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

TL;DR

This paper demonstrates that several categories in pointfree topology and algebraic logic, such as McKinsey-Tarski algebras and Boolean algebras with operators, lack certain colimits, indicating they are not equivalent to varieties.

Contribution

It proves the non-existence of colimits in these categories, answering a specific open question and clarifying their structural limitations.

Findings

Category of McKinsey-Tarski algebras is not equivalent to a variety.

Categories of BAOs, Heyting algebras, and frames are not cocomplete.

These categories are not equivalent to prevarieties or varieties.

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.