The n-Category Café

It sounds to me like you may be looking for something to convince an unreasonable skeptic! I feel like all these examples ought to be enough to convince a reasonable one… (-:

I suppose it’s obvious by now that I’m using a specific request to drive home the need for ‘small but striking examples’ in favor of category theory.

Last fall, Eugenia Cheng told me of a visit to some university to give a colloquium talk. The host greeted her with the observation that he doesn’t regard category theory as a field of research. OK, he was probably a bit extreme, but milder versions of that view are quite common. Now, one possible response is to regard all such people as unreasonable and talk just to friends (who of course are the reasonable people!). This is not entirely bad, because that might be a way to buy time and gain enough stability to eventually prove the earth-shattering result that will show everyone! Another way is to take up the skepticism as a constructive challenge. This I suppose is what everyone here is doing at some level, anyways.

Other than the derived loop space, which is not exactly small, Urs’ examples are all of the simple subtle sort that can, over time, contribute to a really important change in scientific outlook and maybe even the infrastructure of a truly glorious theory. For example, I agree wholeheartedly about the horrors of the old tensor formalism. But it’s not unreasonable to ask for more striking accessible evidence of utility when it comes to the current state of category theory.

The importance of small insights and language that gradually accumulate into the edifice of a coherent and powerful theory is the usual interpretation of Grothendieck’s ‘rising sea’ philosophy. However, the process is hardly ever smooth along the way, especially the question of acceptance by the community. I’m not a historian, but I’ve studied arithmetic geometry long enough to have some sense of the changing climate surrounding etale cohomology theory, for example, over the last several decades. The full proof of the Weil conjectures took a while to come about, as you know. Acceptance came slowly with many bits and pieces sporadically giving people the sense that all those subtleties and abstractions are really worthwhile. Fortunately, the rationality of the zeta function was proved early on. However, there was a pretty concrete earlier proof of that as well using $p$-adic analysis, so I doubt it would have been the big theorem that convinced everyone. One real breakthrough came in the late sixties when Deligne used etale cohomology to show that Ramanujan’s conjecture on his tau function could be reduced to the Weil conjectures. There was no way to do this without etale cohomology and the conjecture in question concerned something very precise, the growth rate of natural arithmetic functions. This could even be checked numerically, so impressed people in the same way that experimental verification of a prediction does in physics. Clearly something deep was going on. Of course there were many other indications. The construction of entirely new representations of the Galois group of $\mathbb{Q}$ with very rich properties, the unification of Galois cohomology and topological cohomology, a clean interpretation of arithmetic duality theorems that gave a re-interpretation of class field theory, and so on.

For myself, being a fan of you folks here, I believe this kind of process is going on in category theory. But I don’t think you have to be too unreasonable to doubt it. In a similar vein, I don’t agree with Andrew Wiles’ view that physics will be irrelevant for number theory, but also think his pessimism is perfectly sensible.

I suppose I’m trying to make the obvious point that the presence of pessimists can be very helpful to the development of a theory, in so far as the optimists interact with them in constructive ways. I haven’t been coming to this site much lately, because the bit of internet time I have tends to be absorbed by Math Overflow. But I did catch David’s recent post on Frank Quinn’s article, which ended up as a catalyst for my MO question.

At the Boston conference following the proof of Fermat’s last theorem, Hendrik Lenstra is said to have said something like this: ‘When I was young, I knew I wanted to solve Diophantine equations. I also knew I didn’t want to represent functors. Now I have to represent functors to solve Diophantine equations!’ So should we conclude that he was foolish to avoid representable functors for so long? I wouldn’t.

This response to the MO question brings up the importance of knowing the specific isomorphism between some Hilbert spaces given by the Fourier transform. This is an excellent example, especially when we consider how it relates to the different realizations of the representations of the Heisenberg group and the attendant global issues, say as you vary over a family of polarizations. But I couldn’t resist recalling Irving Segal’s insistence that ‘There’s only *one* Hilbert space!’ Obviously, he knew, among many other things, the different realizations of the Stone-Von-Neumann representation as well as anyone, so you can take your own guess as to the reasoning behind that proclamation. He certainly may have lost something through that kind of philosophical intransigence. But I suspect that he, and many around him, gained something as well.

