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.
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References
D. W. Anderson: Axiomatic homotopy theory, in:Algebraic Topology, Waterloo, 1978, Lect. Notes in Math.741, Springer-Verlag, 1979, pp. 520–547.
H. J. Baues:Algebraic Homotopy, Cambridge Univ. Press, 1989.
J. Bénabou: Some remarks on 2-categorical algebra (Part I),Bull. Soc. Math. Belgique 41 (1989), 127–194.
K. S. Brown: Abstract homotopy theory and generalised sheaf cohomology,Trans. Amer. Math. Soc. 186 (1973), 419–458.
A. Dold and D. Puppe: Homologie nicht-additiver Funktoren, Anwerdungen,Ann. Inst. Fourier Grenoble 11 (1961), 201–312.
M. Grandis: On the categorical foundations of homological and homotopical algebra,Cahiers Top. Géom. Diff. Catég. 33 (1992), 135–175.
M. Grandis:Homological Algebra in Non Abelian Settings, in preparation.
P. Gabriel and M. Zisman:Calculus of Fractions and Homotopy Theory, Springer-Verlag, 1967.
A. Heller: Stable homotopy categories,Bull. Amer. Math. Soc. 74 (1968), 28–63.
A. Heller:Homotopy Theories, Mem. Amer. Math. Soc.383 (1988).
R. Hartshorne:Residues and Duality, Lect. Notes in Math.20, Springer-Verlag, 1966.
P. J. Huber: Homotopy theories in general categories,Math. Ann. 144 (1961), 361–385.
P. J. Huber: Standard constructions in abelian categories,Math. Ann. 146 (1962), 321–325.
D. M. Kan: Abstract homotopy I–IV,Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092–1096;42 (1956), 255–258; 419–421; 542–544.
D. M. Kan: A combinatorial definition of homotopy groups,Ann. of Math. 61 (1958), 288–312.
K. H. Kamps: Faserungen und Cofaserungen in Kategorien mit Homotopiesystem, Dissertation, Saarbrücken 1968, 1–142.
K. H. Kamps: Über einige formale Eigenschaften von Faserungen undh-Faserungen,Manuscripta Math. 3 (1970), 237–255.
K. H. Kamps: Zur Homotopietheorie von Gruppoiden,Arch. Math. (Basel) 23 (1972), 610–618.
H. Kleisli: Homotopy theory in abelian categories,Canad. J. Math. 14 (1962), 139–169.
S. Mac Lane:Categories for the Working Mathematician, Springer-Verlag, 1971.
H. J. Marcum: Homotopy equivalences in 2-categories, in:Groups of Self-Equivalences and Related Topics, Montréal 1988, Lect. Notes in Math.1425, Springer-Verlag, 1990, pp. 71–86.
M. Mather: Pull-backs in homotopy theory,Can. J. Math. 28 (1976), 225–263.
T. Müller: Zur Theorie der Würfelsätze, Dissertation, Hagen, 1982.
D. Puppe: Homotopiemengen und ihre induzierten Abbildungen, I,Math. Z. 69 (1958), 299–344.
D. Puppe: On the formal structure of stable homotopy theory, in:Colloquium on Algebraic Topology, Math. Inst., Aarhus Univ., 1962, 65–71.
D. G. Quillen:Homotopical Algebra, Lect. Notes in Math.43, Springer-Verlag, 1967.
S. Rodriguez: Homotopía en categorías aditivas,Rev. Acad. Cienc. Zaragoza 43 (1988), 77–92.
R. Street: Categorical structures, in:Handbook of Algebra, Vol. 2, Elsevier Science Publishers, North Holland, to appear.
J. L. Verdier:Catégories Dérivées, Séminaire de Géometrie algébrique du Bois Marie SGA 4 1/2, Cohomologie étale, Lect. Notes in Math.569, Springer-Verlag, 1977, 262–311.
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Grandis, M. Homotopical algebra in homotopical categories. Appl Categor Struct 2, 351–406 (1994). https://doi.org/10.1007/BF00873039
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DOI: https://doi.org/10.1007/BF00873039