i_k\}_{k = 1,2}, \{p_k\}_{k = 1,2})$ consisting of the object $X_1 \oplus X_2$ and morphisms $i_k: X_k \to X_1 \oplus X_2$ and $p_k: X_1 \oplus X_2 \to X_k$ such that $p_j \circ i_k = 1$ if $j = k$ and $p_j \circ i_k = 0$ if $j \neq k$. A bunch of things about finite (co-)products in additive categories become easy to prove using this characterisation. For example, the fact that biproducts can be characterised in a way not using any universal property tells us that if any full subcategory $\mathcal{A}' \subseteq \mathcal{A}$ contains equivalently finite products, coproducts or biproducts, then these must coincide with the ones in $\mathcal{A}$.

• Group objects: I found it cool to see early on that awkward definitions such as “A Lie group is a group which is also a manifold” can be defined as a group object in the category of differentiable manifolds.

• Showing that two definitions are equivalent often means one has found an equivalence of categories: In fact, noting this provides the opportunity to explain that generally there is a stronger and a weaker way in which two definitions may be equivalent, due to the fact that the corresponding equivalence of categories may or may not even be an isomorphism.
For example, some people define a differentiable manifold to be either a set or a topological space together with either a maximal atlas or an equivalence class of compatible atlases. All four resulting definitions induce isomorphisms of categories, as an atlas uniquely determines the topology on a manifold and a maximal atlas uniquely determines its equivalence class. If we however define a manifold to be a set or topological space together with a not necessarily maximal atlas, then we obtain a category which is only equivalent to the other ones.
My favourite example of this is that the forgetful functor from analytic Lie groups to topological Lie groups is an isomorphism of categories.
(On a related note, one might want to emphasise that isomorphisms of categories do actually occur in nature, citing e.g. the classical Galois correspondence.)

• The yoga of internal hom: Often one starts of with some category $\mathcal{C}$ with either some bifunctor $\mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$ such that there ought be some corresponding monoidal structure on $\mathcal{C}$ or one starts of with some monoidal category and one may hope that this category has internal hom.
Here are some examples (which one can understand without introducing all the machinery of monoidal and enriched categories):

• In the category of modules over any ring $R$ each hom-set may be endowed with the structure of an $R$-module in an obvious way. The corresponding monoidal structure is then given by the tensor product.
• In a suitably chosen category of topological spaces one may find a topology on every hom-set giving us internal hom with respect to products. Plugging in the unit interval we see that a homotopy of maps may equivalently be seen as a path in the appropriate hom space or as a bunch of neatly arranged spaghetti in the target space. One could go even one step further and consider the fundamental groupoids of hom spaces to obtain a category enriched in groupoids.
• Looking then at pointed topological spaces we have an obvious topology on hom-sets from the previous example and the smash product is the monoidal structure making this into internal hom; we thus see that the smash product is just like the tensor product!
• Internal hom allows us to motivate the definition of natural transformations intrinsically: Given a bifunctor $F: \mathcal{C}_1 \times \mathcal{C}_2 \to \mathcal{D}$ we obtain a functor $\mathcal{C}_1 \to \mathcal{D}, X \mapsto F(X,Y)$ for any object $Y$, so we obtain a map $\text{Obj}(\mathcal{C}_2) \to \text{Obj}(\mathbf{Cat}(\mathcal{C}_1, \mathcal{D}))$. Setting $\mathcal{C}_2 = (\bullet \to \bullet)$ forces us to figure out what natural transformations have to be, and as an added bonus we also obtain a similar triple of characterisations of natural transformations as we did for homotopies of continuous maps above.
• The two different ways of defining a group action comes from internal hom. If we look at the following diagram, we see that we obtain the definition of a group action as a map $G \times X \to X$ satisfying a bunch of conditions by figuring out what the question mark is. This is also a nice little application of “the first lemma of category theory”. $\begin{array}{ccc} \mathbf{Set}(G \times X, X) & \cong & \mathbf{Set}(G, \mathbf{Set}(X,X)) \\ \cup & & \cup \\ ? & \cong & \mathbf{Mon}(G, \mathbf{Mon}(X,X)) \\ = & & = \\ ? & \cong & \mathbf{Grp}(G, \text{Aut}(X)) \end{array}$
• The chain rule in analysis is just a shadow of the fact that differentiation is functorial in the right context. The right context is of course that of differentiable manifolds.

