# Yang-Baxter systematically

Recommended reading:
- ``Braided tensor categories'' by Joyal and Street
[Adv. Math. 102 (1993), 20-78],
- ``2-Categories and Zamolodchikov tetrahedra equations'' by
Kapranov and Voevodsky [Proc. Symp. Pure Math. 56 (1994), 177-260],
- ``Newton polytopes of the classical resultant and discriminant'' by
Gelfand, Kapranov and Zelevinsky [Adv. Math. 84 (1990), 237-254].

In a braided monoidal category, the Yang-Baxter hexagon commutes for each
triple of objects. There are two different proofs for this, subdividing the
hexagon in two different ways. I looked at the converse: given a monoidal
category together with Yang-Baxter system, a collection of objects Y and a
collection of arrows R_{A,B}:
A B =>
B A,
for A, B in Y such that the Yang-Baxter hexagon commutes for each triple of
objects from Y, does this give a braided monoidal category?
Using the 2-functor st: **MCat** -> **MCat** I constructed a
braided monoidal category from a monoidal category with a Yang-Baxter system,
with naturality of the braiding in each variable being proven via the two
subdivisions of the Yang-Baxter hexagon respectively.
This construction gives the object part of a biequivalence between the 2-category
of braided monoidal categories whose only arrows are composites of R's and
the 2-category of Yang-Baxter systems where Y consists of all objects of the
monoidal category.
See the paper
``Higher-dimensional Yang-Baxter and Zamolodchikov equations''
(in preparation).

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