The formal theory of monoidal monads

Details The formal theory of monoidal monads The formal theory of monoidal monads

2010-12-02

arxiv.org/abs/1012.0547v1

Mathematics

Category Theory

15 pages

Scientific paper

We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. As we explain at the end of the paper a similar phenomenon occurs in many other situations.

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