Projective covers, doctrines of algebras and the relational quotient completion

Francesco Dagnino; Fabio Pasquali

Home/Archive/arXiv

arXiv · 2608.01915

Projective covers, doctrines of algebras and the relational quotient completion

Francesco Dagnino·Fabio Pasquali

View original source ↗Read source document ↗Export metadata ↓

Abstract

The extensional quotient completion of relational doctrines provides a common generalization of both the exact completion of categories with weak finite limits and the elementary quotient completion of existential elementary doctrines. In this paper, we study projective objects in relational doctrines with quotients, characterizing those obtained through the extensional quotient completion as those admitting a projective cover. We apply this result to doctrines of algebras for monads on relational doctrines with quotients, describing in which cases these arise as the extensional quotient completion of their restriction to (appropriate subcategories of) free algebras. This extends a similar result for monadic categories over exact ones, covering also more examples such as monads over the category of metric spaces giving rise to variants of quantitative algebras.

Explore related subjects

math.CT

Keep this discovery

BibTeX RIS

Francesco Dagnino, Fabio Pasquali. 2026-08-03. Projective covers, doctrines of algebras and the relational quotient completion. https://arxiv.org/abs/2608.01915

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A model structure for cartesian 2-fibrations

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

math.CT↗

arXiv2026

A Natural Fuzzy Order on Fuzzy Numbers

This paper introduces a natural fuzzy order on fuzzy numbers that extends the natural orders on real numbers and interval numbers. We investigate its completeness properties and show that the space of uniformly bounded fuzzy numbers is conically complete and conically cocomplete, and that it is complete if and only if the underlying continuous t-norm is the G\"odel t-norm. Moreover, it is proved that this space constitutes a $[0,1]$-enriched domain if and only if the underlying continuous t-norm satisfies the (S) condition. These results provide a foundation for ordering fuzzy numbers.

math.CT↗

arXiv2026

Noetherian forms of free non-symmetric operads

In this paper, we study certain categories of labeled finite rooted ordered trees over a fixed set of labels where each label is equipped with an arity: a fixed number of children that the vertex with the given label must have. Equivalently, these are expression trees for operations in a free non-symmetric operad. A morphism between these trees matches a pruning of one tree (a prefix) with an entire subtree of another (a suffix). We characterize such categories, up to isomorphism, in terms of suitable exactness properties. It turns out that these categories exhibit strong algebraic behavior, in the sense that every such category, when appended with a strict initial object, has a particularly nice noetherian form.

math.CT↗