Page 32 - Textos de Matemática Vol. 46

2 2
J . A D A´ M E K
commutes. Indeed, E is the “projection to j” E(A, (di)i2I ) = (A, dj ).
From the triangles (2) we deduce that (A,dj) satisfies (3). Thus, E is a well-defined, faithful functor. It is surjective on objects: for every algebra (A,a) in C define, given i 2 I,
ai =a·fA :TiA!Aforanyconnectingmorphismf:Ti !Tj.
Then ai is well-defined due to (3) and, since f is a monad morphism, (A, ai) is an Eilenberg-Moore algebra for Ti. Finally, to prove that E is an isomorphism, we verify that it is full. Let
k : (A,a) ! (B,b)
be a homomorphism in C. Then we need to prove that for every i 2 I this is a homomorphism from (A,ai) to (B,bi), where again bi = b·fB. Use the following diagram
ai
fA // a //✏✏
TiA TjA A Tik Tjk k
✏✏ fB TiB
// ✏✏ b
TjB B
bi
// ✏✏ OO
Corollary 3.10. Every diagram of monos-preserving monads with arbitrarily large joint pre-fixpoints has a colimit in Monad (A).
Indeed, apply the usual construction of colimits as a coequalizer of a parallel pair between coproducts; see [11]. Given a diagram D in Monad (A) with monos-preserving objects Si = (Si,μi,⌘i) for i 2 I having arbitrarily large joint pre-fixpoints, then also every collection of monads indexed by I ⇥ J, whereJ isanarbitrarilysetandSi =S(i,j) forall(i,j)2I⇥J,hasarbitrarily large joint pre-fixpoint. (Indeed, for every object A and every cardinal ↵ put ↵0 = ↵+cardJ. By applying Definition 3.4 to A and ↵0 for the former collection indexed by I, we get the required condition for the new collection.) Thus, the two coproducts needed to construct colimD as a coequalizer in Monad (A) exist.
⇤