A bivariant Yoneda lemma and (∞,2)–categories of correspondences

Macpherson, Andrew W

doi:10.2140/agt.2022.22.2689

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A bivariant Yoneda lemma and $(\infty, 2)$–categories of correspondences

Andrew W Macpherson

Algebraic & Geometric Topology 22 (2022) 2689–2774

DOI: 10.2140/agt.2022.22.2689

Abstract

A well-known folklore result states that if you have a bivariant homology theory satisfying a base change formula, you get a representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a $2$ –category of correspondences is the universal way to attach to a given $1$ –category a set of right adjoints that satisfy a base change formula.

Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for bivariant functors. Moreover, conditioned on the existence of a $2$ –dimensional Grothendieck construction, I provide a proof of the aforementioned universal property. The methods, morally speaking, employ the “internal logic” of higher category theory: they make no explicit use of any particular model.

Keywords

bivariant homology, higher category, 2–category, correspondences, motive, Cartesian fibration

Mathematical Subject Classification

Primary: 18A40, 18D20, 18G99, 55P65, 55U40

References

Publication

Received: 15 August 2020

Revised: 16 March 2021

Accepted: 4 June 2021

Published: 13 December 2022

Authors

Andrew W Macpherson
Yokosuka City Kanagawa prefecture Japan
https://awmacpherson.com

Volume 22, issue 6 (2022)