Motivated by a desire to gain a better understanding of the ``dimension-by-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 |-> 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 $\omega$-category monad on globular sets and the free operad-with-contraction monad on the category of collections.
Keywords: Descent theory, monads, globular sets
2000 MSC: 18C15, 18D10, 18D30
Theory and Applications of Categories,
Vol. 16, 2006,
No. 24, pp 668-699.