Hoàng Xuân Sính's Thesis: Categorifying Group Theory
John C. Baez
TL;DR
Hoàng Xuân Sính's thesis develops Gr-categories (2-groups) as monoidal categories with inverses at both object and morphism levels and shows their equivalence classes are classified by quadruples $(G,A,\rho,[a])$, with $[a]$ a class in $H^3(G,A)$. It connects these algebraic structures to topology via classifying spaces $BG$ and the fundamental 2-group $\Pi_2(X)$, establishing that Gr-categories correspond to connected pointed 2-types and that $BG$ encodes the Postnikov data $\pi_1(BG)=G$, $\pi_2(BG)=A$, and $[a]\in H^3(G,A)$. The thesis also shows that strict Gr-categories are equivalent to crossed modules $(G,H,t,\rho)$ and that restrained Pic-categories correspond to 2-term chain complexes, linking categorified symmetry to classical invariants like Picard groups and line bundles. By bridging algebra, topology, and later physics, the work provides a foundational framework for higher categorical symmetry and higher gauge theory, illuminating how 2-group structures govern symmetries of symmetries and their topological realizations.
Abstract
During what Vietnamese call the American War, Alexander Grothendieck spent three weeks teaching mathematics in and near Hanoi. Hoàng Xuân Sính took notes on his lectures and later did her thesis work with him by correspondence. In her thesis she developed the theory of "Gr-categories", which are monoidal categories in which all objects and morphisms have inverses. Now often called "2-groups", these structures allow the study of symmetries that themselves have symmetries. After a brief account of how Hoàng Xuân Sính wrote her thesis, we explain some of its main results, and its context in the history of mathematics.