Completeness for monads and theories

Abstract

We generalize the correspondence between theories and monads with arities of arXiv:1101.3064 to $\infty$-categories. Additionally, we introduce the notion of complete theories that is unique to the $\infty$-categorical case and provide a completion construction for a certain class of theories. Along the way we also develop the necessary technical material related to the flagged bicategory of correspondences and lax functor in the $\infty$-categorical context.