Braidings the post-modern way

Sjoerd Crans, CTRC Seminar 10 November 1998

Braidings (on monoidal categories) are usually defined as the concept that goes between monoidal and symmetric monoidal: it is not that A tensor B equals B tensor A, but there is a natural isomorphism between these, satisfying certain axioms. Braidings on monoidal 2-categories are usually motivated by replacing axioms for braidings on monoidal categories by isomorphisms, which then have to satisfy certain axioms themselves. (Some people call this ``categorification''.) This list of axioms is quite long, has been put together by trial and error, and does not give any clues for higher dimensions.

Braidings can also be defined by observing that a braided monoidal category is a tricategory with (one object and) one arrow, from which it is formally distinguished by looping and delooping operations Omega2 and Sigma2 respectively (to wit, one deloops a braided monoidal category to a one-arrow tricategory). Invoking the coherence theorem for tricategories, a braiding on C is essentially 0-composition in a Gray-category Sigma2 (C). Because this composition is ``governed by'' Gray's tensor product of 2-categories, and noticing that Sigma2 (C) (*,*) = Sigma (C), this gives braiding as a (2-)functor

R: Sigma (C) tensorG Sigma (C) -> Sigma (C) .

The axioms on a braiding are now the familiar axioms for composition: functoriality in each variable, and associativity ((pseudo-)naturality of R is already contained in it being a functor).

Now this definition is easily generalized to higher dimensions: calling the structures to which weak n-categories are equivalent n-dimensional teisi - these are hypothetical above dimension 5, and the equivalence is hypothetical above dimension 3 - one defines a braided nD tas as an (n+2)D tas with one arrow, and assuming that there is a tensor product of teisi ``governing'' composition, the braiding is - again - a functor Sigma (C) tensor Sigma (C) -> Sigma (C), which is (natural,) functorial, and associative.

See the paper ``On braidings, syllepses, and symmetries''.

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