Connected monads weakly preserve products

09/05/2019

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by   H. Peter Gumm, et al.

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If F is a (not necessarily associative) monad on Set, then the natural transformation F(A× B)→ F(A)× F(B) is surjective if and only if F(1)=1. Specializing F to F_V, the free algebra functor for a variety V, this result generalizes and clarifies an observation by Dent, Kearnes and Szendrei.