A model for the assembly map of bordism-invariant functors

Abstract

We study oplax colimits of stable categories, of hermitian categories and of Poincaré categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincaré categories which is also a Verdier projection, whose kernel we explicitly describe. As a direct application, we generalize the Shaneson splitting for bordism-invariant functors of Poincaré categories proved by Calmès-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle to allow for twists. We also show our methods can tackle their general twisted Shaneson splitting of Poincaré-Verdier localizing invariants which specifies to a twisted Bass-Heller-Swan decomposition for the underlying stable categories, generalizing part of recent work of Kirstein-Kremer.

42 pages; comments welcome! v2: fixed Corollary 4.3