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

Let $\mathsf{PreOrd}(\mathbb C)$ be the category of internal preorders in anexact category $\mathbb C$. We show that the pair $(\mathsf{Eq}(\mathbb C),\mathsf{ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{PreOrd}(\mathbbC)$, where $\mathsf{Eq}(\mathbb C)$ and $\mathsf{ParOrd}(\mathbb C)$) are thefull subcategories of internal equivalence relations and of internal partialorders in $\mathbb C$, respectively. We observe that $\mathsf{ParOrd}(\mathbbC)$ is a reflective subcategory of $\mathsf{PreOrd}(\mathbb C)$ such that eachcomponent of the unit of the adjunction is a pullback-stable regularepimorphism. The reflector $F:\mathsf{PreOrd}(\mathbb C)\to\mathsf{ParOrd}(\mathbb C)$ turns out to have stable units in the sense ofCassidy, H\'ebert and Kelly, thus inducing an admissible categorical Galoisstructure. In particular, when $\mathbb C$ is the category $\mathsf{Set}$ ofsets, we show that this reflection induces a monotone-light factorizationsystem (in the sense of Carboni, Janelidze, Kelly and Par\'e) in$\mathsf{PreOrd}(\mathsf{Set})$. A topological interpretation of our results inthe category of Alexandroff-discrete spaces is also given, via the well-knownisomorphism between this latter category and $\mathsf{PreOrd}(\mathsf{Set})$.

A new Galois structure in the category of internal preorders