How braidings the post-modern way give rise to old-fashioned braidings, and the failure of this in higher dimensions

Sjoerd Crans, CTRC Seminar 19 January 1999

A braided monoidal category C is a Gray-category Sigma2 (C) with one object and one arrow. The braiding is a functor R: Sigma (C) tensorG Sigma (C) -> Sigma (C) (where tensorG is Gray's tensor product of 2-categories) which satisfies unit and associativity axioms. Spelling this out, one gets arrows RA,B: A tensor B -> B tensor A, which satisfy naturality and functoriality. In fact, R-,? is a natural transformation - tensor ? => ? tensor -, and there is a commuting hexagon (the Yang-Baxter equation), which can be proven in two ways from naturality and functoriality.

A braided monoidal 2-category C is a 4D tas Sigma2 (C) with one object and one arrow. The braiding is a functor R: Sigma (C) tensor Sigma (C) -> Sigma (C) (where tensor is the tensor product of Gray-categories [tpgc]) which satisfies unit and associativity axioms. Spelling this out, one gets arrows RA,B: A tensor B -> B tensor A, and the naturality axioms of one dimension lower become 2-arrows Rf,B, RA,g, which themselves again satisfy naturality. In fact, R induces a pseudo-natural transformation C tensorG C -> C, not - tensor ? => ? tensor -, but (? tensor -)~ => ? tensor -, where (? tensor -)~ is the `reversal' of ? tensor -, obtained from it by swapping the coordinates and introducing some inverses which is necessary to make it into a functor C tensorG C -> C. Spelling out functoriality of R in both variables one gets an equation of the shape of Kapranov and Voevodsky's resultohedron N2,2. The two proofs of the Yang-Baxter equation in one dimension lower become 2-arrows, which Kapranov and Voevodsky call S+ and S-, and spelling out associativity of R one gets an equation between these 2-arrows.

A braided monoidal Gray-category C is a 5D tas Sigma2 (C) with one object and one arrow. The braiding is a functor R: Sigma (C) tensor Sigma (C) -> Sigma (C) (where there is no monoidal structure on 5D-Teisi but it is possible to define a 5D tas C tensor D via a presentation) which satisfies unit and associativity axioms. Spelling this out, one gets arrows RA,B: A tensor B -> B tensor A, 2-arrows Rf,B, RA,g, and the naturality axioms of one dimension lower become 3-arrows Ralpha,B, RA,beta, Rf,g, which themselves satisfy naturality and functoriality. However, the braiding does not induce a pseudo-natural transformation C tensor C -> C: (? tensor -)~ => ? tensor -, because in defining this there is a dimension-raising horizontal composition.

One dimension higher ? tensor - is not even reversible.

See the papers ``On braidings, syllepses, and symmetries'' and ``Localizations of transfors''.

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