Page 36 - Textos de Matemática Vol. 46

2 6 J . A D A´ M E K
Theorem 4.4 (See [3]). A monad over Set has coproducts with all monads i↵ it is trivial.
Moreover, all monads over Set except 1 and 10 are consistent, i.e., the components of the monad unit are monic.
Definition 4.5. A monad (S, μ, ⌘) is called separated if its unit has a comple- ment in the following sense:
(i) S preserves monomorphisms and
(ii) there exists an endofunctor S¯ of Am such that the restriction of S to
Am fulfils
with the unit ⌘ as the left-hand injection.
S = Id + S¯ (1)
Examples 4.6. (1) The exception monad ME is separated: here M¯E is the constant functor of value E.
(2) Every free monad FH which preserves monomorphisms is separated. (In particular, if A has stable monomorphisms, all free monads on monos- preserving functor are separated.) Here F¯ H = H · FH : use Remarks 2.16 and 2.17.
(3) All consistent monads on Set (i.e., all except 1 and 10) are separated. See [3], Proposition IV.5.
(4) Ideal monads of Elgot [7] are separated if they preserve monomorphisms. Recall that an ideal monad S = (S, μ, ⌘) is one for which an endofunctor S¯ of A exists such that (i) S = Id + S¯ in [A, A] with the left-hand injection ⌘ and (ii) μ restricts to a natural transformation μ¯ : S¯S ! S¯.
(5) In particular, the free completely iterative monad S on an endofunctor H given by the greatest fixpoint
SA = ⌫X · (A + HX)
is separated, with S¯ = H · S, whenever it preserves monomorphisms, see
[1].
Notation 4.7. Let Si (i 2 I) be separated monads. For every object A of A
define an endofunctor HA of AIm as follows: a
HA(Xi)i2I = (S¯iYi)i2I where Yi = A + Xj (2)
j2I,j6=i
If HA has an initial algebra, we denote its components by Si⇤A:
μHA = (Si⇤A)i2I