We construct a flat model structure on the category ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ of additive functors from a small preadditive category $\mathcal {Q}$ satisfying certain conditions to the module category ${_{R}\mathsf {Mod}}$ over an associative ring $R$, whose homotopy category is the $\mathcal {Q}$-shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal {Q}$, and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.

We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals.

We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $\Pi ^1_1$-${\mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.

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.

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the existence of a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $ .

Pirashvili’s Dold–Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are projective. We give an unpointed analogue of this result: namely, we compute the morphisms between the tensor powers of the corresponding functor in the unpointed context. We also calculate the Ext groups between such objects, in particular showing that these functors are not projective; this is an important difference between the pointed and unpointed contexts. This work is motivated by our functorial analysis of the higher Hochschild homology of a wedge of circles.

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.

We describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. It describes ZFC principles distinguishing among the ordinals of the form $\omega _n$ . More precisely, this framework correlates each $\omega _n$ with an $(n+1)$ -dimensional generalization of Todorcevic’s walks technique, and begins to account for that technique’s “unreasonable effectiveness” on $\omega _1$ .

We show in contrast that on higher cardinals $\kappa $ , the existence of these principles is frequently independent of the ZFC axioms. Finally, we detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process.

Abstract prepared by Jeffrey Bergfalk.

E-mail: jeffrey.bergfalk@univie.ac.at

The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of $k_\omega $ -groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the $X_0$ -topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.

There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\], given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\], an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$, there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$. Examples related to inflation categories and weighted projective lines are discussed.

The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e., objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$. We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.

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.

Given a $2$-commutative diagram

in a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

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.

Given a partial action $\unicode[STIX]{x1D703}$ of a group on a set with an algebraic structure, we construct a reflector of $\unicode[STIX]{x1D703}$ in the corresponding subcategory of global actions and study the question when this reflector is a globalization. In particular, if $\unicode[STIX]{x1D703}$ is a partial action on an algebra from a variety $\mathsf{V}$, then we show that the problem reduces to the embeddability of a certain generalized amalgam of $\mathsf{V}$-algebras associated with $\unicode[STIX]{x1D703}$. As an application, we describe globalizable partial actions on semigroups, whose domains are ideals.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

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].