Given a monad T on A and a functor G: A → B, one can construct a monad G_#T on B subject to the existence of a certain Kan extension; this is the pushforward of T along G. We develop the general theory of this construction in a 2-category, giving two universal properties it satisfies. In the case of monads in CAT, this gives, among other things, two adjunctions between categories of monads on A and B. We conclude by computing the pushforward of several familiar monads on the category of finite sets along the inclusion FinSet → Set, which produces the monad for continuous lattices, among others. We also show that, with two trivial exceptions, these pushforwards never have rank.
Keywords: codensity, pushforward, monad, limit completion, Kan extension, powerset, filter, ultrafilter
2020 MSC: 18C15
Theory and Applications of Categories, Vol. 44, 2025, No. 30, pp 927-963.
Published 2025-09-21.
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