# Natural Isomorphism: how can $A \otimes B \simeq B \otimes A$ and yet $A \otimes B \neq B \otimes A$

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.

• A simple example would be the non-strict symmetric monoidal category of vector spaces equipped with the usual tensor product. – JoeyBF Jun 13 '14 at 12:36
• Question: for two sets $A$ and $B$, is $A \times B = B \times A$? – user64687 Jun 13 '14 at 12:44
• Or, to put it more emphatically: the vast majority of "real-world" examples of symmetric monoidal categories do not satisfy $A \otimes B = B \otimes A$. – Zhen Lin Jun 13 '14 at 12:47
• It doesn't make sense to ask if two objects are equal. The correct notion of equality is isomorphism (+ coherence issues). See also ncatlab.org/nlab/show/principle+of+equivalence and ncatlab.org/nlab/show/vertical+categorification – Martin Brandenburg Jun 14 '14 at 13:31

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).