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Homotopical algebra and triangulated categories

Published online by Cambridge University Press:  24 October 2008

Marco Grandis
Marco Grandis
Affiliation:
Dipartimento di Matematica, Universitià di Genova, Via L. B. Alberti 4, I-16132 Genova, Italy e-mail: [email protected]
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Abstract

We study here the connections between the well known Puppe-Verdier notion of triangulated category and an abstract setting for homotopical algebra, based on homotopy kernels and cokernels, which was expounded by the author in [11, 13[.

We show that a right-homotopical category A (having well-behaved homotopy cokernels, i.e. mapping cones) has a sort of weak triangulated structure with regard to the suspension endofunctor σ, called σ-homotopical category. If A is homotopical and h-stable (in a sense related to the suspension-loop adjunction), this structure is also h-stable, i.e. satisfies ‘up to homotopy’ the axioms of Verdier[29[ for a triangulated category, excepting the octahedral one which depends on some further elementary conditions on the cone endofunctor of A. Every σ-homotopical category can be stabilized, by two universal procedures, respectively initial and terminal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 259 - 285
Copyright
Copyright © Cambridge Philosophical Society 1995

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