Suppose $V := \bigoplus\_{i \in \mathbb{N}}V\_i$ and $W := \bigoplus\_{i \in \mathbb{N}}W\_i$ are $\mathbb{N}$-graded vector spaces. Then their graded tensor power is defined by 
$V \bigotimes W := \oplus\_{n \in \mathbb{N}} \oplus\_{i+j=n} V\_i \otimes W\_j$ and there is a natural isomorphism $\sigma\_{V,W}: V \bigotimes W \to W \bigotimes V$ defined on homogeneous elements
by the 'sign convention'


$$\sigma\_{V,W}(v \otimes w) = (-1)^{\deg(v)\*\deg(w)}w \otimes v$$ 


What is the origin of this natural isomorphism? Is this the only one or are there others defined by other sign conventions?



Origin of the sign convention in the Tensor product of graded vector spaces
*Is this the only one or are there others defined by other sign conventions?*


As has already been mentioned - there are essentially two but the Koszul one is amongst other things, the only one which makes the tensor product of complexes a complex.


Since I have been interested in this topic of late, I would point you towards the following internal (within MathOverflow) references:


* For a recent discussion see:
[Tensor product of linear mappings versus chain complexes](https://mathoverflow.net/questions/93464/tensor-product-of-linear-mappings-versus-chain-complexes)
* In particular Simon Letner and Qiaochu Yauns' answers provide a nice conceptual backdrop.
* I also found Theo Johnson-Freyd's answer here quite illuminating:
[References for sign conventions in homological algebra](https://mathoverflow.net/questions/53315/references-for-sign-conventions-in-homological-algebra)
* Theo Johnson-Freyd's answer in this thread is also instructive:
[Sign convention for derivations in CDGAs](https://mathoverflow.net/questions/95795/sign-convention-for-derivations-in-cdgas)


