Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements

Abstract

Given a monoidal ∞-category C equipped with a monoidal recollement, we give a simple criterion for an object in C to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them. Predicated on this, we then characterize dualizability in any monoidally stratified ∞-category in terms of stratumwise dualizability and a projection formula for the links.Using our criterion, we prove a 1-dimensional bordism hypothesis for symmetric monoidal recollements. Namely, we provide an algebraic enhancement of the 1-dimensional framed bordism ∞-category that corepresents dualizable objects in symmetric monoidal recollements.We also give a number of examples and applications of our criterion drawn from algebra and homotopy theory, including equivariant and cyclotomic spectra and a multiplicative form of the Thom isomorphism.