Traces of higher categories

Traces of higher categories

[Jamie Vicary]

Dear all,

The trace of a category C is a well-known construction: it is the set
given by taking the 'trace' of the identity functor on C in the
bicategory of profunctors. Concretely, it is the set given by
endomorphisms in C, modulo the equivalence relation generated by
f.g~g.f for all pairs of functions f and g with opposite types.
Abstractly, it is the coend of the identity functor on C.

In general, I'm sure the trace of an n-category would be an
(n-1)-category. I'm interested in what happens when you iterate this
all the way down, and hence obtain a set from any n-category.

Can someone give me a concrete description of the set one obtains in
this way from a strict n-category? I want something that I can get my
hands on, and try out on my favourite strict n-categories, rather than
an abstract statement about taking higher coends.

What if I had a strict omega-category? Its trace would presumably be
another omega-category. Should I expect to be able to somehow trace it
down to a set?

With best wishes,
Jamie.


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