Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.08610

The saturated prefilter monad

Hongliang Lai·Dexue Zhang·Gao Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper considers some extensions of the notion of filter to the quantale-valued context, including saturated prefilter, $\top$-filter and bounded saturated prefilter. The question is whether these constructions give rise to monads on the category of sets. It is shown that the answer depends on the structure of the quantale. Specifically, if the quantale is the unit interval equipped with a continuous t-norm, then these constructions give rise to monads if and only if the implication operator corresponding to that t-norm is continuous at each point off the diagonal.

Explore related subjects

math.CTmath.GN

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hongliang Lai, Dexue Zhang, Gao Zhang. 2020-08-23. The saturated prefilter monad. https://arxiv.org/abs/2003.08610

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

KEEP EXPLORING

Related papers

arXiv2026

Mixing Extriangulated Model Structures

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be a weakly idempotent complete extriangulated category. We generalize Cole's Theorem to construct a mixed admissible model structure $\mathcal{M}_m$ from two compatible admissible model structures relative to proper classes $ξ_1\subseteqξ_2$ of $\mathscr{C}$. We then explicitly characterize the cofibrant objects of $\mathcal{M}_m$. Finally, we apply these results to exact and triangulated categories, recovering and extending recent work on mixed model structures.

math.CT↗
arXiv2026

Characterizing (Co)Free Dagger Categories

For any category, there exists both a free dagger category and a cofree dagger category over it. A natural question to ask is: given a dagger category, how can we tell if it is free or cofree without specifying an external base category? In this paper, we provide characterizations of both free dagger categories and cofree dagger categories via internal dagger category structure. To characterize cofree dagger categories, we use rectangular bands and show that a dagger category is cofree if and only if it is enriched over rectangular bands. For free dagger categories, we define the notion of a zigzag dagger category, and then show that a dagger category is free if and only if it is a zigzag dagger category. We also show that free dagger categories can be characterized as the coalgebras of the induced comonad from the free dagger category adjunction, and similarly that cofree free dagger categories can be characterized as the algebras of the induced monad from the cofree dagger category adjunction.

math.CT↗
arXiv2026

A Counterexample to the Open Question on Object Ideals

We give a counterexample to completeness descent from ideal cotorsion pairs to their objects. A radical-square-zero algebra on the two-cycle gives a finite-dimensional example. The construction is intrinsically non-weakly-idempotent-complete.

math.CT↗