# Extending braidings to tensor powers

Given a braiding $\Psi: X \otimes Y \to Y \otimes X$ for two objects $X,Y$ in a monoidal category, it seems reasonable to assume that $\Psi$ extends uniquely to a braiding $X^k \otimes Y^l \to Y^l \otimes X^k$, for all $k,l \in {\mathbb N}$. The proof would surely be based upon the Yang--Baxter property of $\Psi$ and the fact that one can express any permutation as a product of transpositions. However, I can't seem to write it down exactly.

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## 1 Answer


The challenge is how to efficiently prove that, given any $k$ and $l$ and any possible choice of way you build your braiding, that you get the same map. This is the sort of thing that is known as a "coherence theorem." These are often stated in the fashion: Every diagram in (some category) commutes, where the category is a sort of "free category" whose morphisms are the structural ones that you are talking about.

There are coherence theorems for the associativity morphisms in a monoidal category (see MacLane's book, Chapter VII, Section 2) which is what allows you to drop parentheses when doing iterated tensor products. And similarly there is a coherence theorem for morphisms built from the braidings in a braided monoidal category. This is spelled out in detail in Joyal and Street's 1993 article Braided Tensor Categories.

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