The question is in the title.


To make the statement more precise, is is true that for any given monoidal category $(\mathcal C, I, \otimes)$ there exists at least one braiding $\beta$? In other words, does the forgetful functor from braided monoidal categories admit a section (not a left adjoint!)?


I strongly suspect the answer to be 'no', or at least 'yes but not functorially'.



Does every monoidal category admit a braiding?
No, sometimes there are even $x,y$'s with no abstract isomorphism $x\otimes y \cong y\otimes x$.


Here are two families of examples:


* Monoids, viewed as discrete categories. The tensor product is just the multiplication, and the existence of a braiding would simply mean that the monoid is commutative, which it typically isn't.
* Functor categories : if $C$ is a category, then $Fun(C,C)$ is monoidal with respect to $\circ$, but it's quite rare for endofunctors to commute.


