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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

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.

bivariant homology, higher category, 2–category, correspondences, motive, Cartesian fibration
Mathematical Subject Classification
Primary: 18A40, 18D20, 18G99, 55P65, 55U40
Received: 15 August 2020
Revised: 16 March 2021
Accepted: 4 June 2021
Published: 13 December 2022
Andrew W Macpherson
Yokosuka City
Kanagawa prefecture