Search results for "Snake lemma"

showing 2 items of 2 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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On Fibrations Between Internal Groupoids and Their Normalizations

2018

We characterize fibrations and $$*$$ -fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. As an application, we deduce the internal version of the Brown exact sequence for $$*$$ -fibrations from the internal version of the Gabriel–Zisman exact sequence. We also analyse fibrations and $$*$$ -fibrations in the category of arrows and study when the normalization functor preserves and reflects them. This analysis allows us to give a characterization of protomodular categories using strong homotopy kernels and a generalization of the Snake Lemma.

Normalization (statistics)Pure mathematicsInternal groupoid Fibration Strong h-pullback Protomodular categoryGeneral Computer ScienceFibrationSnake lemmaStrong h-pullbackMathematics::Algebraic Topology01 natural sciencesTheoretical Computer ScienceMathematics::Algebraic GeometryMathematics::K-Theory and HomologyMathematics::Category Theory0103 physical sciences0101 mathematicsMathematics::Symplectic GeometryMathematicsExact sequenceInternal groupoidAlgebra and Number TheoryFunctorHomotopy010102 general mathematicsFibrationInternal versionSettore MAT/02 - AlgebraProtomodular categoryTheory of computation010307 mathematical physicsApplied Categorical Structures
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