Let $C$ be a symmetric monoidal category. Fix an object $X$, let $S$ denote the symmetry $X \otimes X \to X \otimes X$. Also define $X^{\otimes n}$ by induction on $n$: $X^{\otimes 0} = 1$, $X^{\otimes (n+1)} = X^{\otimes n} \otimes X$. Now it is "absolutely clear" that we have a canonical action of the symmetric group $\mathfrak{S}_n$ on $X^{\otimes n}$, i.e. a homomorphism of groups $\mathfrak{S}_n \to \text{Aut}(X^{\otimes n})$. But what is a precise and short definition?

A transposition $\sigma_i = (i,i+1)$ acts as the composite of isomorphisms $X^{\otimes n} \cong X^{\otimes (i-1)} \otimes ((X \otimes X) \otimes X^{\otimes (n-i-1)}) \stackrel{S}{\cong} X^{\otimes (i-1)} \otimes ((X \otimes X) \otimes X^{\otimes (n-i-1)}) \cong X^{\otimes n}$

Now $\mathfrak{S}_n$ is freely generated by these $\sigma_i$ modulo the relations a) $\sigma_i^2 = 1$, b) $\sigma_i \sigma_j = \sigma_j \sigma_i$ for $i \neq j \pm 1$, c) $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$. It should be possible to test these relations above, using the coherence between the symmetry and the associator on $C$, but obviously it will be tedious.

I would like to define the action in element notation via $x_1 \otimes ... \otimes x_n \mapsto x_{\sigma(1)} \otimes ... \otimes x_{\sigma(n)}$, but is this possible at all? Perhaps using a weak-monadic representation of monoidal categories?

In my research I often have to define such morphisms which are "obvious" for the usual symmetric monoidal categories, but are nasty to write down and manipulate in the general case. After having checked many examples, meanwhile I have "trust" in these element definitions. But I wonder if any general machinery has been established for this. Note that in my former question I asked about checking equality, but in this question about construction in monoidal categories.

Quantum Fields and Strings. They develop precisely the categorical machinery necessary to write formulas like the one you want. – Theo Johnson-Freyd Jul 9 '11 at 15:01