# The Unapologetic Mathematician

## 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 $\mathbb{F}$-algebra is just like a monoid but enriched over the category of vector spaces over $\mathbb{F}$. That is, it’s a one-object category with an $\mathbb{F}$-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 $\mathbf{Vect}_\mathbb{F}$. So what’s a functor $\rho$ from, say, a group $G$ (considered as a category) to $\mathbf{Vect}_\mathbb{F}$? First the single object of the category $G$ picks out some object $V\in\mathbf{Vect}_\mathbb{F}$. That is, $V$ is a vector space over $\mathbb{F}$. Then for each arrow $g$ in $G$ — each group element — we have an arrow $\rho(g)\in\hom_\mathbb{F}(V,V)$. Since $g$ has to be invertible, this $\rho(g)$ must be invertible — an element of $\mathrm{GL}(V)$.

What about an algebra? Now our source category $A$ and our target category $\mathbf{Vect}_\mathbb{F}$ are both enriched over $\mathbf{Vect}_\mathbb{F}$. It only makes sense, then, for us to consider $\mathbb{F}$-linear functors. Such a functor $F$ again picks out a single vector space $V$ for the single object of $A$ (considered as a category). Every arrow $a$ in $A$ gets sent to an arrow $\alpha(a)\in\hom_\mathbf{F}(V,V)$. This mapping is linear over the field $\mathbb{F}$.

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 $\mathcal{T}ang$ of tangles (or $\mathcal{OT}ang$ of oriented tangles, $\mathcal{F}r\mathcal{T}ang$ of framed tangles, or $\mathcal{F}r\mathcal{OT}ang$ of framed, oriented tangles). This is a monoidal category with duals, as is $\mathbf{Vect}_\mathbb{F}$, 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 $\mathbf{Vect}_\mathbb{F}$, since that would trivialize the structure.

So what is a representation of the category $\mathcal{T}ang$? 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 $\mathcal{T}ang$. This is my work: to translate existing knot theoretical ideas into this algebraic language, where I believe they find a better home.

October 27, 2008