The Monadic Tower for $\infty$-Categories

TL;DR

This paper demonstrates that right adjoint functors between presentable ∞-categories can be decomposed into a coreflection and monadic functors, with a functorial iterated colimit presentation, linking to homology localization.

Contribution

It introduces a canonical decomposition of right adjoint functors in ∞-categories into coreflections and monadic functors, with a new functorial colimit framework.

Findings

Decomposition of right adjoint functors into coreflection and monadic parts

Functorial iterated colimit presentation of coreflections

Connections to homology localization and completion

Abstract

Every right adjoint functor between presentable $\infty$-categories is shown to decompose canonically as a coreflection, followed by, possibly transfinitely many, monadic functors. Furthermore, the coreflection part is given a presentation in terms of a functorial iterated colimit. Background material, examples, and the relation to homology localization and completion are discussed as well.