Page 20 - Textos de Matemática Vol. 46

1 0 J . A D A´ M E K
objD). It is a monad morphism. Indeed, the axiom q · ⌘S = qT follows, since (pd) is a monocone, from
pd ·(q·⌘S)=qd ·⌘S =⌘d =pd ·⌘T. Analogously, the axiom q · μS = μT · q ⇤ q follows from
pd ·(μT ·(q⇤q))=μd ·(pd ⇤pd)·(q⇤q)=pd ·μT ·(q⇤q)
⇤
Corollary 2.3. Limits of monads over a complete category A are computed object-wise (on the level of A).
Proposition 2.4. The forgetful functor of Monad (A) creates absolute coequal- izers.
Remark. Recall that this means that given a parallel pair of monad mor- phisms p, q : S ! T whose coequalizers c in [A, A]:
p
q
is absolute (that is, preserved by every functor with domain [A, A]), there exists a unique monad structure on C making c a monad morphism. Moreover, c is a coequalizer of p and q in Monad (A).
Proof. The unit of C is, necessarily,
⌘C =c·⌘T.
To define the multiplication μC : C · C ! C , use the endofunctor of [A, A] defined by X 7! X · X on objects and by f 7! f ⇤ f on morphisms. Since c ⇤ c is the coequalizer of p ⇤ p and q ⇤ q, we have a unique μC for which c preserves multiplication:
S
))55 T c // C
S · S
p⇤p ,, c⇤c //
C · C
22 T · T
q⇤q
μS μT μC
✏✏ p ✏✏ ✏✏
**44 T c // C
(A) is easy. ⇤
Definition 2.5. An object Z is a fixpoint of an endofunctor H if HZ ' Z, and it is a pre-fixpoint of H if HZ is a subobject of Z.
We say that H has arbitrarily large pre-fixpoints provided that for every object X there exists a pre-fixpoint Z of H with Z ' Z + X.
S
The verification that (C,⌘C,μC) is a monad and c is a coequalizer in Monad
q