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LOCALLY CONSTANT FUNCTORS
, 803
"... To Michael Batanin, for his nice questions Abstract. We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interp ..."
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To Michael Batanin, for his nice questions Abstract. We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. At last, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors. Contents 1. Homology with locally constant coefficients 1 2. Model structures for locally constant functors 6 3. Locally constant coefficients in Grothendieck derivators 8
THE FUNDAMENTAL ISOMORPHISM CONJECTURE VIA NONCOMMUTATIVE MOTIVES
"... Abstract. Given a group, we construct a fundamental additive functor on ..."
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Abstract. Given a group, we construct a fundamental additive functor on
DOI: 10.1007/s0022200700612 Triangulated categories without models
, 2007
"... Abstract. We exhibit examples of triangulated categories which are neither ..."
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Abstract. We exhibit examples of triangulated categories which are neither
F. Hörmann Homotopy Limits and Colimits in Nature A Motivation for Derivators
, 2014
"... preliminary version 21.10.14 An introduction to the notions of homotopy limit and colimit is given. It is explained how they can be used to neatly describe the “old ” distinguished triangles and shift functors of derived categories resp. cofiber and fiber sequences in algebraic topology. One of the ..."
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preliminary version 21.10.14 An introduction to the notions of homotopy limit and colimit is given. It is explained how they can be used to neatly describe the “old ” distinguished triangles and shift functors of derived categories resp. cofiber and fiber sequences in algebraic topology. One of the goals is to motivate the language of derivators from the perspective of classical homological algebra. Another one is to give elementary proofs (one bruteforce in the exercises, and one a bit more abstract) that in the category of unbounded chain complexes of an (AB4, resp. AB4*) abelian category all homotopy limits (resp. colimits) exist and that this situation leads to a (stable) derivator. The heart of these proofs is an explicit formula for homotopy limits and colimits, the BousfieldKan formula. Later it is explained how these results fit in the framework of model categories. We sketch proofs that any model category gives rise to a derivator. We also rediscuss BousfieldKan’s formula and outline the proof that it is valid in any simplicial model category (even a slightly weaker structure). In the end the homotopy theory of (homotopy) limits and colimits is discussed. In particular we explain that any derivator is a module over H (the derivator associated with the homotopy theory of spaces). The reader is assumed to have seen some algebraic topology and/or homological algebra (here in
unknown title
, 2009
"... We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of ..."
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We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors. 1. Homology with locally constant coefficients 1·1. Given a model category1V with small colimits, and a small category A, we will write [A,V] for the category of functors from A to V. Weak equivalences in [A,V] are the termwise weak equivalences. We denote by Ho([A,V]) the localization of [A,V] by the class of weak equivalences. 1·2. We denote by LC(A,V) the full subcategory of the category Ho([A,V]) whose objects are the locally constant functors, i.e. the functors F: A V
Fibered Multiderivators and (co)homological descent
, 2015
"... homological descent, fundamental localizers, wellgenerated triangulated categories The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi)derivator is developed, which similarly enhances fibrations of (monoidal) triangulat ..."
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homological descent, fundamental localizers, wellgenerated triangulated categories The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The main motivation is a descent theory for Grothendieck’s six operations.