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## "Braided monoidal categories", Andre Joyal, Ross Street, 1986

*Reviewed December 19, 2023*

*Citation:* Joyal, AndrĂ©, and Ross Street. "Braided monoidal categories." Mathematics Reports 86008 (1986).

*Web:* http://web.science.mq.edu.au/~street/JS1.pdf

*Tags:* Mathematical, Foundational, Monoidal-categories

In this unpublished work, Joyal and Street introduce the notion of a "braided monoidal category". The notion
of a symmetric monoidal category was known earlier, having been introduced by Saunders MacLane in 1963:

> MacLane, Saunders. "Natural associativity and commutativity." Rice Institute Pamphlet-Rice University Studies 49.4 (1963).

The insight of Joyal and Street's work is that there a great deal of interesting examples of monoidal categories
which have a braiding but are not symmetric. A large portion of the paper is spent listing such examples.

After setting up coherence axioms and giving examples, the rest of this paper is spent on the classification
of compact braided monoidal groupoids, where by *compact*> Joyal and Street mean that for each object $A$
there exists a dual object $A^*$ such that $A \otimes A^*$ is isomorphic to the tensor identity. The classification theorem
is that compact braided monoidal groupoids correspond to triples (G,M,k), where (G,M) are abelian groups and k:G->M is a quadratic function.
In working up to this result, the authors discuss the (by now well-known) 2-categorical interpretation of H^2 cohomology classes.