Applied Categorical Structures

, Volume 2, Issue 4, pp 351–406 | Cite as

Homotopical algebra in homotopical categories

  • Marco Grandis
Article

Abstract

We develop here a version of abstract homotopical algebra based onhomotopy kernels andcokernels, which are particular homotopy limits and colimits. These notions are introduced in anh-category, a sort of two-dimensional context more general than a 2-category, abstracting thenearly 2-categorical properties of topological spaces, continuous maps and homotopies. A setting which applies also, at different extents, to cubical or simplicial sets, chain complexes, chain algebras, ... and in which homotopical algebra can be established as a two-dimensional enrichment of homological algebra.

Actually, a hierarchy of notions ofh-,h1-, ...h4-categories is introduced, through progressive enrichment of thevertical structure of homotopies, so that the strongest notion,h4-category, is a sort of relaxed 2-category. After investigating homotopy pullbacks and homotopical diagrammatical lemmas in these settings, we introduceright semihomotopical categories, ash-categories provided with terminal object and homotopy cokernels (mapping cones), andright homotopical categories, provided also with anh4-structure and verifying second-order regularity properties forh-cokernels.

In these frames we study the Puppe sequence of a map, its comparison with the sequence of iterated homotopy cokernels and theh-cogroup structure of the suspension endofunctor. Left (semi-) homotopical categories, based on homotopy kernels, give the fibration sequence of a map and theh-group of loops. Finally, the self-dual notion of homotopical categories is considered, together with their stability properties.

Mathematics Subject Classifications (1991)

55U35 18D05 18G55 55P 55R05 

Key words

Homotopical algebra abstract homotopy theory 2-categories homotopy homotopy (co-)limits homotopy (co-)kernels mapping cone suspension Puppe sequence (co-)fibrations 

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Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Marco Grandis
    • 1
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

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