Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Functoriality of the Schmidt Construction

Version 1 : Received: 24 September 2021 / Approved: 27 September 2021 / Online: 27 September 2021 (11:45:35 CEST)

How to cite: Climent Vidal, J.; Cosme Llópez, E. Functoriality of the Schmidt Construction. Preprints 2021, 2021090445 (doi: 10.20944/preprints202109.0445.v1). Climent Vidal, J.; Cosme Llópez, E. Functoriality of the Schmidt Construction. Preprints 2021, 2021090445 (doi: 10.20944/preprints202109.0445.v1).

## Abstract

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, 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, labeled 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 in 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 in Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0.

## Keywords

Many-sorted partial algebra; free completion; category of completions; weakly initial object; comma category of objects over a completion; Schmidt construction; Schmidt homomorphism; twisted morphism category; Schmidt endofunctor; functoriality of the Schmidt construction

## Subject

MATHEMATICS & COMPUTER SCIENCE, General & Theoretical Computer Science

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