Examples of codim-2 objects in extended TQFT

Question

I'm scratching my head trying to understand what an extended TQFT associates to $(n-2)$-hypersurfaces.

Here's some intuition that I've developed. For an $(n-1)$-hypersurface chopped into $(n-2)$-hypersurfaces with the initial and final set to null $(n-2)$-hypersurfaces, $Z$ associates a vector space. Hence whatever objects are associated to $(n-2)$-hypersurfaces must: 1. Form a monoidal category. 2. Have a notion of "trace" that evaluates the vector space.

The only object from mathematics that I can think of that has the second property is a cell complex (it's cohomology is a vector space).

However, the nlab article suggests that the sought object is an linear additive category. I don't quite understand this. Here's a few questions that I have:

1. So is it correct that different types of $(n-2)$-hypersurfaces are assigned different categories?
2. It must be that linear additive categories themselves form a monoidal category, but I fail to see how.
3. What is the unit linear additive category in the monoidal category of linear additive categories?
4. Say I have a morphism from a unit to the unit in the monoidal category of linear additive categories, which corresponds to a $(n-1)$-hypersurface. How do I trace it to get the vector space?

Answer 1

1. I am not sure what “type” refers to, but different codimension 2 surfaces are indeed assigned different categories.
2. The monoidal product can be taken to be the Deligne tensor product, or some variant thereof.
3. The category of vector spaces (possibly finite-dimensional).
4. In concrete terms, by the Eilenberg–Watts theorem such a morphism corresponds to an A-A-bimodule M, where A is a ring whose category of modules is equivalent to a given category. Then the 0th Hochschild homology of M computes the trace of M.