• This is not really an example, but I found it amazing when I learned that arbitrary (co-)limits may be constructed from (co-)products and (co-)equalizers. This made it seem much less unreasonable to me that so many categories would have arbitrary (co-)limits.

A book you may want to check out for some inspiration is Algebra: Chapter 0 by Paolo Aluffi. Aluffi offers the most gentle introduction to category theory that I know of based on some very elementary examples.

### Re: A Call for Examples

This is a great list, thanks. I recall also being very happy when I first understood how the universal properties distinguish the cartesian, smash, and wedge products of based spaces.

A small omission: the hom-set between two $R$-modules is always an abelian group, but it’s only an $R$-module if the ring is commutative.

I like this example because many of the natural examples of enriched categories are self-enriched, which can be confusing when you’re first introducing the definition.

Thanks also for mentioning the Aluffi reference, which I wasn’t aware of. I’ll check it out.

Posted by: Emily Riehl on December 29, 2014 8:33 PM | Permalink | Reply to this

### Re: A Call for Examples

I’ve also enjoyed the more-particular example of product and coproduct in the division ordering of $\mathbb{N}$, which also seems to be the oldest example of an object defined by a universal property (being in Euclid).

Posted by: Jesse C. McKeown on December 30, 2014 4:11 PM | Permalink | Reply to this

### Re: A Call for Examples

Just noticed a little typo. $\mathbf{Mon}(X,X)$ should be $\mathbf{Set}(X,X)$.

### Re: A Call for Examples

I found it difficult to follow the details of Urysohns lemma & Tietzes Extension theorem; but I did notice (much, much later) that Tietzes theorem has a categorical interpretation, and thus a categorical interpretation of normality; and from this Urysohns Lemma is an easy deduction.

Since I never really managed to understand point-set topology, I’ve no idea whether its important ‘sociologically’…

Another example, which is more ‘advanced’ but because of the generality that Grothendieck worked in made it actually easy, is the first part of his Kansas notes where he showed how fibre-bundles (which in his definition is simply a morphism) can be decomposed and then put back together again. I found his explanation much easier to follow for example than the explanation in Michors book, Natural Operations in Differential geometry, or Steenrods book on Bundles which completely confused me.

Posted by: Mozibur Ullah on December 29, 2014 7:53 PM | Permalink | Reply to this

### Re: A Call for Examples

Great suggestions.

For any bystanders, the Kansas notes can be found here. The basic theory of fiber bundles is developed in chapter 1.

Posted by: Emily Riehl on December 29, 2014 9:31 PM | Permalink | Reply to this

### Re: A Call for Examples

Following this example, you can interpret the topology on a simplicial complex as a quotient of the coproduct of the simplices which constitute it. Then you can prove that such a complex is Hausdorff and normal, following the same categorical interpretation of Urysohn’s lemma alluded to above.

In a similar vein weak, strong, and limit topologies will be of interest to any students with a background in analysis.

Posted by: Josh Drum on January 4, 2015 12:33 AM | Permalink | Reply to this

### Re: A Call for Examples

Van Kampen’s theorem, which looks very intimidating and arbitrary when described only in gory explicitness, really just says that the fundamental group(oid) functor preserves (certain, homotopy) pushouts.

Posted by: Mike Shulman on December 30, 2014 12:25 AM | Permalink | Reply to this

### Re: A Call for Examples

Indeed! I’m certainly planning to mention this example.

Posted by: Emily Riehl on December 30, 2014 7:12 PM | Permalink | Reply to this

### Re: A Call for Examples

I think this reformulation hides something. There’s an abstract-nonsense argument you can write down as follows: the fundamental groupoid functor is left adjoint (in a suitable $\infty$-categorical sense) to the inclusion of homotopy 1-types into homotopy types, and consequently it preserves homotopy colimits. This does not buy you van Kampen’s theorem because you still need to know something which is not abstract nonsense about when an actual pushout of actual topological spaces is also a homotopy pushout.

Posted by: Qiaochu Yuan on January 2, 2015 10:31 AM | Permalink | Reply to this

### Re: A Call for Examples

I didn’t mean to make a claim about the proof of van Kampen’s theorem, only its statement. When proving it, the abstract nonsense does buy you a lot, but I agree, not everything.

