Search results for "18G50"

showing 3 items of 3 documents

The snail lemma for internal groupoids

2019

Abstract We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers.

Pure mathematicsExact sequenceLemma (mathematics)Internal groupoid Snail lemma Fibration Snake lemmaAlgebra and Number TheoryFunctorMathematics::Operator Algebras010102 general mathematicsFibrationMathematics - Category Theory01 natural sciences18B40 18D35 18G50Settore MAT/02 - AlgebraMathematics::K-Theory and HomologyMathematics::Category Theory0103 physical sciencesFOS: MathematicsCategory Theory (math.CT)Regular category010307 mathematical physics0101 mathematicsMathematics::Symplectic GeometryMathematics
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Bipullbacks of fractions and the snail lemma

2017

Abstract We establish conditions giving the existence of bipullbacks in bicategories of fractions. We apply our results to construct a π 0 - π 1 exact sequence associated with a fractor between groupoids internal to a pointed exact category.

Pure mathematicsLemma (mathematics)Exact sequenceInternal groupoidAlgebra and Number Theory010102 general mathematicsMathematics - Category TheoryBicategory of fraction18B40 18D05 18E35 18G5001 natural sciencesMathematics::Algebraic TopologySettore MAT/02 - AlgebraExact categoryMathematics::K-Theory and HomologyMathematics::Category Theory0103 physical sciencesFOS: MathematicsBipullbackSnail lemmaCategory Theory (math.CT)010307 mathematical physics0101 mathematicsMathematics
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Split extensions, semidirect product and holomorph of categorical groups

2006

Working in the context of categorical groups, we show that the semidirect product provides a biequivalence between actions and points. From this biequivalence, we deduce a two-dimensional classification of split extensions of categorical groups, as well as the universal property of the holomorph of a categorical group. We also discuss the link between the holomorph and inner autoequivalences.

Semidirect product18D05categorical groupsGroup (mathematics)split extensionssplit extension18D10Context (language use)18G5018D35AlgebraMathematics (miscellaneous)HolomorphMathematics::Category TheoryholomorphUniversal propertysemidirect productcategorical groupLink (knot theory)Categorical variableMathematics
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