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During the Vietnam war, Grothendieck taught math to the Hanoi University mathematics department staff, out in the countryside. Hoàng Xuân Sính took notes and later did a PhD with him - by correspondence! She mailed him her hand-written thesis.

What was it about?

(1/n)

First, some background. Hoàng Xuân Sính was born in 1933, one of seven children of fabric merchant. She got her PhD in 1975, and later became the first female math professor in Vietnam. In 1988 she started the first private university in Vietnam.

(2/n)

In 2003 she was awarded France's Ordre des Palmes Académiques. She is still alive! I hope someone has interviewed her, or does it now. Her stories must be very interesting.

But what about her thesis?

(3/n)

Her thesis classified Gr-categories, which are now called '2-groups' for short. A 2-group is the categorified version of a group: it's a monoidal category where every object and morphism is invertible.

(An object X is invertible if there's Y with X⊗Y ≅ Y⊗X ≅ I.)

(4/n)

From a 2-group you can get two groups:

the group G of isomorphism classes of objects, and

the group H of automorphisms of the unit object I.

H is abelian, and G acts on H. But there's one more thing! The associator can be used to get a map

a: G³ → H

(5/n)

The pentagon identity for the associator implies that

a: G³ → H

obeys an equation. And this equation is familiar in the subject of group cohomology: it says a is a '3-cocycle' on the group G with coefficients in H.

So, we can classify 2-groups using cohomology!

(6/n)

To prove this, Sính needed to show something else too: cohomologous 3-cocycles give equivalent 2-groups. (Equivalent as monoidal categories, that is.)

This connection between 2-groups and cohomology is no coincidence! It's best understood using a bit more topology.

(7/n)

Any connected space with a basepoint, say X, has a fundamental group. But it also has a fundamental 2-group! This 2-group has G = π₁(X) and H = π₂(X). And if all the higher homotopy groups of X vanish, this 2-group knows *everything* about the homotopy type of X!

(8/n)

So, Sính's thesis helped nail down the complete structure of 'homotopy 2-types': that is, homotopy types of spaces with πₙ(X) = 0 for n > 2. The most exciting, least obvious part of this is the 3-cocycle on π₁(X) with values in π₂(X), coming from the associator.

(9/n)

So, Sính's thesis illuminated one of the simplest - yet still important - special cases of Grothendieck's 'homotopy hypothesis', namely that homotopy n-types correspond to n-groupoids.

You can see it along with a summary in English here:

pnp.mathematik.uni-stuttgart.d

(10/n)

That website also has a few nice photos of Grothendieck in Vietnam. He started teaching the Hanoi University math department staff in the countryside near Hanoi. Later they moved to Đại Từ.

I think the woman in front of him is Hoàng Xuân Sính.

(11/n, n = 11)

@johncarlosbaez this is amazing and something simple and powerful to give to someone who asks what category theory gives us other than generality nonsense. Thanks for sharing it.

@jesusmargar - Thanks! Indeed that's one reason I've been working on this: to show category theory can be practical.

@AlejandroP - "associator" means a couple different things. For one, the cartesian product of sets is not associative, but there's a bijection

$$ (X \times Y) \times Z \to X \times (Y \times Z) $$

and this is called the associator.

For more, try this:

en.wikipedia.org/wiki/Associat

@johncarlosbaez ohh is similar to the commutator, Which makes sense of the name

👍

@AlejandroP - yes, and there's another meaning of associator, used in nonassociative algebras: there it means

$$ (xy)z - x(yz) $$

so it's a lot like a commutator!

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