Generalized existential completions and their regular and exact completions

TL;DR

This paper explores generalized existential completions of doctrines to better understand regular and exact completions of categories, providing new characterizations and examples including realizability toposes.

Contribution

It introduces the concept of full existential completion and links it to regular and exact completions, offering new characterizations and applications.

Findings

Regular and exact completions are special cases of full existential completions.

Characterizations of regular/exact completions via generalized existential completions.

Includes examples like realizability toposes and supercoherent localic toposes.

Abstract

This paper aims to apply the tool of generalized existential completions of conjunctive doctrines, concerning a class $\Lambda$ of morphisms of their base category, to deepen the study of regular and exact completions of existential elementary Lawvere's doctrines. After providing a characterization of generalized existential completions, we observe that both the subobjects doctrine $\mathrm{Sub}_{C}$ and the weak subobjects doctrine $\Psi_{\mathcal{C}}$ of a category $\mathcal{C}$ with finite limits are generalized existential completions of the constant true doctrine, the first along the class of all the monomorphisms of $\mathcal{C}$ while the latter along all the morphisms of $\mathcal{C}$. We then name full existential completion a generalized completion of a conjunctive doctrine along the class of all the morphisms of its base. From this we immediately deduce that both the regular and the exact completion of a finite limit category are regular and exact completions of full existential doctrines since it is known that both the regular completion $(\mathcal{D})_{ reg / lex}$ and the exact completion $(\mathcal{D})_{ex / lex}$ of a finite limit category $\mathcal{D}$ are respectively the regular completion $\mathrm{Reg}(\Psi_{\mathcal{D}})$ and the exact completion $\mathcal{T}_{\Psi_{\mathcal{D}}}$ (as an instance of the tripos-to-topos construction) of the weak subobjects doctrine $\Psi_{\mathcal{D}}$ of $\mathcal{D}$. Here we prove that the condition of being a generic full existential completion is also sufficient to produce a regular/exact completion equivalent to a regular/exact completion of a finite limit category. Then, we show more specialized characterizations from which we derive known results as well as remarkable examples of exact completions of full existential completions, including all realizability toposes and supercoherent localic toposes.