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From coextensive varieties to the Gaeta topos

William Zuluaga

TL;DR

This work analyzes how central elements govern decompositions in coextensive varieties, linking universal algebra to topos theory via the Gaeta topology. It proves that the central-elements assignment $Z(-)$ is functorial and representable by $\mathcal{V}(\mathbf{0}\times \mathbf{0},-)$ in any coextensive variety $\mathcal{V}$, and that the finitely presented part of the theory is itself coextensive. Under $(\vec{0},\vec{1})$-density, the Gaeta topos classifies central-free $\mathcal{V}$-models, with a precise internal-logic characterization of indecomposability and a dual presentation of the Gaeta basis in the finite presentation site. The results provide a unified framework for understanding direct-product decompositions across diverse algebraic varieties and establish a topos-theoretic classifier for central-free models, bridging categorical, universal-algebraic, and model-theoretic perspectives.

Abstract

In this paper, we show that in every coextensive variety V, the assignment that maps each algebra to its set of central elements is both functorial and representable. Furthermore, we prove that the full subcategory of finitely presented algebras in V is coextensive. Finally, we establish that if V is additionally (0, 1)-dense, the Gaeta topos classifies central-free V-models.

From coextensive varieties to the Gaeta topos

TL;DR

This work analyzes how central elements govern decompositions in coextensive varieties, linking universal algebra to topos theory via the Gaeta topology. It proves that the central-elements assignment is functorial and representable by in any coextensive variety , and that the finitely presented part of the theory is itself coextensive. Under -density, the Gaeta topos classifies central-free -models, with a precise internal-logic characterization of indecomposability and a dual presentation of the Gaeta basis in the finite presentation site. The results provide a unified framework for understanding direct-product decompositions across diverse algebraic varieties and establish a topos-theoretic classifier for central-free models, bridging categorical, universal-algebraic, and model-theoretic perspectives.

Abstract

In this paper, we show that in every coextensive variety V, the assignment that maps each algebra to its set of central elements is both functorial and representable. Furthermore, we prove that the full subcategory of finitely presented algebras in V is coextensive. Finally, we establish that if V is additionally (0, 1)-dense, the Gaeta topos classifies central-free V-models.
Paper Structure (8 sections, 23 theorems, 58 equations)

This paper contains 8 sections, 23 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and basic results
  4. Central Elements
  5. The central-elements functor
  6. The Gaeta topos and coextensive varieties
  7. Central-free models in a topos
  8. The characterization

Key Result

Lemma 2.1

Let $\mathbf{A}$ be an algebra and let $\theta\in \mathsf{Con}(\mathbf{A})$. Consider the sets and Then, the assignment $(\lambda,\mu)\mapsto (\lambda/\theta,\mu/\theta)$ defines a bijection between $P_{\theta}$ and $Z_{\theta}$.

Theorems & Definitions (41)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Theorem 3.4: Z2021
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 31 more