User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:12:05Z http://mathoverflow.net/feeds/user/6250 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25588/free-monad-or-monad-defined-from-an-adjunction Free monad or monad defined from an adjunction. unknown (yahoo) 2010-05-22T16:22:19Z 2013-01-13T21:43:31Z <p>My first question here. </p> <p>Accordingly to M. Barr "Coequalizers and free triples" by a free triple (or free monad) generated by an endofunctor $R: X\rightarrow{X}$ we mean a triple $T=(T,\eta,\nu)$ and a natural transformation $p: R\rightarrow{T}$ such that if $T'= (T',\eta',\nu')$ is another triple and $p': R\rightarrow{T'}$ is a natural transformation, then $p'=\tau{p}$, where $\tau:T\rightarrow{T'}$ is a map of triples.</p> <p>On the other hand we have always a monad in the form $(GF,\eta,G\epsilon{F})$ when we have an adjunction $F\dashv{G}$ where $\eta$ is the unit and $\epsilon$ de counit of the adjunction.</p> <p>Can the endofunctor $R$ have the form of an adjunction (for example between a <em>free</em> and a <em>forgetful functor</em>) in the first definition? When does it happen and which is the relationship between the adjunction and the free monad after all? </p> http://mathoverflow.net/questions/25588/free-monad-or-monad-defined-from-an-adjunction Comment by 2010-05-23T08:41:32Z 2010-05-23T08:41:32Z Being for exemple in the form $U\circ{F}$ where $F\dashv{U}$.