Walter Tholen

Semi-abelian categories

Abstract 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 sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to “old” exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semi-abelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar non-abelian structures.

Facets of Descent, II

Methods of internal-category theory are applied to show that the split epimorphisms in a category C are exactly the morphisms which are effective for descent with respect to any fibration over C (or to any C-indexed category). In the same context, composition-cancellation rules for effective descent morphisms are established and being applied to (suitably defined) locally-split epimorphisms.

Metric, topology and multicategory—a common approach

Abstract For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T, V ) -algebra and show that various old and new structures are instances of such algebras. Lawveres presentation of a metric space as a V -category is included in our setting, via the Betti–Carboni–Street–Walters interpretation of a V -category as a monad in the bicategory of V -matrices, and so are Barrs presentation of topological spaces as lax algebras, Lowens approach spaces, and Lambeks multicategories, which enjoy renewed interest in the study of n -categories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.

One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawveres presentation of metric spaces and Barrs presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.

Exponentiable morphisms, partial products and pullback complements

In a category K with finite limits, the exponentiability of a morphisms s is (rather easily) characterised in terms of K admitting partial products (essentially those of Pasynkov) over s; and that of a monomorphism is characterised in terms of the new concept of a pullback complement (a universal construction of a pullback diagram whose top and right sides are given). Then, characterisations, previously given by the first author for the category Sp of topological spaces, of the notions of totally reflective subcategory and of hereditary factorisation system are shown to be instances of simple results on adjointness and factorisations.

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.

Monoidal Topology: A Categorical Approach to Order, Metric, and Topology

Preface 1. Introduction Robert Lowen and Walter Tholen 2. Monoidal structures Gavin J. Seal and Walter Tholen 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen Bibliography Tables Index.

Separation versus connectedness

Abstract For a closure operator c in the sense of Dikranjan and Giuli, the subcategory Δ(c) (▽(c)) of objects X with c-closed (c-dense) diagonal δX: X → X × X is known to give a general notion of separation (connectedness, respectively), with the expected closure properties under products and subspaces (images), etc. The purpose of this note is to fully characterize the notions of connectedness and disconnectedness in the sense of Arhangelskiǐ and Wiegandt and of separation by Pumplun and Rohrl in this context. Briefly, an AW-connectedness is a subcategory of type ▽(c) with c a regular closure operator, and an AW-disconnectedness is of type Δ(c) with c a coregular closure operator, as introduced in this paper. The latter subcategory is in particular PR-separated, i.e., a subcategory of type Δ(c) with c weakly hereditary. Categorical proofs and new applications are provided for the characterization theorems originally given by Arhangelskiǐ and Wiegandt in the context of topological spaces.

Effective descent maps of topological spaces

Abstract The basic technique in A. Joyals and M. Tierneys work on “An extension of the Galois theory of Grothendieck” is descent theory for morphisms of locales (in a topos). They showed that open surjections are effective descent morphisms in the category of locales. I. Moerdijk gave an axiomatic proof of this result which shows that the same result holds true also in the category Top of topological spaces. G. Janelidze and W. Tholen proved that every locally sectionable map in Top is an effective descent morphism, and that effective descent morphisms are universal quotient maps in Top. In this paper, we give • • bu a complete characterization of effective descent maps in bdTop, • bu an example of a universal quotient map in bdTop which is not an effective descent morphism. This is done by first transfering the problem into a friendlier environment than Top, namely into the topological quasitopos hull of Top, the category of pseudotopological spaces. Here effective descent morphisms are simply quotient maps. Although the notion of effective descent morphism depends on the category, it is possible to reinterpret the pseudotopological characterization in purely topological terms, under extensive use of filter theory.

Functorial factorization, well-pointedness and separability

Abstract 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 paper presents two important factorization structures in the context of pointed endofunctors: concordant–dissonant and inseparable–separable.