Related papers: A syntactic characterization of weakly Mal'tsev va…
We explore a hierarchy of notions in categorical algebra: Mal'tsev categories (where every reflexive relation is symmetric); naturally Mal'tsev categories (where every reflexive graph underlies a unique internal groupoid structure, also…
Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and the beauty of the notion is revealed through a lot of characterizations of different flavour. Depending on the context, one can…
A classification theorem for three different sorts of Mal'tsev categories is proven. The theorem provides a classification for Mal'tsev category, naturally Malt'sev category, and weakly Mal'tsev category in terms of classifying classes of…
Meet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev…
Matrix conditions extend linear Mal'tsev conditions from Universal Algebra to exactness properties in Category Theory. Some can be stated in the finitely complete context while, in general, they can only be stated for regular categories. We…
We study the difference between internal categories and internal groupoids in terms of generalised Mal'tsev properties---the weak Mal'tsev property on the one hand, and $n$-permutability on the other. In the first part of the article we…
We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular…
The original Lawvere condition asserts that every reflexive graph admits a unique natural structure of internal groupoid. This property was identified by P. T. Johnstone, following a question by A. Carboni and a suggestion by F. W. Lawvere,…
This paper provides a short introduction to the notion of regular category and its use in categorical algebra. We first prove some of its basic properties, and consider some fundamental algebraic examples. We then analyse the algebraic…
We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new…
Matrix properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications…
In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various properties of such groups and establish in…
Considering classical first-order logic with equality, we give a "fully syntactic" construction of the (weak) syntactic category $\text{Syn}(T)$ associated to a consistent theory $T$; we show it is a consistent coherent category; and we…
Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric…
We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras…
We provide a Maltsev characterization of congruence distributive varieties by showing that a variety $\mathcal {V}$ is congruence distributive if and only if the congruence identity $\alpha \cap (\beta \circ \gamma \circ \beta ) \subseteq…
Many definitions of weak and strict $\infty$-categories have been proposed. In this paper we present a definition for $\infty$-categories with strict associators, but which is otherwise fully weak. Our approach is based on the existing type…
Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator. This means a strong generator which is abstractly finite and regularly projective. An analogous…
We introduce the notion of Mal'tsev reflection which allows us to set up a partial notion of Mal'tsevness with respect to a class $\Sigma$ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what…
This paper is the first in a series of articles devoted to the construction and study of synthetic deformations of $\infty$-categories in the unstable context: that is, deformations of $\infty$-categories that categorify spectral sequence…