English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper aims to apply the tool of generalized existential completions ofconjunctive doctrines, concerning a class $\Lambda$ of morphisms of their basecategory, to deepen the study of regular and exact completions of existentialelementary Lawvere's doctrines. After providing a characterization ofgeneralized existential completions, we observe that both the subobjectsdoctrine $\mathrm{Sub}_{C}$ and the weak subobjects doctrine$\Psi_{\mathcal{C}}$ of a category $\mathcal{C}$ with finite limits aregeneralized existential completions of the constant true doctrine, the firstalong the class of all the monomorphisms of $\mathcal{C}$ while the latteralong all the morphisms of $\mathcal{C}$. We then name full existentialcompletion a generalized completion of a conjunctive doctrine along the classof all the morphisms of its base. From this we immediately deduce that both theregular and the exact completion of a finite limit category are regular andexact completions of full existential doctrines since it is known that both theregular completion $(\mathcal{D})_{ reg / lex}$ and the exact completion$(\mathcal{D})_{ex / lex}$ of a finite limit category $\mathcal{D}$ arerespectively the regular completion $\mathrm{Reg}(\Psi_{\mathcal{D}})$ and theexact completion $\mathcal{T}_{\Psi_{\mathcal{D}}}$ (as an instance of thetripos-to-topos construction) of the weak subobjects doctrine$\Psi_{\mathcal{D}}$ of $\mathcal{D}$. Here we prove that the condition ofbeing a generic full existential completion is also sufficient to produce aregular/exact completion equivalent to a regular/exact completion of a finitelimit category. Then, we show more specialized characterizations from which wederive known results as well as remarkable examples of exact completions offull existential completions, including all realizability toposes andsupercoherent localic toposes.

Generalized existential completions and their regular and exact completions