Is there a generalization of the notion of braided monoidal category that does not force the braiding $\gamma$ to be an isomorphism? I mean, it is of course possible to define such a kind of category, but is this a common notion with an established name?



Is there a generalization of braided monoidal category without the isomorphism requirement?
Day, Panchadcharam, and Street have a [paper](http://science.mq.edu.au/%7Estreet/laxcentre.pdf) on lax braidings, though I don't think it could be called a *common* notion. Anyway, "lax" seems to be the obvious terminology to try here.


