Page 95 - Textos de Matemática Vol. 45

7.4. The direction functor in Mon 87 h: A → A′ determines a morphism in AbSPtY (Mon), i.e. a monoid homo-
morphism h 1 making the following diagram commute A φ Y h 1 // A′ ψ Y
ccpφ ιψ;; YY
ιφY
if and only if h is a Y -module homomorphism, i.e. if and only if we have:
h(φ(y)(a)) = ψ(y)(h(a)) for all (a, y) ∈ A φ Y .
Proof. Straightforward calculation.
Theorem 7.4.3. Given any special Schreier extension with abelian kernel A and direction (A φ Y, Y, pφY , ιφY ):
A=K[f]//k //Xf////Y
and any surjective homomorphism h : A A′ of Y -modules, where the struc- ture of Y -module of A′ is given by the morphism ψ : Y → End(A′), there is a special Schreier extension with abelian kernel A′ and direction given by (A′ ψ Y , Y, pψY , ιψY ), called direct image of the special Schreier extension along h, together with a monoid homomorphism h¯ which makes the following diagram commutative
A//kf //Xf////Y
##Y{{ pψY
h A′ //
h¯
//X′ ////Y
f′
Given any morphism l of special Schreier extensions:
A//kf //Xf////Y
kf′
K(l)

phism, there is a unique morphism μ of special Schreier extensions A′//kf′ //X′ f′ ////Y
l

//X′′ ////Y
f′′
with a factorization mh = K(l), where m: A′ → A′′ is a Y -module homomor-
A′′ //
kf′′
mμ
A′′ //
kf′′

//X′′ ////Y f′′