On linear exactness properties (Q2035813)

The authors elaborate on the conceptual framework they introduced in [\textit{P.-A. Jacqmin} and \textit{Z. Janelidze}, Adv. Math. 377, Article ID 107484, 56 p. (2021; Zbl 1452.18005)]. There, an abstract formalism was developed, based on the notion of sketch, in order to give an abstract account of exactness properties on a small finitely complete category \(\mathbb{C}\) that are preserved under pro-completion (completion under co-filtered limits, given by the embedding \( \mathbb{C} \hookrightarrow \mathrm{Lex}(\mathbb{C},\mathrm{Set})^{op}).\) Here the same formalism is used in order to obtain exactness properties in regular categories such that, when the category is algebraic, turn out to be equivalent to the existence of certain Mal'tsev terms in the corresponding algebraic theory. The main characterization theorem of the article (Theorem 3.3) asserts the equivalence of suitable exactness conditions and the existence of Mal'tsev terms and equations in the context of essentially algebraic (hence locally finitely presentable) regular categories. The theorem exploits a reformulation of those exactness conditions on a finitely cocomplete regular category in terms of a certain morphism, in the image of a left Kan extension with values in the finitely cocomplete category, being a regular epimorphism (Theorem 2.1).