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

TL;DR

This paper establishes a criterion for dualizability in stratified monoidal categories, characterizes it in terms of strata, and proves a 1-dimensional bordism hypothesis for symmetric monoidal recollements, with applications in algebra and homotopy theory.

Contribution

It introduces a simple criterion for dualizability in monoidal recollements and stratified categories, and proves a 1-dimensional bordism hypothesis in this context.

Findings

Criterion for dualizability based on factors and projection formula

Characterization of dualizability in stratified categories via stratumwise dualizability

Application to algebraic and homotopy-theoretic examples

Abstract

Given a monoidal $\infty$-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 $\infty$-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 $\infty$-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.