I will mention one example I didn't get to last week: two equivalent categories of locally convex topological vector spaces. The dualizing object is the topological field C (you could just look at real spaces and use R and get the same results). The first category is the weakly topologized spaces--they have the weak topology for their continuous linear functionals. The second is known as the Mackey spaces, but like the other examples have the strongest possible topology for their continuous linear functionals. Then I will describe the "And then a miracle happened" of the Chu and chu constructions that were based on Mackey's "pairs".

Anyone interested in more detail on last week's introduction should look here: http://www.math.mcgill.ca/barr/papers/#dsac

particularly these two papers:

- Topological *-autonomous categories (TAC 2006)
- On duality of topological abelian groups (unpublished note)

Some of the other papers in that list might be interesting, but those are the most relevant to what I said on Feb 2nd.