I am reading Braided Monoidal Categories by Joyal and Street. They say cateogories with tensor product arise naturally such as the category of Abelian Groups and that of Banach Spaces.
Is there any example where: $A \otimes B \simeq B \otimes A $ and yet $A \otimes B \neq B \otimes A $ ? Maybe I am not reading far enough into the paper.
The examples in section 2 use categories of ribbons or braids. I would like to see instances of "ribbon-ness" or "braiding" arising from Algebra or Topology.
In general it is very rare for a braided monoidal category to commute "on the nose". For a very concrete example, the usual category of sets and functions is cartesian monoidal, so braided (even symmetric) monoidal, but in general $A \times B \neq B \times A$.
As for the braids, there is a PRO (monoidal category with $\mathbb{N}$ as the set of objects and $+$ as product) where the morphisms are the braid groups. This construction is even the universal braided monoidal category (Edit: It's actually the free braided monoidal category on one object. I was just repeating what my category theory professor said.), and appears at a number of places in physics for example. The braid groups arise from topology as the fundamental group of a certain configuration space (see the formal treatment and the book Braid Groups by Christian Kassel and Vladimir Tuarev).
The ribbons arise when, intuitively, the braids are allowed to twist on themselves, just like the half-twist for the Möbius band. Unfortunately, I am not very familar with this added generalization.