Related papers: Varieties
The theory of support varieties gives a rich supply of examples of thick subcategories of the stable module category of a finite group algebra. We study direct sum decompositions of such categories. We give examples where there are finer…
We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can…
Algebraic theories, sometimes called equational theories, are syntactic notions given by finitary operations and equations, such as monoids, groups, and rings. There is a well-known category-theoretic treatment of them that algebraic…
We review the definition of quiver varieties and their relation to representation theory of Kac-Moody Lie algebras. Target readers are ring and representation theorists. We emphasize important roles of first extension groups of the…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
Underlying the theory of inferences, a primary task of logic is language analysis. Such a task can be understood as depending on a general theory of representation, taking as a starting point the idea that some entities (`` representations…
We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic…
An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
We define a notion which contains numerous basic notions of Analysis as special cases, for example limit, continuity, differential, Riemann and Lebesgue integral, root and exponential functions. Properties like additivity or linearity of…
We introduce regular languages of morphisms in free monoidal categories, with their associated grammars and automata. These subsume the classical theory of regular languages of words and trees, but also open up a much wider class of…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
This the first of a series of articles dealing with abstract classification theory. The apparatus to assign systems of cardinal invariants to models of a first order theory (or determine its impossibility) is developed in [Sh:a]. It is…
It is proved that every prevariety of algebras is categorically equivalent to a "prevariety of logic", i.e., to the equivalent algebraic semantics of some sentential deductive system. This allows us to show that no nontrivial equation in…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…
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…
In this work we show that the classical subject of general valuation theory and Zariski-Riemann varieties has a much wider scope than commutative algebra and desingularization theory. We construct and investigate birational projective limit…