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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.
@article{TAC_2006_16_a23,
author = {Pieter Hofstra and Federico De Marchi},
title = {Descent for {Monads}},
journal = {Theory and applications of categories},
pages = {668--699},
publisher = {mathdoc},
volume = {16},
year = {2006},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2006_16_a23/}
}
Pieter Hofstra; Federico De Marchi. Descent for Monads. Theory and applications of categories, Tome 16 (2006), pp. 668-699. http://geodesic.mathdoc.fr/item/TAC_2006_16_a23/