Page 19 - Textos de Matemática Vol. 46

COLIMITS OF MONADS 9
Notation 2.1. (a) Given a category A we write [A, A] for the category of endofunctors on A and natural transformations between them. And Monad (A) denotes the category of monads and monad morphisms. The obvious forgetful functor is denoted by V : Monad (A) ! [A, A].
(b) We use ⇤ to denote the parallel (horizontal) composition of natural trans- formations: given a : F ! F0 and b : G ! G0, where all functors are endo- functor of A, we have a⇤b : FG ! F0G0 given by a⇤b = aG0·Fb = F0b·aG. Recall also the interchange law:
(c ⇤ d) · (a ⇤ b) = (c · a) ⇤ (d · b).
Proposition 2.2. The forgetful functor of Monad (A) creates limits.
Remark. Recall that creation of limits means that for every diagram D in Monad (A) with a limit cone pd : T ! WDd of the underlying diagram in [A, A] there exists a unique structure of a monad on T for which each pd is a monad morphism. Moreover, the resulting cone is a limit in Monad (A). The following two (easy) proofs work, more generally, for the category of monads over an arbitrary monoidal category. We have not found a reference for them, we thus present those proofs.
Proof. For the given diagram
D:D! Monad(A)
denote the objects by
Dd = (Td, μd, ⌘d) (d 2 objD).
Given a limit cone pd : T ! Td, the unit of the monad on T is, necessarily, the
unique natural transformation
⌘T :Id!T withpd·⌘T =⌘d (d2objD).
(Recall that pd’s are required to preserve unit.) And the multiplication μT : T · T ! T is, necessarily, the unique natural transformation for which the squares
μT
Td·Td μd
T · T
pd ⇤pd
✏✏
// T pd
// ✏✏ Td
commute for all d 2 objD. The verification of the monad axioms is easy. To verify that this is a limit cone, let qd : (S,μS,⌘S) ! (Td,μd,⌘d) be a cone of D. There exists a unique natural transformation q : S ! T with qd = pd · q(d 2