北京雁栖湖应用数学研究院

In ordinary linear algebra, splitting a projection produces a subspace. In higher algebra, algebras and monads behave as higher-dimensional idempotents, and splitting them produces categories and higher categories of modules. This process—unitary higher idempotent completion—is the central theme of this two-semester sequence.
Beginning with the complex numbers, we develop the staircase
C -> Hilb -> 2Hilb -> 3Hilb
The first semester introduces finite-dimensional Hilbert spaces and operator algebras, modules and bimodules, Morita equivalence, semisimple unitary categories, and ordinary unitary Cauchy completion, culminating in the construction and classification of 2-Hilbert spaces. The second semester categorifies this story through unitary multitensor categories, algebra objects, module categories, 2-categories, and unitary condensation completion, culminating in the construction and representation theory of 3-Hilbert spaces.
All constructions will be finite-dimensional, with an emphasis on concrete examples, graphical calculus, and complete proofs of the central reconstruction theorems. Finite-dimensional linear algebra and mathematical maturity are assumed; no prior category theory, operator algebra, or higher algebra is required.

Unitary Higher Algebra I: From Hilbert Spaces to 2-Hilbert Spaces
This course develops the finite-dimensional foundations of unitary higher algebra. We begin with Hilbert spaces, adjoint operators, positivity and projections, and then study finite-dimensional C^*-algebras, traces, modules, bimodules and Morita equivalence. Category theory will be introduced from the beginning, including functors, natural transformations, adjunctions, linear categories, semisimplicity, and additive and idempotent completion.
The unifying question is how ordinary linear algebra changes when vector spaces are replaced by categories. This leads to unitary categories and 2-Hilbert spaces: finite semisimple unitary categories equipped with compatible trace data. The course culminates in the reconstruction of 2-Hilbert spaces as categories of unitary modules over finite-dimensional traced C^*-algebras, and in the equivalence
H^*Alg\cong 2Hilb
The course is self-contained beyond finite-dimensional linear algebra and may be taken independently of the second semester.