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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. The potential audience for this work may extend outside the boundaries of modern homotopy theory, so our treatment is structured to use as little technology as possible. In particular, many of the results are stated without using model categories. We also illustrate some applications to fixed-point theory.
We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable in
We give a new formula for $p$-typical real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for $p$-typical topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this,
The main result of this note is a parametrized version of the Borsuk-Ulam theorem. We show that for a continuous family of Borsuk-Ulam situations, parameterized by points of a compact manifold W, its solution set also depends continuously on the para
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or, equivalently
We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra.