George Janelidze

• University of Cape Town

We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder X, is defined as a preorder A equipped with a...

The purpose of this paper is to initiate a development of a new non-pointed counterpart of semi-abelian categorical algebra. We are making, however, only the first step in it by giving equivalent definitions of what we call ideally exact categories, and showing that these categories admit a description of quotient objects by means of intrinsically...

For a commutative semiring S, by an S-algebra we mean a commutative semiring A equipped with a homomorphism from S to A. We show that the subvariety of S-algebras determined by the identities 1+2x=1 and x^2=x is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices unde...

A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider cent...

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum \({\mathrm {Spec}}(R)\) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ide...

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum $\mathrm{Spec}(R)$ of a unital commutative ring $R$ is always a spectral (=coherent) topological space. In this generalization, which includes several other known ones, the role of ideals...

The paper is devoted to a kind of “very non-abelian” spectral categories. Under strong conditions on a category \mathcal X , we prove, among other things, that, for a given faithful localization \mathcal C \to \mathcal X , we have canonical equivalences Spec (\mathcal{C})\sim\mathcal{X}\sim (category of injective objects in \mathcal{C}) , and that...

For a category with finite limits and a class of monomorphisms in that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable -essential monomorphisms in to construct a spectral category . We show that it has finite limits and that the canonical functor prese...

The paper is devoted to a kind of `very non-abelian' spectral categories. Under strong conditions on a category $\mathcal{X}$, we prove, among other things, that, for a given faithful localization $\mathcal{C}\to\mathcal{X}$, we have canonical equivalences $\mathrm{Spec}(\mathcal{C})\sim\mathcal{X}\sim(\mathrm{Category\,\,of\,\,injective\,\,objects...

Given a monad T on the category of sets, we consider reflections of Alg(T) into its full subcategories formed by algebras satisfying natural counterparts of topological separation axioms T0, T1, T2, Tts, and Tths; here ts stands for totally separated and ths for what we call totally homomorphically separated, which coincides with ts in the (compact...

We make three independent observations on characterizing effective descent morphisms in the category of topological spaces. The first of them proposes a new modification of known characterizations of effective descent morphisms of general spaces, while the other two are devoted to locally finite and Hausdorff spaces, respectively. The Hausdorff cas...

We generalize the van Kampen theorem for unions of non-connected spaces, due to R. Brown and A. R. Salleh, to the context where families of subspaces of the base space $B$ are replaced with a `large' space $E$ equipped with a locally sectionable continuous map $p:E\to B$.

We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) actions of unitary magmas on unitary magmas. The class of split extensions is pullback stable but...

For a category $\mathcal{C}$ with finite limits and a class $\mathcal{S}$ of monomorphisms in $\mathcal{C}$ that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable $\mathcal{S}$-essential monomorphisms in $\mathcal{C}$ to construct a spectral category $\m...

We investigate additional properties of protolocalizations, introduced and studied by Borceux, Clementino, Gran, and Sousa, and of protoadditive reflections, introduced and studied by Everaert and Gran. Among other things, we show that there are no non-trivial (protolocalizations and) protoadditive reflections of the category of groups, and establi...

We make various observations on infinitary addition in the context of the series monoids introduced in our previous paper on real sets. In particular, we explore additional conditions on such monoids suggested by Tarski’s Arithmetic of Cardinal Algebras, and present a monad-theoretic construction that generalizes our construction of paradoxical rea...

We describe an alternative way of constructing some of the monads, recently introduced by E. Colebunders, R. Lowen, and W. Rosiers for the purposes of categorical topology.

In the original publication of the article, Eq. 3.24 was published incorrectly. The corrected equation is given in this correction article. The original article has been corrected. © 2018 Springer Science+Business Media B.V., part of Springer Nature

The purpose of this paper is to introduce Frattini theory in semi-abelian categories. We extend several basic group-theoretic results about Frattini subgroups and Frattini extensions, specifically selected from an article of Moori and Rodrigues, to a categorical context involving either Hoehnke radicals or our notion of Frattini subobjects.

After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions: \begin{itemize} \item what is a set with half an element? \item what is a set with $\pi$ elements? \end{itemize} The category of these extended positive real sets is equ...

After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions: \begin{itemize} \item what is a set with half an element? \item what is a set with $\pi$ elements? \end{itemize} The category of these extended positive real sets is equ...

We investigate additional properties of protolocalizations, introduced and studied by F. Borceux, M. M. Clementino, M. Gran, and L. Sousa, and of protoadditive reflections, introduced and studied by T. Everaert and M. Gran. Among other things we show that there are no non-trivial (protolocalizations and) protoadditive reflections of the category of...

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that...

It is shown that the Huq and the Smith commutators do not coincide in the variety of near-rings.

We introduce new notions of weighted centrality and weighted commutators corresponding to each other in the same way as centrality of congruences and commutators do in the Smith commutator theory. Both the Huq commutator of subobjects and Pedicchio's categorical generalization of Smith commutator are special cases of our weighted commutators; in fa...

