Comparison of Abelian categories recollements
We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
math.CT↗
arXiv subjects
Teimuraz Pirashvili
Publications and source records attributed to Teimuraz Pirashvili.
Shukla cohomology and additive track theories
The approach of Quillen to the cohomology of general algebraic systems is based on simplicial methods, which are usually quite hard to deal with. The aim of this work is to give the foundation of Shukla cohomology (of algebras over a commutative ring) based on chain algebras. We also give an application to the problem of strengthening additive track theories, which is based on the comparison homomorphism between Shukla and Mac Lane cohomology theories.
math.KT↗
Algebra cohomology over a commutative algebra revisited
The aim of this paper is to give a relatively easy bicomplex which computes the Shukla, or Quillen cohomology in the category of associative algebras over a commutative algebra $A$, in the case when $A$ is an algebra over a field.
math.KT↗
Strengthening track theories
Using cohomology of categories with coefficients in natural systems it is proved that a groupoid enrichad category with pseudoproducts is pseudoequivalent to one with strict products.
math.CT↗
Sets with two associative operation
In this paper we consider dimonoids, which are sets equipped with two associative binary operations. Dimonoids in the sense of J.-L. Loday are xamples of duplexes. The set of all permutations, gives an example of a duplex which is not a dimonoid. We construct a free duplex generated by a given set via planar trees and then we prove that the set of all permutations form a free duplex on an explicitly described set of generators. We also consider duplexes coming from planar binary trees and vertices of the cubes. We prove that these duplexes are free with one generator in appropriate variety of duplexes.
math.CO↗
Linear extensions and nilpotence for Maltsev theories
Relationship is clarified between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear extensions by bifunctors from the so called abelian theories. The latter theories are described; they are slightly more general than theories of modules over a ring.
math.CT↗
On the PROP corresponding to bialgebras
A PROP is a symmetric monoidal category, whose set of objects is the set of natural numbers and on objects the monoidal structure is given by the addition. An algebra over a PROP is a symmetric strict monoidal functor to the tensor category of vector spaces. We give an explicite construction of the PROP whose category of algebras is equivalent to the category of bialgebras (= associative and coassociative bialgebras).
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André-Quillen homology via functor homology
We obtain André-Quillen homology for commutative algebras using relative homological algebra in the category of functors on finite pointed sets
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On Kan fibrations for Maltsev algebras
We prove that any surjective homomorphism of simplicial Maltsev algebras is a Kan fibration.
math.AT↗
Vanishing line for the descent spectral sequence
For a surjective and proper map f: X -> Y there is a spectral sequence, called descent spectral sequence, abutting to the cohomology of Y with coefficients in a sheaf F. We prove that if the fibers of the map f satisfy some connectivity condition then the spectral sequence has a vanishing line.
math.AG↗