7.5. Braided and Symmetric Monoidal Categories
Braided and symmetric monoidal categories serve as some of the most fruitful and
most studied environments of monoidal categories. The formulation of these categories
may seem mysterious and random, but they have been recognized as important
in their applications to physics. Specifically, braided monoidal categories were
first defined by Joyal-Street in an attempt to abstract the solutions to the
Yang-Baxter equation, an important equation of matrices in statistical mechanics.
It turns out that braided monoidal categories are exactly the categorical environment
one needs to describe the category of representations of a Hopf algebra
Before we dive into what exactly braided monoidal categories are, we'll introduce the concept of braids.
The
where
- 1.
whenever - 2.
.
Relations (1) and (2) are imposed in order to reflect physical reality. Below the relations are pictured on a three strands.
The first two braids represent
A Braided Monoidal Category
\end{statement}
such that the following diagrams commute for all objects
\end{statement}
Note that just because we have a twist morphism, it is not necessarily the case
that
The canonical example of a braided monoidal category is the braid category
- Objects. All integers
. - Morphisms. For any two integers
, we have that
Composition in this category is simply braid composition. We can introduce a tensor
product
With an identity object being the empty braid, we see that
where on objects the addition is simply permuted; on morphisms, however,
\end{figure}
It is a simple exercise to show that this satisfies the hexagon axioms; the task is simplified due to the fact that the associators are identities. While this category may seem like a boring example, it plays a critical role in demonstrating coherence for braided monoidal categories, something we will do later.
Let
We can additionally introduce a braiding on this category for each invertible elements
whenever
which clearly commutes. The second hexagon axiom is also easily seen to be satisfied:
Thus we see that
If
defined pointwise for each
One can then check that this natural transformation satisfies the braided hexagon
axioms since the braiding
A Symmetric Monoidal Category
Symmetric monoidal categories are basically monoidal categories which collapse the information which braided monoidal categories have the potential to encode. Their environment is much simpler, but at the cost of information.
Recall from the previous examples that
but these braidings have the property that
Recall from
In this category, we can introduce a symmetric braiding
One thing to notice is that this is the underlying permutation of braid given in
Figure
A PROP, an acronym coined by Mac Lane
for "Product and Permutation Category",
is a symmetric monoidal category
Consider the category
Here, we necessarily include
Next, consider the set of morphisms
Let
where
Hence
Now consider the morphisms
but note that
Hence we must have that
Now we show that this is a monoidal category. Define the natural isomorphisms
We can describe these functions in further detail.
Observe that
Elements of
On the other hand, elements of
Now we can explicitly define
and
and
Note for both
All of these establish a bijection, and hence an isomorphism.
Now to demonstrate that they are natural, consider
commutes, which we can do by a case-by-case basis.
First we follow the path
and then show it is equivalent to the other path.
If the input is , we see that . If this is fed into , the output will be , whose output will be .
However, suppose we first put
$$
[ ( f+ g) + h]((y, 1), 0) = ([f + g](y, 1), 0) = ((g(y), 1), 0).
$$
However, we also could have obtained this value by first starting with
$f + (g + h)$. In this case,
$$
[f + (g + h)]((y, 0), 1) = ([g + h](y, 0), 1) = ((g(y), 0), 1).
$$
Plugging this into $\alpha_{n',m',p'}$, we then get that
$$
\alpha_{n',m',p'}((g(y), 0), 1) = ((g(y), 1), 0).
$$
Hence the two paths are equivalent.
* **$\bm{y \in S_p}$.** Suppose $x = (y, 1)$, Then we have that
$\alpha_{n, m, p}((y,1), 1) = (y, 1)$. Sending this into $(f + g)+ h$, we get
$$
[(f + g) + h](y, 1) = (h(y), 1).
$$
However, we could have achieved this value by first plugging $((y, 1),1)$ into
$f + (g + h)$:
$$
[f + (g + h)]((y, 1), 1) = ([g + h](y, 1), 1) = ((h(y), 1), 1).
$$
Then sending this into $\alpha_{n',m',p'}$, we get
$$
\alpha_{n',m',p'}((h(y), 1), 1) = (h(y), 1).
$$
Thus the two paths are equivalent.
[Show naturality works for
Now we show that these natural isomorphisms satisfy the monoidal properties. Specifically, we'll show that the diagram
must commute. To do this, we consider how these functions are realized in Set.
If we consider
. If , then we see that . Sending this into , we get that .
On the other hand, we could obtain this value by directly sending
On the other hand, we can start instead be evaluating
Thus we see that this diagram holds for all naturals