Search results for "Normal subobject"

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Normalities and Commutators

2010

We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and we define a new notion of normality in terms of this commutator. Our main result is to extend to any semi-abelian category the following well-known characterization of normal subgroups: a subobject K is normal in A if. and only if, {[A, K] <= K. (C) 2010 Elsevier Inc. All rights reserved.}

Normal subgroupPure mathematicsmedia_common.quotation_subjectCharacterization (mathematics)law.inventionSemi-abelianNormal subobjectlawCommutatorMathematics::Category TheorySubobjectFOS: MathematicsIdeal (order theory)Category Theory (math.CT)Algebraic numberCategorical variableNormalityMathematicsmedia_commonDiscrete mathematicsAlgebra and Number TheoryCommutator (electric)Mathematics - Category TheoryIdealSettore MAT/02 - Algebra08A30 18A20 08A50
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A NOTE ON THE CATEGORICAL NOTIONS OF NORMAL SUBOBJECT AND OF EQUIVALENCE CLASS

2021

In a non-pointed category E, a subobject which is normal to an equivalence relation is not necessarily an equivalence class. We elaborate this categorical distinction, with a special attention to the Mal'tsev context. Moreover, we introduce the notion of fibrant equipment, and we use it to establish some conditions ensuring the uniqueness of an equivalence relation to which a given subobject is normal, and to give a description of such a relation.

Settore MAT/02 - AlgebraMal'tsev and protomodular categoriesunitalnormal subobjectequivalence classconnected pair of equivalence relations
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