We discuss the problem formulated in the title. We solve it only in two very special cases: for maps with finite codomains and for maps that are open and order-open, or, equivalently, open and order-closed.

We introduce new notions of weighted centrality and weighted commutators corresponding to each other in the same way as centrality of congruences and commutators do in the Smith commutator theory. Both the Huq commutator of subobjects and Pedicchio's categorical generalization of Smith commutator are special cases of our weighted commutators; in fa...

In this paper we show that the theorem, by Cagliari and Mantovani, stating that in the category of compact Hausdorff spaces every étale map is exponentiable, can be formulated in a general category Alg(T) of Eilenberg–Moore TT-algebras, for a monad TT, and proved in case TT satisfies the so-called Beck–Chevalley condition. For that, Alg(T) is embed...

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh–Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical cl...

This paper begins a systematic study of weakly cartesian properties of monads that determine familiar varieties of universal algebras. While these properties clearly fail to hold for groups, rings, and many other related classical algebraic structures, their analysis becomes non-trivial in the case of semimodules over semirings, to which our main r...

We formulate two open problems related to and, in a sense, suggested by the Reiterman-Tholen characterization of effective descent morphisms of topo-logical spaces.

We show that the category of regular epimorphisms in a Barr exact Goursat category is almost Barr exact in the sense that (it is a regular category and) every regular epimorphism in it is an eective descent morphism.

We clarify the role of Hofmann's Axiom in the old-style definition of a semi-abelian category. By removing this axiom we obtain the categorical counterpart of the notion of an ideal-determined variety of universal algebras – which we therefore call an ideal-determined category. Using known counterexamples from universal algebra we conclude that the...

We clarify the relationship between separable and covering morphisms in general categories by introducing and studying an intermediate class of morphisms that we call strongly separable.

Regarding categories as simplicial sets via the nerve functor, we extend the notion of a factorization system from morphisms in a category, to 1-simplexes in an arbitrary simplicial set. Applied to what we call the simplicial set of short exact sequences, it gives the notion of Kurosh–Amitsur radical. That is, we present a unified approach to facto...

We clarify the relationship between ideals, clots, and normal subobjects in a pointed regular category with finite coproducts.

We examine Galois theory of the T0-reflection, for topological spaces and in a more general context. We show that it admits the monotone-light factorization system and the so-called theory of locally semisimple coverings. As expected, the class of locally semisimple coverings coincide with the class of light maps.

We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads dir...

A general description of the Galois group of a “pointed” normal extension in categorical Galois theory is examined under the presence of a suitable commutator operation. In particular, using the Hopf formula for the second homology group of a group, the connection between Galois theory and group homology is clarified.

Using the reflection of the category C of compact 0-dimensional topological spaces into the category of Stone spaces we introduce a concept of a fibration in C. We show that: (i) effective descent morphisms in C are the same as the surjective fibrations; (ii) effective descent morphisms in C with respect to the fibrations are all surjections.

We generalize the definition of satellites with respect to presheaves (and copresheaves) with trace in the sense of Inassaridze; a presheaf with trace is replaced by a graph with a pair of diagrams defined on it. We show that the right satellite functor is left adjoint to the left satellite functor, and that a functor having a right (left) adjoint...

We generalize the definition of satellites with respect to presheaves (and co-presheaves) with trace in the sense of [1]; a presheaf with trace is replaced by a graph with a pair of diagrams defined on it. We show that the right satellite functor is left adjoint to the left satellite functor, and that a functor having a right (left) adjoint preserv...

We study what kinds of limits are preserved by the greatest semilattice image functor from the category of all semigroups to its subcategory of all semilattices.

In a pointed category with kernels and cokernels, we characterize torsion–free classes in terms of their closure under extensions. They are also described as indexed reflections. We obtain a corresponding characterization of torsion classes by formal categorical dualization.

We clarify the relationship between basic constructions of semi-abelian category theory and the theory of ideals and clots in universal algebra. To name a few results in this frame, which establish connections between hitherto separated subjects, 0-regularity in universal algebra corresponds to the requirement that regular epimorphisms are normal;...

The continuous maps p : E → B for which p* : Top/B → Top/E reflects isomorphisms are shown to coincide with the universal quotient maps as characterized by Day and Kelly. Monadicity of p* turns out to be a local property. This is used to prove the main result of the paper, namely that p* is monadic for every locally sectionable map p : E → B. There...

We give a new version of Galois theory in categories in which normal extensions are replaced by arbitrary extensions for which the “pullback functor” is monadic, and their Galois groupoids are replaced by internal pregroupoids; we obtain the “fundamental theorem of Galois theory” using just simple remarks on internal precategories and change of uni...

A category C is additive if and only if, for every object B of C, the category Pt(C,B) of pointed objects in the comma category (C,B) is canonically equivalent to C. We reformulate the proof of this known result in order to obtain a stronger one that uses not all objects of B of C, but only a conveniently defined generating class S.I fC is a variet...

