For predual categories C andD we establish isomorphisms between opfibrations representing local varieties of languages in C, local pseudovarieties of D-monoids, and finitely generated profinite D-monoids. The global sections of these opfibrations are shown to correspond to varieties of languages in C, pseudovarieties of D-monoids, and profinite equational theories of D-monoids, respectively. As...

Before embarking on the proof, we first explain Theorem 1 in a bit more detail, in particular identifying the two categories that are relevant to our discussion. So let Gk denote the category whose objects are profinite group extensions of Gk by a pro-p group, and whose morphisms are outer open Gkhomomorphisms. Thus, an object of Gk is a profinite group G together with a continuous surjection π...

A group Γ is called boundedly generated (BG) if it is the settheoretic product of finitely many cyclic subgroups. We show that a BG group has only abelian by finite images in positive characteristic representations. We use this to reprove and generalise Rapinchuk’s theorem by showing that a BG group with the FAb property has only finitely many irreducible representations in any given dimension ...

We show that a large number of equations are preserved by DedekindMacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that a large class of varieties of Heyting algebras and FL-algebras admits completions.

Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups.

We extend Haran’s Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 in [FrJ].

The main goal of this paper is to prove that the idempotent completions of the triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason theorem on dense subcategories and a relation to the negative K-theory.