In the context of integral transforms on sheaves one thinks of a span as a generalized linear map. The span trace is the corresponding generalization of the notion of a trace of a linear map.
This is just the general trace of an endomorphism which is definable in any compact/autonomous symmetric monoidal (2-)category, of which is an example (as described below).
In the context of FQFT a useful aspect of the span trace is that it is manifestly dual to the co-span co-trace, which, as described there, corresponds under the interpretation of spans as cobordisms to gluing of the two ends of a cobordism.
a span with identical left and right index object , the simplest way to define its span trace is as by regarding it as a map , then pulling back along the diagonal morphism .
This can be expressed in terms of the bicategory Span in several ways. For instance, we can regard it as the composite of the result
of dualizing one leg of the span with the span
i.e. the pullback
regarded as a span from the point to the point
More generally, the trace of a multispan over identical of its index objects is the composite with the multispan
Trace of Set-valued matrices
Let the ambient category be Set, let be a finite set and an -matrix of finite sets, regarded under groupoid cardinality as a groupoidified -matrix with entries in .
The trace of the span
is the pullback
which is the coproduct set . Under groupoid cardinality this is indeed the trace of the matrix represented by .
Loop objects from homotopy span traces
Let be a category of fibrant objects with interval object . Recall that for every object of its free loop space object is the part of the path object which consists of closed paths, i.e. the pullback.
This can be understood as the homotopy span trace of the identity span on
where the homotopy span trace is computed like the span trace but with the pullback replaced by a homotopy pullback:
According to the example described at homotopy limit and using that we assume that we are in a category of fibrant objects we can compute this homotopy limit, up to weak equivalence, as the ordinary limit of the weakly equivalent pullback diagram
where we replace by its path object using the factorization of as guaranteed to exist in a category of fibrant objects, where is a fibration:
But by the above .
Categorical trace from homotopical span trace
The categorical trace on a 1-endomorphism in a 2-category is the homotopy trace on the span given by that endomorphism.
This should be true quite generally, but here are the details just for the special case that connects to the above example:
Let Grpd with the standard interval object . This is a category of fibrant objects with respect to the folk model structure.
Notice that natural transformations between two functors are in bijection with commuting diagrams
Now, the homotopy trace on the span corresponding to an ednofunctor is
Since we are in a category of fibrant objects the assumptions of the example discussed at homotopy limit apply and the above homotopy limit is again computed, up to weak equivalence, by the ordinary limit of
By the above, every cone over the pullback diagram with a functor , on the right defines a natural transformation . By the universal property of the limit, it represents the collection of these transformations.
That the canonical trace on is compatible with the interpretation of spans as linear maps in the context of groupoidification, and that it corresponds under duality (in terms of the co-span co-trace) to the gluing of ends of cobordisms was mentioned in
More discussion is in