Coherence theorems and conformal field theory. (English) Zbl 0789.18006

Seely, R. A. G. (ed.), Category theory 1991. Proceedings of an international summer category theory meeting, held in Montréal, Québec, Canada, June 23-30, 1991. Providence, RI: American Mathematical Society. CMS Conf. Proc. 13, 321-328 (1992).
A symmetric monoidal category \(M\) is a category equipped with a (binary) tensor product which is associative up to a natural isomorphism \(a\) and commutative up to such a \(c\). Moreover, there is an object \(I\) which up to natural isomorphisms acts as a left and right unit. A coherence theorem (due independently to the author and Stasheff (1971)) stated that if a suitable pentagon involving \(a\) and four factors commutes, then every larger diagram on \(a\) commutes. A corresponding theorem involving \(a\) and \(c\) requires also that the \(a\) hexagon commutes, and that \(c^ 2=1\). A “braided” monoidal category omits this last condition.
This paper describes the recent surprising use of these results in conformal field theory, string theory, the Yang-Baxter equations and Tannakian categories. For a recent example of the first, see the paper “Braided monoidal 2-categories” by M. Kapranov and V. Voevodsky [J. Pure Appl. Algebra 92, 241-267 (1994)].
For the entire collection see [Zbl 0771.00047].
Reviewer: S.Mac Lane


18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18A15 Foundations, relations to logic and deductive systems
81T05 Axiomatic quantum field theory; operator algebras
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory