Search results for "semi-abelian category"

showing 3 items of 3 documents

Peiffer product and peiffer commutator for internal pre-crossed modules

2017

In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator 〈X, X〉 is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varieties, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.

Large classPure mathematicssemi-abelian categoryCrossed module01 natural scienceslaw.inventionMathematics (miscellaneous)law0103 physical sciencesFOS: MathematicsSemi-abelian categoryCategory Theory (math.CT)0101 mathematicsAlgebraic numberAssociative propertyMathematicsPeiffer commutator010102 general mathematicsCoproductCommutator (electric)Mathematics - Category Theorycrossed moduleProduct (mathematics)010307 mathematical physicscrossed module; Peiffer commutator; semi-abelian category
researchProduct

A Push Forward Construction and the Comprehensive Factorization for Internal Crossed Modules

2014

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids.

Exact sequenceAlgebra and Number TheoryGeneral Computer ScienceSemi-abelian categoryAccessible categoryPushoutCrossed moduleCrossed modulecrossed module push forward comprehensive factorizationTheoretical Computer ScienceAlgebraSettore MAT/02 - AlgebraComprehensive factorizationFactorizationMathematics::Category TheoryLie algebraPush forwardAbelian groupComprehensive factorization; Crossed module; Push forward; Semi-abelian categoryCategorical variableMathematicsApplied Categorical Structures
researchProduct

Butterflies in a Semi-Abelian Context

2011

It is known that monoidal functors between internal groupoids in the category Grp of groups constitute the bicategory of fractions of the 2-category Grpd(Grp) of internal groupoids, internal functors and internal natural transformations in Grp, with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category C, and we show th…

Discrete mathematicsPure mathematicsButterflyFunctorInternal groupoidWeak equivalenceGeneral MathematicsSemi-abelian categoryFunctor categoryContext (language use)Mathematics - Category TheoryBicategory of fractionBicategoryMathematics::Algebraic TopologyWeak equivalence18D05 18B40 18E10 18A40Surjective functionMorphismMathematics::Category TheoryFOS: MathematicsCategory Theory (math.CT)Abelian groupMathematics
researchProduct