The n-Category Café

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!

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.

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.

I have a little story I like to tell about 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.

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 $\mathbb{N}$?

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.

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.

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.

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.

And how about the Curry-Howard isomorphism?

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?

https://www.quora.com/What-is-category-theory-23346/answer/Sameer-Gupta

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

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.

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.

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.

The union of two graphs is easy to guess and clear as a definition. But the product could be guessed with category theory and is very visual and concrete.