Posted by: Mike Shulman on January 2, 2015 6:09 PM | Permalink | Reply to this

### Re: A Call for Examples

And how about the Curry-Howard isomorphism?

Posted by: Mike Shulman on December 30, 2014 5:34 AM | Permalink | Reply to this

### Re: A Call for Examples

I know this is meant to refer to some correspondence between type theory, $\lambda$-calculus, and cartesian closed categories, but I’ve never understood the precise mathematical content of the statement (likely because I’m only really comfortable with the last of these).

Can you explain it to a novice?

Posted by: Emily Riehl on December 30, 2014 7:18 PM | Permalink | Reply to this

### Re: A Call for Examples

A simple way of putting it is that the logical rules governing connectives like “and”, “or”, and “implies” are also special cases of the general exponential laws mentioned above. For instance, after categorifying John’s equation $a^{b+c} = a^b \times a^c$ to an isomorphism $A^{B+C} \cong A^B \times A^C$ in a cartesian closed category, we can then conclude, by interpreting $\times$ as “and”, $+$ as “or”, and exponentials as “implies”, that the logical statements “if B or C, then A” and “if B then A, and also if C then A” are equivalent.

Posted by: Mike Shulman on December 30, 2014 10:55 PM | Permalink | Reply to this

### Re: A Call for Examples

The main confusing thing about the “Curry-Howard isomorphism” is that it doesn’t — as far as I can tell — mainly state something is isomorphic to something else. So, I prefer the term Curry–Howard correspondence (and so does Wikipedia).

Posted by: John Baez on December 31, 2014 5:46 AM | Permalink | Reply to this

### Re: A Call for Examples

I had been dubbed “isomorphism” a long time ago because it established an equivalence between phenomena from logic and constructions from computation which extend the proof-as-program correspondance. Namely between (1) the deduction theorem of Hilbert-style proof systems and abstraction in combinatory logic (2) cut elimination in natural deduction and $\beta$-reduction in $\lambda$-calculus.

All of this is a bit passé nowadays, so the word “correspondance” is increasingly prefered.

Posted by: Arnaud Spiwack on January 3, 2015 10:56 AM | Permalink | Reply to this

### Re: A Call for Examples

One reason why I got interested in Category Theory, is that it suggested that such a basic concept such as Addition and Multiplication was worth looking at again. After all multiplication, as it names suggests is a ‘multiple’ of addition.

In category theory, its shown that a deeper consideration is duality. ie Addition is dual to multiplication.

And its deeper not just in this context; but also for other categories such as groups and so on.

I know this is basic stuff to sophisticated CT’s on this site; but such a new discovery about the basics - to my mind is profound.

Another consideration I found valuable is that it upends the traditional distinction between sets and functions; where functions are ‘derived’ concepts from sets; that is functions are not basic. Instead the notion of a morphism becomes basic.

At least for me, this is important in mathematical philosophy where the prevailing mainstream notion is the timeless platonic realm; by making the idea of process or morphism basic suggests the possibility of a philosophy of mathematics that is implicitly Heraclitian (changeful) rather than Parmenidian (changeless); I’d be curious how this works, if at all, with Whiteheads ‘Process and Reality’.

To have a basic philosophy of mathematics that is changeful, might be important to the problem of time in physics - but I know very little about either area. Has any work been done in this area?

Posted by: Mozibur Ullah on December 30, 2014 11:05 AM | Permalink | Reply to this

### Re: A Call for Examples

Posted by: sameer gupta on December 30, 2014 11:45 AM | Permalink | Reply to this

### Re: A Call for Examples

Posted by: Emily Riehl on December 30, 2014 7:20 PM | Permalink | Reply to this

### Re: A Call for Examples

I am intrigued by the notion “Cambridge-style”. Is this a well-known classification? What does it mean?

Posted by: Tom Ellis on January 1, 2015 3:47 PM | Permalink | Reply to this

### Re: A Call for Examples

No, I made it up as a playful way of acknowledging my personal history.

My first exposure to category theory was as a Part III student at Cambridge. The course is offered every year and, judging from the lecture notes that I periodically stumble across online, seems to be of uniformly high quality.

What I meant is that I’m hopeful that the course won’t simply provide a passing acquaintance with categorical language but that my students will be able to probe some of the central notions (e.g., representably) from several different perspectives (the Yoneda lemma, universal objects, …).

