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Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 22 (6 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Crossed complexes, and free crossed resolutions for amalgamated sums and HNNextensions of groups
 Georgian Math. J
, 1999
"... Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the p ..."
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Cited by 7 (6 self)
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Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNNextensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1
THE PORDER OF TOPOLOGICAL TRIANGULATED CATEGORIES
"... p annihilates objects of the form Y/p. In this paper we show that the porder of a topological ..."
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Cited by 1 (1 self)
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p annihilates objects of the form Y/p. In this paper we show that the porder of a topological
Crossed complexes, free crossed resolutions and graph products of groups’, (submitted
"... The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show ..."
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Cited by 1 (1 self)
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The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions of graph products of groups, and so obtain computations of higher homotopical syzygies in this case. 1
TOPOLOGICAL TRIANGULATED CATEGORIES
"... Many triangulated categories arise from chain complexes in an additive or abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying additive categories. Stable homotopy theory produces examples of triangu ..."
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Many triangulated categories arise from chain complexes in an additive or abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying additive categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘nonadditive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, and formalize this in Definition 1.4 via homotopy categories of stable cofibration categories. The purpose of this paper is to explain some systematic differences between algebraic and topological triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. The precise statements use the norder of a triangulated category, for n a natural number (see Definition 2.1). The norder is a nonnegative integer (or infinity), and it measures, roughly speaking, ‘how strongly ’ the relation n · Y/n = 0 holds for the objects Y in a given triangulated category (where Y/n denotes a cone of multiplication by n on Y). Our main results are: • The norder of every algebraic triangulated category is infinite (Theorem 3.3). • For every prime p, the porder of every topological triangulated category is at least p − 1 (Theorem 8.2). • For every prime p, the porder of the plocal stable homotopy category is exactly p − 1
TORSION INVARIANTS FOR TRIANGULATED CATEGORIES
"... The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory ..."
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The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘nonadditive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the norder of an object in a triangulated category, for n a natural number (see Definition 1.1). The norder is a nonnegative integer (or infinity), and an object Y has positive norder if and only if n · Y = 0; the norder can be thought of
www.elsevier.com/locate/top Secondary derived functors and the Adams spectral sequence
, 2005
"... Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secon ..."
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Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E3term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E3term explicitly by an algorithm.
Generalized Homotopy Theory
"... Usually, in homotopy theory with cofibrations, cofibrant objects and a suspension functor are necessary to obtain homotopy groups and exact sequences of them. In this sense H.J. Baues defines in his book “Algebraic Homotopy” [1] the concept of “Category with a Natural Cylinder”. His definition requi ..."
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Usually, in homotopy theory with cofibrations, cofibrant objects and a suspension functor are necessary to obtain homotopy groups and exact sequences of them. In this sense H.J. Baues defines in his book “Algebraic Homotopy” [1] the concept of “Category with a Natural Cylinder”. His definition requires