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"2-categories and Zamolodchikov tetrahedra equations", Mikhail Kapranov, Vladimir Voevodsky, 1994

Reviewed November 10, 2023

Citation: Kapranov, Mikhail M., and Vladimir A. Voevodsky. "2-categories and Zamolodchikov tetrahedra equations." Proc. Symp. Pure Math. Vol. 56. No. Part 2. 1994.

Web: https://www.math.ias.edu/vladimir/node/71

Tags: Mathematical, Monoidal-categories

This paper introduced the notion of 2-vector space which is appropriate for quantum topology. Their main definition (definition 5.17) is that a 2-vector space is a lax module category over the ring category Vec which is module-equivalent to Vec^n for some n. 1-Morphisms are module functors, and 2-morphisms are module natural transformations. This definition is nice because, as a category, we can just worry about Vec^n. The rest of the 2-vector space theory comes in the extra structure, which allows you to do a higher category theory analogue of linear algebra.

Of course, there are more abstract ways of quantifying the notion of a 2-vector space which feel much more "natural" to mathematicians. Of particular note is the definition given in Martin Neuchl's PhD thesis:

> Neuchl, Martin. Representation theory of Hopf categories. Diss. Verlag nicht ermittelbar, 1997.

He defines a 2-vector space (definition 2.12) as a semisimple k-linear abelian category with finitely many isomorphism classes of simple objects. Of course, he doesn't use this language exactly but his words are equivalent to this, and he proves in the following lemma (lemma 2.13) that every semisimple k-linear abelian category is defined by the cardinality of its basis, and hence they are all equivalent to Vec^n for some n. This "semisimple k-linear abelian category" perspective is useful because fusion categories are defined as being "semisimple k-linear abelian", so this drastically reduces the amount of background needed for the definition of a fusion category.

Nowadays there are various references for higher linear algebra, which are more concise than Kapranov-Voevodsky's. When this paper was written the subject of higher category theory was very new, so the bulk of this paper is an introduction of the theory of 2-categories, with great detail on the axioms and coherence relations. The winner for conciseness and relevance to quantum topology is Yetter's preprint,

> Yetter, D. "Categorical linear algebra—A setting for questions from physics and low-dimensional topology." Kansas State University preprint (1993).

The best treatment of 2-categorical representation theory is Baez's big paper,

> Baez, John, et al. Infinite-dimensional representations of 2-groups. Vol. 219. No. 1032. American mathematical society, 2012.

Just like standard linear algebra, 2-linear algebra can be quantified in terms of matrices. Of course, the entries of these 2-morphism matrices will themselves be matrices. Hence, 2-morphisms are matrices of matrices. While certainly adding a great deal of complexity, these matrices-of-matrices can be implemented just as explicitly in software. This has been done successfully by David Roberts in his mathematica package "TwoVect":

> Roberts, Daniel A. "User Guide for TwoVect. m." (2012).