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.