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Braided Monoidal Category

A braided monoidal category is a monoidal category equipped with natural isomorphisms c_(X,Y):X tensor Y->Y tensor X satisfying the two hexagon identities. For a strict monoidal category these identities read

c_(X,Y tensor Z)=(1_Y tensor c_(X,Z))(c_(X,Y) tensor 1_Z),

and

c_(X tensor Y,Z)=(c_(X,Z) tensor 1_Y)(1_X tensor c_(Y,Z)).

In general, the associativity isomorphisms enter these formulas. The braiding is symmetric when c_(Y,X)c_(X,Y)=1_(X tensor Y) (Etingof et al. 2015, pp. 195-197). Vector spaces over a fixed field with c(x tensor y)=y tensor x give a symmetric example. A braided monoidal functor preserves the braidings as well as the monoidal structure (Etingof et al. 2015, p. 196).

See also

Braided Monoidal Functor, Monoidal Category, Monoidal Functor, Tensor Category

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References

Etingof, P.; Gelaki, S.; Nikshych, D.; and Ostrik, V. Tensor Categories. Providence, RI: Amer. Math. Soc., 2015.

Cite this as:

Weisstein, Eric W. "Braided Monoidal Category." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BraidedMonoidalCategory.html

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