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arXiv · 2005.10496

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

Andrew W. Macpherson
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Abstract

Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an 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.

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Andrew W. Macpherson. 2021-08-10. A bivariant Yoneda lemma and $(\infty,2)$-categories of correspondences. https://doi.org/10.2140/agt.2022.22.2689

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