A note on the relationship between action accessible and weakly action representable categories

TL;DR

This paper investigates the relationship between action accessible and weakly action representable categories, demonstrating that the known implications do not reverse and exploring inheritance properties in subcategories.

Contribution

It provides counterexamples showing that action accessibility and normalizer existence do not imply weakly action representability, and analyzes inheritance properties in Birkoff subcategories.

Findings

Weakly action representability does not follow from action accessibility.

Existence of all normalizers does not imply weakly action representability.

Weakly action representability is not inherited by Birkoff subcategories.

Abstract

The main purpose of this paper is to show that the converse of the known implication weakly action representable implies action accessible is false. In particular we show that both action accessibility, as well as the (at least formally stronger) condition requiring the existence of all normalizers do not imply weakly-action-representability even for varieties. In addition we show that in contrast to both action accessibility and the condition requiring the existence of all normalizers, weakly-action representability is not necessarily inherited by Birkoff subcategories.