Titlebook: Local Homotopy Theory; John F. Jardine Book 2015 Springer-Verlag New York 2015 algebraic K-theory.higher category theory.homotopical algeb

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Introduction,plicial presheaves and simplicial sheaves, and its chain complex and stable homotopy theoretic variants. An overview of the logical flow of the theory is presented, in relation to its origins and applications in Algebraic Geometry, Algebraic K-theory, and Algebraic Topology.

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拘留

Some Topos Theoryculus of coverings which generalizes the algebra of open covers of a topological space, and can exist in much more generality..The examples of most interest for us come from Algebraic Geometry, and include the Zariski, flat, étale and Nisnevich topologies..These are discussed here, along with the ba

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Local Weak Equivalencescial presheaves which induces isomorphisms in all possible sheaves of homotopy groups. A local fibration is a map which has a suitably defined local right lifting property with respect to all inclusions of horns in simplices. It is fundamental result that a map is both a local weak equivalence and a

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Local Model Structures injective model structures for both of these categories, and the Quillen equivalence between them..The projective local model structure for simplicial presheaves, and all model structures intermediate between it and the injective model structure, are constructed. A useful "solution set condition" m

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Cocyclesn topological and algebraic settings, but the theory is much more general..The primary theorem, that one can recover morphisms in the homotopy category from path components in cocycle categories, applies to a large selection of model structures. This result is one of the most useful formal ideas in

Ancestor

Localization Theoriesion of the localization of the injective model structure on a simplicial presheaf category, which is a model structure in which the members of a set of cofibrations are formally inverted. Examples include the motivic model structure for the category of simplicial presheaves on the Nisnevich site of

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SOB

Non-abelian Cohomologyids and presheaves of groupoids enriched in simplicial sets. The corresponding injective fibrant objects are stacks, .-stacks and higher stacks, respectively, and the fibrant model construction is a candidate for stack replacement in each case..Torsors, in all cases, can be defined by the local acyc

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