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COLIMITS OF MONADS 11
Example 2.6. A monos-preserving endofunctor H of the category SetS of many-sorted sets has arbitrarily large pre-fixpoints i↵ for every cardinal ↵ there exists a pre-fixpoint of H all components of which have at least ↵ elements.
Lemma 2.7. Every accessible endofunctor of a cocomplete category with monic coproduct injections has arbitrarily large pre-fixpoints.
Proof. If H is accessible, then every object B generates a free H-algebra B¯ and B¯ = B + HB¯, see [2]. Given an object A let B be an infinite copower of A. ThentheequalityA+B'BimpliesA+B¯'B¯,andB¯isapre-fixpoint. ⇤
Notation 2.8. (a) For an endofunctor H of A an algebra is a pair (A, a) consisting of an object A and a morphism a : HA ! A. Homomorphisms of algebras are defined by the usual commutative square. The resulting category is denoted by AlgH.
(b) μH denotes the initial algebra (if it exists). By Lambek’s Lemma [10] its algebra structure is invertible, thus, μH is a fixpoint of H.
(c) If H has free algebras, i.e., the forgetful functor AlgH ! A has a left adjoint, then FH denotes the corresponding monad over A. And ⌘ˆ : Id ! FH denotes its unit, whose components are the universal arrows of the free algebras.
Theorem 2.9 (Barr [6] and Kelly [9]). If an endofunctor H has free algebras, then FH is a free monad on H. The converse holds whenever the base category is complete.
Example 2.10. The power-set functor P has no fixpoint, hence, it does not generate a free monad.
Construction 2.11 (see [2]). For every object X of A define the free-algebra chain W : Ord ! A (with objects Wi and morphisms wi,j : Wi ! Wj for all ordinals i  j) uniquely up to natural isomorphism by the following transfinite induction:
The objects are given by
Wo = X
and
Wi+1 =X+HWi,
Wj = colim Wi for limit ordinals j.
i<j
The morphisms are as follows:
w0,1 : X ! X + HX, coproduct injection
wi+1,j+1 = idX + Hwi,j
















































































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