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Sheaf Of Spectra
In algebraic topology, a presheaf of spectra on a topological space ''X'' is a contravariant functor from the category of open subsets of ''X'', where morphisms are inclusions, to the good category of commutative ring spectra. A theorem of Jardine says that such presheaves form a simplicial model category, where ''F'' →''G'' is a weak equivalence if the induced map of homotopy sheaves \pi_* F \to \pi_* G is an isomorphism. A sheaf of spectra is then a fibrant/cofibrant object in that category. The notion is used to define, for example, a derived scheme In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) ... in algebraic geometry. References External links * Algebraic topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Category Of Commutative Ring Spectra
In the mathematical field of algebraic topology, a commutative ring spectrum, roughly equivalent to a E_\infty-ring spectrum, is a commutative monoid in a goodsymmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of symmetric spectra category of spectra. The category of commutative ring spectra over the field \mathbb of rational numbers is Quillen equivalent to the category of differential graded algebras over \mathbb. Example: The Witten genus may be realized as a morphism of commutative ring spectra MString → tmf. See also: simplicial commutative ring, highly structured ring spectrum and derived scheme In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) .... Terminology Almost all reasonable categories of commutative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Model Category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results. Motivation Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of ''R''-modules for a commutative ring ''R''. Homotopy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Sheaf
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site. Example: Consider the étale site of a scheme ''S''. Each ''U'' in the site represents the presheaf \operatorname(-, U). Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). Example: Let ''G'' be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf BG. For example, one might set B\operatorname = \varinjlim B\operatorname. These types of examples appear in K-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Scheme
In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) is a scheme and (2) \pi_k \mathcal is a quasi-coherent \pi_0 \mathcal-module. The notion gives a homotopy-theoretic generalization of a scheme. A derived stack is a stacky generalization of a derived scheme. Differential graded scheme Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme. By definition, a differential graded scheme is obtained by gluing affine differential graded schemes, with respect to étale topology. It was introduced by Maxim Kontsevich "as the first approach to derived algebraic geometry." and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine. Connection with differential graded rings and examples Just as affine algebraic geometry is equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |