Smash powers and traces on parametrized spectra: an application of rigidity

Abstract

Abstract The bicategory of parametrized spectra has a remarkably rich structure. We can take traces in this bicategory, giving classical invariants that count fixed points. We can also take $$C_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -equivariant external smash powers and equivariant traces, which give significant generalizations of the classical invariants that count periodic points. Unfortunately, the existence of these smash powers and traces for parametrized spectra depends on technical statements about the bicategory that can be difficult to verify directly, especially if one wants the construction to have a direct geometric interpretation. In this paper, we demonstrate the effectiveness of two tools—rigidity and deformable functors—by using them to establish formal structures on this bicategory directly from point-set level data.