If I have an endomorphism f:X->X of a finite set, then it induces a map Z[X]->Z[X], and the trace of that counts fixed points of f
this is a de-homotopification of the lefshetz fixed point theorem in a way that i would like to understand better
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iirc one of the monoidal duality papers explains this special case pretty well. it goes something like, Z[X] is a symmetric monoidal functor from (finite sets + finite spans) to f.g. abelian groups, so sends traces to traces, and traces of spans count fixed points
nlab page here, and note that this does not in any obvious way generalize to lefschetz, taking traces of spans of spaces gets you this like twisted loop space thing
ncatlab.org/nlab/show/span
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I think Dold's paper introducing duality actually proves Lefschetz that way.
The point being that a monoidal functor preserves dualizable objects and thus traces, and Dold shows that the trace in some category of spaces (maybe actually the Spanier Whitehead category) is just
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