Page 29 - Textos de Matemática Vol. 46

COLIMITS OF MONADS 19
follows from the following diagram
S S μi // S iii
ui Si ""
fi Si RSi ✏✏||pfSi
TfSi
ui
""μR ⌥⌥
RR //R fi
pfR )) T ✏✏ R
pf Tf pf
Rui
Tf ui
Tf fi
f
"" ✏✏
TfTf μTf
// ✏✏↵↵ Tf
All the inner parts but the upper one (to be proved commutative) commute: recall fi = pf · ui, use the fact that pf is a monad morphism for the lower square, and use the naturality of pf for pfR·Rui = Tfui ·pfSi. Since fi is a monad morphism, the outward square also commutes. This, together with the collective monicity of all pf ’s, proves that the upper square commutes.
For every cocone f = (fi)i2I the monad morphism pf is the desired factor- ization: fi = pf · ui, see (1). This is unique since whenever r : R ! Tf is a monad morphism with fi = r·ui for all i, then f = r·[ui]. From (1) we see that[ui]=e·⌘ˆ,thusr·e·⌘ˆ=f=pf ·e·⌘ˆwhichimpliesr·e=pf ·ebythe universal property of ⌘ˆ; hence r = pf since e is epic. ⇤
Remark 3.8. (a) Kelly described colimits of monads, see [9], Section 27 as follows:
Let D be a diagram in Monad (A) with objects Ti = (Ti,μi,⌘i) for i 2 I.
Form the category CD of all pairs (A,(ai)i2I) where A is an object of A and ai : TiA ! A is an Eilenberg-Moore algebra for Ti (i 2 I) such that for every connecting morphism f : i ! j of the indexing category the