TL;DR
This paper explores a categorical framework based on Waldhausen A-theory, drawing analogies to mixed Tate motives in arithmetic geometry, and investigates the structure of a Hopf object with implications for differential topology.
Contribution
It introduces a novel categorical perspective linking Waldhausen A-theory to motives, revealing structural similarities and properties of associated Hopf objects.
Findings
The Hopf object S ∧_A S has properties akin to a motivic group.
The rational Lie algebra associated is free on odd-degree generators.
Analogies suggest new connections between topology and arithmetic geometry.
Abstract
A category of correspondences based on Waldhausen A-theory has interesting analogies, in the context of differential topology, to categories of mixed Tate motives studied in arithmetic geometry. In particular, the Hopf object S \wedge_A S (regarding A(*) as a kind of local ring over the sphere spectrum) has some similarities to a motivic group for this category; its associated rational Lie algebra is free, on odd-degree generators...
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology