We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$ . The structure in our axiomatization is a triple $(O,L,s)$ , where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon O\to O$ . In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.

On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des $\unicode[STIX]{x1D6E4}$-espaces et le scindement de Snaith.

We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.

Ringel’s right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline–Parshall–Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.

Let $E$ be a (right) Hilbert module over a $C^{\ast }$-algebra $A$. If $E$ is equipped with a left action of a second $C^{\ast }$-algebra $B$, then tensor product with $E$ gives rise to a functor from the category of Hilbert $B$-modules to the category of Hilbert $A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules, Compos. Math.FirstView (2016), 1–33, 2].

We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n + 1 functors of n variables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities so-encoded.

We present the notion of “cyclic double multicategory” as a structure in which to organise multivariable adjunctions and mates. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.

In this article we study injective representations of infinite quivers. We classify the indecomposable injective representations of trees and describe Gorenstein injective and projective representations of barren trees.

It is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the representation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable injective representations are described in terms of the injective indecomposable $R$-modules and the injective indecomposable $R[\text{x},\,{{\text{x}}^{-1}}]$ -modules.

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

We give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.

We prove that a topological space X has a locally connected regular T1, extension if and only if X is the underlying topological space of a nearness space Y which is concrete, regular and uniformly locally uniformly connected.

We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.