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J . A D A´ M E K
TiA ai //A OO
triangle
(b)
commutes. The morphisms of CD are the morphisms of A which are algebra homomorphisms for every Ti. We have the obvious forgetful functor
UD :CD !A.
Kelly proved that if UD has a left adjoint, then the corresponding monad on A is a colimit of D in Monad (A). The converse also holds if A is a complete category.
Given a discrete diagram D of monads Ti (i 2 I) the category CD has as objects multi-algebras
(A,(ai)i2I)whereai :TiA!AliesinATi
and morphisms are those maps in A that are homomorphisms for each of Ti simultaneously. A coproduct of the monads Ti exists in Monad (A) whenever every object of A generates a free multi-algebra.
(Df)A
aj (2) ""Tj A
Theorem 3.9. Every diagram with a weakly terminal object has a colimit in Monad (A). In particular, Monad (A) has coequalizers.
Proof. Let D : D ! Monad (A) be a diagram with objects Ti = (Ti,μi,⌘i) for i 2 I, and let Tj be weakly terminal, i.e., for every i 2 I there exists a connecting morphism f : Ti ! Tj in D.
(a) Form the full subcategory C of ATj of all algebras a : TjA ! A for Tj such that for every pair f, g : Ti ! Tj of connecting morphisms of D (i 2 I) we have
a·fA =a·gA (3)
This category is closed in ATj under products, which easily follows from the forgetful functor UTj creating limits. It is also closed under subalgebras. More precisely, let m : (A, a) ! (B, b) be a homomorphism in ATj with m