Page 27 - Textos de Matemática Vol. 46

COLIMITS OF MONADS 17
X'X+A' IXtoget
a`a
Proof. Let `Si = (Si, μi, ⌘i), i 2 I, be such a collection. Then the endo- functor S = Si preserves monomorphisms. And it has arbitrarily large
`i2I
pre-fixpoints: given an object A find X with SiX ⇢ X for all i 2 I and
SiX ⇢
i2I Si generates a free monad FS with the
SX = By Corollary 2.15 the functor S =
universal arrow ⌘ˆ : S ! F S; the coproduct injections are denoted by vi :Si !S (i2I)
The forgetful functor Monad (A) ! [A, A] creates limits, see Proposition 2.2, and we conclude that for the slice category FS / Monad (A) the the correspond- ing forgetful functor
U : FS/ Monad (A) ! FS/[A,A]
also creates limits. Now consider an arbitrarily cocone f = (fi) consisting of monad morphisms fi : Si ! Tf (i 2 I). The functor [fi] : S ! Tf generates a uniquemonadmorphismf¯:FS !Tf withf¯·⌘ˆ=[fi]thatwefactorizeasin Lemma 3.3
S
i2I
i2I
X ' X.
⌘S FS
[fi ]
✏✏ ((
// Tf f >>
mf
We get a (possibly large) collection of objects (ef , Rf ) of the slice category FS / Monad (A). This collection has a product in FS/[A,A]. Indeed, recall from Remark 3.2 that A is cowellpowered, thus, for every object A all quotients of the object FSA form a complete lattice. Form the meet eA : FSA ! RA of all quotients (ef )A : FS A ! Rf A ranging through all cocones f . For every cocone f above we have a morphism
q fA : R A ! R f A w i t h ( e f ) A = q fA · e A .
The resulting functor R and natural transformations qf : R ! Rf clearly form a product of all ef in FS /[A, A]. Consequently, there exists a product (e, R) of the objects (ef , Rf ) in FS / Monad (A) as f ranges through all cocones: see Proposition 2.2. For the projections qf : R ! Rf define
pf =mf ·qf :R!Tf.
¯
ef
R >> f