Category Representations
We’ve seen how group representations are special kinds of algebra representations. But even more general than that is the representation of a category.
A group is a special monoid, within which each element is invertible. And a monoid is just a category with a single object. Similarly, an -algebra is just like a monoid but enriched over the category of vector spaces over
. That is, it’s a one-object category with an
-bilinear composition. It makes sense to regard both of these structures as categories of sorts. A representation will then be a functor from one of these categories.
The clear target category is . So what’s a functor
from, say, a group
(considered as a category) to
? First the single object of the category
picks out some object
. That is,
is a vector space over
. Then for each arrow
in
— each group element — we have an arrow
. Since
has to be invertible, this
must be invertible — an element of
.
What about an algebra? Now our source category and our target category
are both enriched over
. It only makes sense, then, for us to consider
-linear functors. Such a functor
again picks out a single vector space
for the single object of
(considered as a category). Every arrow
in
gets sent to an arrow
. This mapping is linear over the field
.
So what do category representations get us? Well, one thing is this: consider a combinatorial graph — a collection of “vertices” with some directed “edges” joining them. A path in the graph is a sequence of directed edges joined tip-to-tail, and the collection of all paths in the graph constitutes the “path category” of the graph (exercise: identify the identity paths). A representation of this path category is what mathematicians call a “quiver representation”, and they’re big business.
More interesting to me is this: the category of tangles (or
of oriented tangles,
of framed tangles, or
of framed, oriented tangles). This is a monoidal category with duals, as is
, and so it only makes sense to ask that our functors respect those structures as well. We don’t ask that it send the braiding to the symmetry on
, since that would trivialize the structure.
So what is a representation of the category ? It is my contention that this is nothing but a knot invariant, viewed in a more natural habitat. A little more generally, knot invariants are the restrictions to knots (and links) of functors defined on the category of tangles, which can often (always?) be decategorified — or otherwise rendered down — into representations of
. This is my work: to translate existing knot theoretical ideas into this algebraic language, where I believe they find a better home.