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