Functoriality of the Schmidt construction

Abstract

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σalgebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair (α, f), where α = (K, γ, α) is a morphism of Σ-completions from F = (C, F, η) to G = (D, G, ρ) and f a homomorphism of D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥαG,0(f): Schα(f) −→ B. We then prove that there exists an endofunctor, ΥαG,0, of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every Σ-completion G = (D, G, ρ), there exists a functor ΥG from the comma category (Cmpl(Σ) ↓ G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of ΥG, sends a morphism of Σ-completion of Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0