I’ll try one more shop-worn example (but one that hasn’t made it yet into undergraduate textbooks, as far as I know). I believe it shows that there are situations where it’s natural to think of groups as one object categories.

Suppose $f:X\to Y$ is part of homotopy equivalence of topological spaces and you want to convince your students that the induced map on fundamental groups $\Pi f:\pi_1 (X,x) \to \pi_1 (Y, f(x))$ is an isomorphism of groups. Here is the way I tried it last month.

A map $f:X\to Y$ defines/extends to a map of fundamental groupoids $\Pi f:\Pi X \to \Pi Y$. A homotopy $\phi$ from $f$ to $g:X\to Y$ defines a natural isomorphism $\Pi\phi: \Pi f \Rightarrow \Pi g$. Hence if $f:X\to Y$ is part of a homotopy equivalence, then $\Pi f:\Pi X \to \Pi Y$ is part of an equivalence of categories. In particular, it’s fully faithful. Since $\pi_1 (X,x) = \Pi X (x,x)$, $\Pi f:\pi_1 (X,x) \to \pi_1 (Y, f(x))$ is a bijection.

I suppose it’s obvious by now that I’m using a specific request to drive home the need for ‘small but striking examples’ in favor of category theory.

That wasn’t obvious to me, as I tried to indicate a few times. It seemed to me that we were talking about why one should not conflate isomorphisms with identities.

But okay, it seems we effectively agreed that only a straw man would do that anyway! ;-)

But category theory is not what happens when we identify a category. Or an isomorphism. Much as number theory is not what happens when I find that the number of apples in my grocery bag is 3.

Category theory is what happens when the Yoneda lemma is applied, the adjoint functor theorem, the monadicity theorem, the Grothendieck construction-theorem and such things.

Of course in order to apply all that, first the categories need to be identified. It’s the observation that many different constructions and statements in math all come down to special cases of such abstract statements in category theory that we care about trying to identify categorical structures as much as possible. It’s because the whole world is divided into abstract structures and their concrete models. Category theory tries to understand the abstract structures shared by all concrete models.

Okay, so let’s shift gears and start giving small but striking examples for category theory .

Here is one that I found small and striking when I became aware of it recently:

Theorem. (Tannaka duality for permutation representations).

Let $G$ be a group and $Rep_{perm}(G)$ its category of permutation representations. Let $F : Rep_{perm}(G) \to Set$ be the forgetful functor that remembers the representing set.

Then $G$ may be recovered as the endomorophisms of this fiber functor $G \simeq End(F)$.

To prove this, apply the Yoneda lemma precisely four times in a row. And nothing else. The details are spelled out here.

This is pretty cool, if you know that Tannaka duality appears in plenty of guises throughout mathematics. That simple Yoneda-power-proof above will not work verbatim for more sophisticated examples, but it determines the basic pattern.

But there are striking examples of where that proof does go through verbatim.

Here is one: Let $X$ be a locally contractible topological space and $Sh_{(\infty,1)}(X)$ the $(\infty,1)$-topos of $\infty$-stacks over it. Let $LConst(X)$ be the $\infty$-groupoid of locally constant $\infty$-stacks on $X$. For $x \in X$ any point, there is a forgetful fiber functor $F_x : LConst(X) \to \infty Grpd$ that produces the fiber of the covering $\infty$-bundle at $X$ given by the locally constant $\infty$-stack (in immediate analogy to the familiar statement obtained by discarding all the $\infty$s here.)

Then:

Corollary ($\infty$-Galois theory) The homotopy groups of $X$ at $x$ are those of the automorphisms of the fiber functor:

$$\pi_n(X,x) \simeq \pi_n \mathbf{B} Aut(F_x) \,.$$

Even if it may not immediately look like it, but the proof of this is, again, given by applying the $(\infty,1)$-Yoneda lemma four times in a row. This is just using that a reprentation of the fundamental $\infty$-group of a space on covering $\infty$-bundles is a “permuation representation”. Details are at $n$Lab: homotopy groups in an $(\infty,1)$-topos – In terms of monodromy and Galois theory.