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

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.

arXiv comment62 pages. Final version. To appear in Algebraic Geom. Topol.. Keywords: bivariant homology, correspondences, spans, (infinity,2)-category, Yoneda, Beck-Chevalley, base change, straightening, Grothendieck construction, biCartesian fibration

References in corpus (11)

• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• Iterated wreath product of the simplex category and iterated loop spaces
• A cartesian presentation of weak n-categories
• Higher Segal spaces I
• Loop Spaces and Connections
• Iterated spans and classical topological field theories
• On the Unicity of the Homotopy Theory of Higher Categories
• Rectification of enriched infinity-categories
• Operations on derived moduli spaces of branes
• A user's guide to co/cartesian fibrations
• Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory

Cited by in corpus (3)

• Cohomology of the moduli stack of algebraic vector bundles
• Characters and transfer maps via categorified traces
• Universality of Barwick's unfurling construction