Seminar Heaven II

Today Michael Hopkins gave an elementary introduction to TFTs from a topologist's perspective, before outlining a theorem for $\omega$ $n$-categories (ie. with groupoid like arrows above dimension $n$) related to the classification of TFTs. He stressed the importance of category theory for tackling this problem. Lurie will also probably speak about this subject next week. The interesting construction is a choice of subcategory chain

$C^{fd} \rightarrow C^{f} \rightarrow C$

where $C$ is any suitable symmetric monoidal category at the target of the (generalised) TFT functor. The category $C^f$ ($f$ for finite) is the collection of all arrows that have both left and right adjoints, and $C^{fd}$ (for fully dualisable) is the category where objects have duals in a suitable sense. In other words, the categories they study generalise categories such as FinVect (finite dimensional vector spaces) to the infinite dimensional path space realm that topologists love.