Descent for Monads

Motivated by a desire to gain a better understanding of the “dimensionby-dimension” decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C 7→ Mnd(C) as an indexed category; on this base category, there is a natural topology. Then we single out a class of monads which are well-behaved with respect to reindexing. The main result is now, that such monads form a stack. Using this, we can shed some light on the free strict ω-category monad on globular sets and the free operad-with-contraction monad on the category of collections.