Poincar\'e Duality Pairs of $\infty$-Categories

Abstract

We introduce a notion of Poincar\'e duality for pairs of $\infty$-categories, extending Poincar\'e-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincar\'e ads of spaces, iterated Poincar\'e cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of $\infty$-categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincar\'e triads to all ads. This theory also lays the foundation for future work by the authors on Poincar\'e cobordism categories, isovariant Poincar\'e spaces and string topology.