Okay, just watched it. Great fun! Principle bundles have moral fibres. They’re the good, good, good, good fibrations! I’ll never forget that one.

On a more serious note, I was quite disturbed by some of the things Serre said regarding proofs in homological algebra and “identifications” like $\mathbb{N} \subset \mathbb{Z}$ which are strictly speaking not identifications, but injective homomorphisms.

I’ve been a bit depressed lately about this stuff. For one thing, I’ve been reading Bott and Tu. You’ll see all sorts of equality signs there where they really mean isomorphism. Sometimes its not even a canonical isomorphism.

Take for instance, Proposition 23.9 about how every bundle “is” a pullback of one over a Grassmannian:

Let $E$ be a rank $k$ complex vector bundle over a differentiable manifold $M$…. then there is a map $f$ from $M$ to some Grassmannian $G_k (\mathbb{C}^n)$ such that $E$ is the pullback under $f$ of the universal quotient bundle $Q$, that is, $E=f^{-1}Q$.

There’s more… at the end of the proof we find:

We can identify $V$ with $\mathbb{C}^n$, and $G_k(V)$ with $G_k(\mathbb{C}^n)$.

The dodgy things here are the statements “$E = f^{-1}Q$”, the use of the words “this map $f$” after the proposition (in violation of Serre’s advice), and the cavalier identification of $V$ with $\mathbb{C}^n$.

The depressing thing is that: * it’s all completely clear * what he’s saying. If we included the gory details about isomorphisms satisfying commutative diagrams of their own, just imagine how long and horrible Bott and Tu’s book would become!

You’ll find algebraic topologists doing this all the time. “$X$ is the colimit of such and such”, or “the classifying space”, and so on.

I know that as $n$-cafe regulars, we have been taught by John and others that “the” is always understood to mean “up to canonical isomorphism satisfying some natural commuting diagram”.

But somehow I still find it very awkward…

This business of “harmless identifications” has been exposed by category theory to be not-so-harmless. Who would have thought that the innocent isomorphism of tensor products of vector spaces,

(1)$V \otimes (W \otimes Q) \cong (V \otimes W) \otimes Q,$

would have been non-trivial? If you think its not: it *is*, because its precisely this associator which distinguishes the representation categories of the dihedral and quaternionic groups $Rep(D_8)$ and $Rep(Q_8)$.
The other way to proceed is by anafunctors. But somehow these * also * give me the heeby jeebies, because its kind of a non-algebraic description. Tensor product is no longer an operation in the strictest sense of the word.

To give a more concrete example: I work with 2-representations of groups on 2-Hilbert spaces. One of the things you need to do right at the beginning is to * choose *, for each morphism $f : H \rightarrow H'$ of 2-Hilbert spaces, a morphism $f^* : H' \rightarrow H$ which is left and right adjoint to $f$ - and yes, you must * choose * the unit and co-unit maps too!

That gives me the heeby-jeebies (axioms of choice nightmares), but it seems the only way to proceed… at least, in an “algebraic” fashion.

As the ladder of higher categories climbs ever higher and higher into the sky, I sense that mathematical terminology is less and less able to cope elegantly with the notion of “equality”. Sometimes it makes me want to quit this game, and go and do some ordinary maths, where there’s not a sinister presence in the corner which cracks its whip everytime you assert the “equality” of two things when you should have said “isomorphism satisfying some diagram” or worse… you know what I mean.

Think about the word “canonical”. What on earth does it mean nowadays? Suppose there exists a left adjoint to a functor $F : C \rightarrow D$. Most mathematicians will tell you that $F^*$ is given “canonically”. When pushed, they’ll admit it only exists up to a unique natural isomorphism which satisfies some diagram. See? In the good old days, the word “canonical” used to be a very strong word, which meant something like “this map * really * exists, independent of any choices”. Nowadays we have to use the term “supercanonical” for such a concept. Imagine how awkward it will be when we’re up to 5-categories!

## Re: How to Write Mathematics Badly

I haven’t found the hour to watch Serre yet, but I still want to mention Mathematics Made Difficult, which I’ve never gotten to look at.