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Effective descent morphisms of ordered families

Maria Manuel Clementino, Rui Prezado

TL;DR

The paper analyzes effective descent morphisms in the lax comma category Ord//X for locally complete X, reducing the problem to the well-understood cases in Ord and in Fam(X). Using a functorial embedding and an obstruction-style approach, it provides a complete characterization for the case where X has a bottom element, expressed via a combination of descent in Ord and in Fam(X) alongside precise compatibility conditions on associated families. It then shows that the bottom hypothesis is unnecessary by decomposing X into connected components and transferring the componentwise results, thereby obtaining the characterization for any locally complete X. An appendix extends the analysis to the antisymmetric setting (Pos), completing the descent-theoretic account in both lax and antisymmetric ordered contexts.

Abstract

We present a characterization of effective descent morphisms in the lax comma category $\mathsf{Ord}//X$ when $X$ is a locally complete ordered set, as well as in the antisymmetric setting.

Effective descent morphisms of ordered families

TL;DR

The paper analyzes effective descent morphisms in the lax comma category Ord//X for locally complete X, reducing the problem to the well-understood cases in Ord and in Fam(X). Using a functorial embedding and an obstruction-style approach, it provides a complete characterization for the case where X has a bottom element, expressed via a combination of descent in Ord and in Fam(X) alongside precise compatibility conditions on associated families. It then shows that the bottom hypothesis is unnecessary by decomposing X into connected components and transferring the componentwise results, thereby obtaining the characterization for any locally complete X. An appendix extends the analysis to the antisymmetric setting (Pos), completing the descent-theoretic account in both lax and antisymmetric ordered contexts.

Abstract

We present a characterization of effective descent morphisms in the lax comma category when is a locally complete ordered set, as well as in the antisymmetric setting.
Paper Structure (5 sections, 16 theorems, 13 equations)

This paper contains 5 sections, 16 theorems, 13 equations.

Table of Contents

  1. State-of-the-art
  2. Familial descent
  3. The characterization
  4. Redundancy of the bottom element
  5. Antisymmetry

Key Result

Theorem 1.1

Let $\mathsf{A}$ and $\mathsf{D}$ be categories with pullbacks, and $F\colon\mathsf{A}\to\mathsf{D}$ a fully faithful, pullback preserving functor. If $f\colon A\to B$ is a morphism in $\mathsf{A}$ such that $F(f)$ is effective for descent in $\mathsf{D}$, then the following conditions are equivalen

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2: JS
  • Theorem 1.3: CLN23
  • Theorem 1.4: CJ23
  • Lemma 2.1: Prez, PrezTh
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • ...and 13 more