Parametrized spectra, a low-tech approach

TL;DR

This paper provides an accessible, low-tech approach to the foundations of parametrized spectra, extending key results with weaker hypotheses and clarifying the bicategory structure and duality, aimed at applications in fixed-point theory.

Contribution

It offers a new, explicit construction of the bicategory of parametrized spectra and a detailed account of coherence and duality results, simplifying the standard theory.

Findings

New construction of the bicategory $ ext{Ex}$ of parametrized spectra

Explicit proof of Costenoble-Waner duality

Clarification of the relationship between index formulas and bicategorical duality

Abstract

We give an alternative treatment of the foundations of parametrized spectra, with an eye toward applications in fixed-point theory. We cover most of the central results from the book of May and Sigurdsson, sometimes with weaker hypotheses, and give a new construction of the bicategory $\mathcal Ex$ of parametrized spectra. We also give a careful account of coherence results at the level of homotopy categories. This is an expository work, akin to a set of lecture notes, encompassing and extending the more novel material in the paper "A convenient category of parametrized spectra." Our primary goal is to give careful and explicit explanations for the standard facts surrounding the bicategory of parametrized spectra and the Reidemeister trace. In particular, our proof of Costenoble-Waner duality gives a more explicit account of how the concrete index formulas of Dold are related to the abstract theory of bicategorical duality from May and Sigurdsson, Ponto, and Shulman.