Posted by: Emily Riehl on January 2, 2015 1:25 AM | Permalink | Reply to this

### Re: A Call for Examples

There are two very categorical constructions that students are likely to know pretty well. The first one is quotients (this function respects $R$, so it’s defined on $A/R$). The second is free objects (left adjoints), of which I guess they have mostly an intuitive grasp and category theory would probably help cement a better understanding.

Posted by: Arnaud Spiwack on January 3, 2015 11:10 AM | Permalink | Reply to this

### Re: A Call for Examples

These are both good suggestions.

A simple but generally useful fact is that the quotient map $A \to A/R$, as an instance of a coequalizer, is always an epimorphism. Thus to define a map $A/R \to X$ it suffices to define a morphism $A \to X$ that respects $R$.

My history is a little shaky but I believe that the projection from a group to its quotient by a normal subgroup was a particular example used by Emmy Noether (who profoundly influenced Mac Lane) when emphasizing the importance of homomorphisms in group theory.

Posted by: Emily Riehl on January 3, 2015 8:04 PM | Permalink | Reply to this

### Re: A Call for Examples

Have not had time to read all the answers, so apologies if this comes up elsewhere, but I am fond of looking at various constructions in functional analysis and going “ooh, that’s an adjunction”.

Case 1: the bidual monad on the category of Banach spaces (with either linear maps of norm at most 1, or all bounded linear maps, as morphisms). I found that the categorical perspective helped keep track of e.g. the two ways of putting the second dual inside the fourth dual. Also, several times I’ve seen people refer to the “Dixmier projection” of the triple dual onto a canonicaly embedded image of the first dual; this is just one of the triangle identities for the underlying adjunction.

Case 2: Every analyst who works with $\ell^1$ of some indexing set knows that describing bounded linear maps from $\ell^1(S)$ to a Banach space $E$ is easy; these “are just the same as” bounded $E$-valued functions on $S$, since any such function extends uniquely to a bounded linear map $\ell^1(S)\to E$ by linearity and continuity. But this is essentially just the observation that the Set-valued functor assigning to a Banach space its closed unit ball has a left adjoint, which also makes it transparent that every Banach space $E$ arises as a quotient of some $\ell^1(S)$ (take the co-unit of the adjunction and do some checking).

(This must also be related to the canonical isometric isomorphism $\ell^1(S) \odot \ell^1(T) \to \ell^1(S\times T)$ where $\odot$ is the “natural” monoidal structure on Banach spaces – the continuous version of this isomorphism, using $L^1$, is usually attributed to work of some guy who gave up a very promising early career in functional analysis in order to go and reinvent algebraic geometry.)

Case 3. A nice (but atypical) example of a commutative unital Banach algebra is $C(X)$, the space of continuous complex-valued functions on a compact Hausdorff space $X$. The Gelfand representation theorem (no star neded here!) is usually presented in analysis courses in the slightly vague form of: every commutative unital Banach algebra A admits some norm-non-increasing unital homomorphism into some $C(\Phi_A)$ where the compact Hausdorff space $\Phi_A$ is abstractly and obscurely defined; and then for all the examples you need to learn for your exam, $\Phi_A$ can be “identified with” some familiar space like the disc/disk or the circle or the interval. It wasn’t until a while after taking such a course that I realized I was much happier saying: the contravariant functor from CpctHff to CBA, $X \mapsto C(X)$, has a left adjoint. This makes it clear and precise what is special about choosing $\Phi_A$ as a carrier space, and also that it is determined uniquely up to homeomorphism by a universal property.

Posted by: Yemon Choi on January 4, 2015 7:45 PM | Permalink | Reply to this

### Re: A Call for Examples

I just thought of another example, which I am very fond of. An action of a group $G$ is of course nothing but a functor $F: G \to \mathbf{Set}$. The quotient $F(\ast)/G$ of the action (where $\ast$ denotes the only object in $G$) is then given by $\colim F$. This can be generalised to actions of categories; i.e. functors $F: \mathcal{C} \to \mathbf{Set}$. One can then again describe $\colim F$ as a set of orbits by noting that it is isomorphic to the connected components of the category of elements of $F$. This description of $\colim F$ is actually quite useful, see e.g. Corollary 2.4.6 in Kashiwara Shapira, which is used e.g. in Proposition 2.5.2.
Related examples include linear representations of groups and modules over a ring.