Zurab Janelidze

Subtractive Categories and Extended Subtractions

We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal’tsev, unital, strongly unital and subtractive categories. However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations of subtractive categories and moreover, they give an enlarged “algebraic tool” for working in a subtractive category—we demonstrate this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects in a general subtractive category is given for the first time in the present paper.

On the Form of Subobjects in Semi-Abelian and Regular Protomodular Categories

Let Gls denote the category of (possibly large) ordered sets with Galois connections as morphisms between ordered sets. The aim of the present paper is to characterize semi-abelian and regular protomodular categories among all regular categories ℂ, via the form of subobjects of ℂ, i.e. the functor ℂ → Gls which assigns to each object X in ℂ the ordered set Sub(X) of subobjects of X, and carries a morphism f : X → Y to the induced Galois connection Sub(X) → Sub(Y) (where the left adjoint maps a subobject m of X to the regular image of fm, and the right adjoint is given by pulling back a subobject of Y along f). Such functor amounts to a Grothendieck bifibration over ℂ. The conditions which we use to characterize semi-abelian and regular protomodular categories can be stated as self-dual conditions on the bifibration corresponding to the form of subobjects. This development is closely related to the work of Grandis on “categorical foundations of homological and homotopical algebra”. In his work, forms appear as the so-called “transfer functors” which associate to an object the lattice of “normal subobjects” of an object, where “normal” is defined relative to an ideal of null morphism admitting kernels and cokernels.

Star-regularity and regular completions

In this paper we establish a new characterisation of star-regular categories, using a property of internal reflexive graphs, which is suggested by a recent result due to O. Ngaha Ngaha and the first author. We show that this property is, in a suitable sense, invariant under regular completion of a category in the sense of A. Carboni and E.M. Vitale. Restricting to pointed categories, where star-regularity becomes normality in the sense of the second author, this reveals an unusual behaviour of the exactness property of normality (i.e. the property that regular epimorphisms are normal epimorphisms) compared to other closely related exactness properties studied in categorical algebra.

On the Structure of Zero Morphisms in a Quasi-Pointed Category

A quasi-pointed category in the sense of D. Bourn is a finitely complete category 𝓒

An order-theoretic perspective on categorial closure operators

\mathcal {C}

A Note on the Abelianization Functor

having an initial object such that the unique morphism from the initial object to the terminal object is a monomorphism. When instead this morphism is an isomorphism, we obtain a (finitely complete) pointed category, and as it is well known, the structure of zero morphisms in a pointed category determines an enrichment of the category in the category of pointed sets. In this note we examine quasi-pointed categories through the structure formed by the zero morphisms (i.e. the morphisms which factor through the initial object), with the aim to compare this structure with an enrichment in the category of pointed sets.

Split Short Five Lemma for Clots and Subtractive Categories

Abstract This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.

Closedness Properties of Internal Relations III: Pointed Protomodular Categories

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1G, 1G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.