Monads and monoids on symmetric monoidal closed categories

Abstract

part of a $/%monad A. In this note, we show that for a monad Tand a monoid A and a Y/'-natural transformation r -- (~ A --> T the following are equivalent (1) q is a Yf-monad map; (2) q is monoidal W-natural transformation; (3) the map Oq: A -~ T I corresponding to q under the bijection mentioned above is a monoid map. As a consequence we show that for every $P-monad T on Y/" there is a monoid (namely TI) and a Y/~-monad map z: TI---> T such that if A is a monoid and ~.: A -~ Tis a Y/--monad map then there exists a unique $/~-monad map fl: A --> TI with T �9 fl = a. We also show that if T is a commutative monad (i.e. the two canonical monoidal structures on T agree [3], p. 7) then TI is a commutative monoid and thus TI is a commutative monad. We also note that there are non-commutative monads T such that TI is a commutative monoid.