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On the lack of colimits in various categories arising in pointfree topology and algebraic logic

Marco Abbadini, Guram Bezhanishvili, Luca Carai

Abstract

We prove that the category of McKinsey-Tarski algebras is not equivalent to a variety of algebras, thus answering a question of Peter Jipsen in the negative. More generally, we show that various categories of BAOs (boolean algebras with an operator), Heyting algebras, and frames with appropriate morphisms between them are not cocomplete. As a consequence, none of these categories is equivalent to a prevariety, let alone a variety.

On the lack of colimits in various categories arising in pointfree topology and algebraic logic

Abstract

We prove that the category of McKinsey-Tarski algebras is not equivalent to a variety of algebras, thus answering a question of Peter Jipsen in the negative. More generally, we show that various categories of BAOs (boolean algebras with an operator), Heyting algebras, and frames with appropriate morphisms between them are not cocomplete. As a consequence, none of these categories is equivalent to a prevariety, let alone a variety.

Paper Structure

This paper contains 5 sections, 28 theorems, 23 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. MT-algebras
  3. MT-algebras with stable morphisms
  4. Closure algebras with stable morphisms
  5. Heyting algebras and frames with lattice morphisms

Key Result

Theorem 2.2

The free countably generated complete boolean algebra does not exist. Thus, ${\mathbf{CBA}}$ is not cocomplete, and hence is not equivalent to a prevariety.

Figures (4)

  • Figure 1: The space $(X,R)$.
  • Figure 2: The space $(X',R')$.
  • Figure 3: The space $(X,R)$.
  • Figure 4: The space $(X',R')$.

Theorems & Definitions (56)

  • Definition 2.1
  • Theorem 2.2: Gai64Hal64
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 46 more