On \(n\)-unital and \(n\)-Mal'tsev categories (Q6631562)

The authors' abstract says: ``Inspired by some properties of the (dual of the) category of \(2\)-nilpotent groups, we introduce the notion of \(2\)-unital and \(2\)-Mal'tsev categories which, in some sense, generalises the notion of unital and Mal'tsev categories, and we characterise their varietal occurrences. This is actually the first step of an inductive process which we begin to unfold.''\N\NBut, in fact, the authors do a lot more than just ``begin to unfold'', including the following:\N\N1. They introduce \(n\)-unital categories. Let \(\mathbb{E}\) be a pointed category with finite limits. For a natural number \(n\geqslant2\), objects \(X_1,\ldots,X_{n+1}\), and \(k\in\{1,\ldots,n+1\}\), let us write \(X(k)\) for the product of all \(X_1,\ldots,X_{n+1}\) except \(X_k\). Let us also write \(\iota_k\) for the morphism \(X(k)\to X_1\times\ldots X_{n+1}\) induced by all the projections \(X(k)\to X_i\) \((i\neq k)\) and the zero morphism \(X(k)\to X_k\). The category \(\mathbb{E}\) is said to be \(n\)-unital, if the morphisms \(\iota_1,\ldots,\iota_{n+1}\) are always jointly strongly epimorphic. In particular, a category is \(1\)-unital if and only if it is unital in the sense of [\textit{D. Bourn}, J. Algebra 256, No. 1, 126--145 (2002; Zbl 1015.18003)]. It is also shown that if \(m\leqslant n\), then every \(m\)-unital category is \(n\)-unital.\N\N2. As shown in [\textit{D. Bourn}, Appl. Categ. Struct. 4, No. 2--3, 307--327 (1996; Zbl 0856.18004)], Mal'tsev categories can be defined via unital ones and the so-called fibration of points. Now \(n\)-Mal'tsev categories are similarly defined via \(n\)-unital ones. And again, a category is \(1\)-Mal'tsev if and only if it is a Mal'tsev category, and if \(m\leqslant n\), then every \(m\)-Mal'tsev category is am \(n\)-Mal'tsev category.\N\N3. Both \(n\)-unital and \(n\)-Mal'tsev varieties of universal algebras are fully characterized syntactically.\N\N4. So-called \(n\)-paralinear categories are introduced and studied.\N\N5. Many other interesting links with what is done in the last six papers the authors refer to are explored.