On the lack of colimits in various categories of BAOs and Heyting algebras

Abstract: We prove that various categories of BAOs (boolean algebras with an operator) with stable morphisms between them are not cocomplete, and that neither are the category of Heyting algebras with bounded lattice morphisms, its full subcategory consisting of frames, and the category of frames with Heyting morphisms. As a consequence, none of these categories is equivalent to a prevariety of algebras, let alone a variety. In particular, we obtain that the category of McKinsey-Tarski algebras is not equivalent to a variety, thus answering a question by Peter Jipsen in the negative.