Internal homs of monads

Steve Lack – 9 May 2018

This talk is about the monoidal 2-category Mnd(K) of monads in a monoidal 2-category K. Internal homs in this monoidal 2-category exist only rarely. But if [(A,t),(A,t)] does exist, for some monad t on an object A, then it becomes a monoidale in Mnd(K), or equivalently an opmonoidal monad in K. If the underlying object of [(A,t),(A,t)] is itself the internal hom [A,A] in K, then we may think of this opmonoidal monad as a bialgebroid.

This describes a universal property of a concrete construction of Ardizzoni-el Kaoutit-Menini, called the coendomorphism bialgebroid of an algebra. Their construction involved the case where K is the monoidal bicategory Mod of rings and bimodules.