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
–category
of correspondences is the universal way to attach to a given
–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
–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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