Abstract: It's a long established principle that an interesting way to think
about numbers is as the sizes of sets or dimensions of vector spaces, or
better yet, the Euler characteristic of complexes. You can't have a
map between numbers, but you can have one between sets or vector
spaces. For example, Euler characteristic of topological spaces is not
functorial, but homology is.
One can try to extend this idea to a bigger stage, by, say, taking a
vector space, and trying to make a category by defining morphisms
between its vectors. This approach (interpreted suitably) has been a
remarkable success with the representation theory of semi-simple Lie
algebras (and their associated quantum groups). I'll give an
introduction to this area, with a view toward applications in
topology; in particular to replacing polynomial invariants of knots
that come from representation theory with vector space valued
invariants that reduce to knot polynomials under Euler characteristic.