Using the ultrafilter-convergence description of topological spaces, we generalize Janelidze-Sobral characterization of local homeomorphisms between finite topological spaces, showing that local homeomorphisms are the pullback-stable discrete fibrations.

We consider a semi-abelian category V and we write Act(G, X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(−, X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all...

We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and in-troduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for th...

As observed by J. Beck, and as we know from M. Barr's and his joint work on triple cohomology, the classical isomorphism Opext » = H2 that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The same is known for many other "group-like" algebraic structu...

We develop an elementary approach to the classical descent problems for modules and algebras, and their generalizations, based on the theory of monads.

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat1-group or crossed module. An advantage of our construction is that th...

This expository paper presents a short review of categorical Galois theory, with special attention to the connection with A. R. Magid’s Galois theory of commutative rings and most recent developments in the theory of generalized central extensions. In the last section some open questions are proposed.

We describe a sufficient condition on a finitely complete and cocomplete lextensive category X, under which the categorical smash product provides a canonical (symmetric, distributive with respect to finite coproducts) monoidal structure on the category (1↓X) of its pointed objects. We also show that the ground category can be reconstructed as the...

We examine basic notions of categorical Galois theory for the adjunction between Π 0 and the inclusion as discrete, in the case of simplicial complexes. Covering morphisms are characterized as the morphisms satisfying the unique simplex lifting property, and are classified by means of the fundamental groupoid, for which we give an explicit "Galois-...

We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.

In the present paper we consider a correspondence weaker than Galois connection and prove that this produces Kurosh–Amitsur radicals in a very general setting including all universal classes of Ω-groups. As a framework we introduce a simple combinatorial structure which uses mappings between complete lattices.

Protomodular categories were introduced by the first author more than ten years ago. We show that a variety V of universal algebras is protomodular if and only if it has 0-ary terms e1,...,en, binary terms t1,...,tn, and (n+1)-ary term t satisfying the identities t(x,t1(x,y),...,tn(x,y)) = y and ti(x,x )= ei for each i =1 ,...,n.

It is shown that the descent constructions of finite preorders provide a simple motivation for those of topological spaces, and new counter-examples to open problems in Topological descent theory are constructed.

It is known that every effective (global-) descent morphism of topological spaces is an effective etale-descent morphism. On the other hand, in the predecessor of this paper we gave examples of: a descent morphism that is not an effective etale-descent morphism; an effective etale-descent morphism that is not a descent morphism. Both of the example...

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat^1-group or crossed module. An advantage of our construction is that t...

An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functor|s...

We develop a general approach to adjunctions satisfying the ad-missibility condition useful for Boolean Galois Theories, i. e. for Galois The-ories whose Galois (pre)groupoids are profinite. Various examples and appli-cations are briefly described.

We develop a new approach to Commutator theory based on the theory of internal categorical structures, especially of so called pseudogroupoids. It is motivated by our previous work on internal categories and groupoids in congruence modular varieties.

Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context, presenting work by Grothendieck in terms of separable algebras and then proceeding to the infinite-dimensional case, which requires considering topological Galois groups. In the core of the book, the authors first...

The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abelian-group and module theory. In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the...

We compare the categorical notion of central extension introduced in an earlier paper of ours, which generalized that used previously in non-abelian homological algebra by Fröhlich and Lue, with the notion of centrality arising from commutator theory in universal algebra, showing that they agree in the most important cases.

. We show that every algebraically--central extension in a Mal'tsev variety --- that is, every surjective homomorphism f : AGamma! B whose kernel--congruence is contained in the centre of A, as defined using the theory of commutators --- is also a central extension in the sense of categorical Galois theory; this was previously known only for variet...

For a given Galois structure on a category and an effective descent morphism in we describe the category of so-called weakly split objects over (E,p) in terms of internal actions of the Galois (pre)groupoid of (E,p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two...

A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, the so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the p...

The Galois theory presented here is at a level of generality essentially between that of G. Janelidze, D. Schumacher, and R. Street (1993, Appl. Categ. Structures1, 103–110) and G. Janelidze (1991, Lecture Notes in Mathematics, Vol. 1488, pp. 157–173). However, our purpose is to concentrate on symmetric monoidal categories and so provide a new appr...

A classical theory gives an equivalence between the category of covering maps of a space and the category of actions on sets of the fundamental groupoid of the space. We give a corresponding theory in dimension 2 for simplicial sets as a consequence of a Generalised Galois Theory. This yields an equivalence between a category of 2-covering maps of...

We introduce the notion of a locally semisimple covering with respect to a class X of objects in a given exact category, and classify these coverings in terms of internal-category actions inside X. This is similar to the classification of ordinary covering spaces of a “good” topological space in terms of its fundamental-group actions. Locally semis...

Using descent theory we give various forms of short five-lemma in pro-tomodular categories, known in the case of exact protomodular categories. We also describe the situation where the notion of a semidirect product can be defined categor-ically.