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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 homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutativ ... 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, 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 space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Category Of Commutative Ring Spectra
In algebraic topology, a commutative ring spectrum, roughly equivalent to a E-infinity ring spectrum, 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 spectrum, symmetric spectra category of spectrum (topology), 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 →Topological modular forms, tmf. See also: simplicial commutative ring, highly structured ring spectrum and derived scheme. Terminology Almost all reasonable categories of commutative ring spectra can be shown to be Quillen equivalent to each other. Thus, from the point view of the stable homotopy theory, the term "commutative ring spectrum" may be used as a synonymous to an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Model Category
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ed ... [...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. Examples 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 appea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Scheme
In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutative ring spectra. From the functor of points point-of-view, a derived scheme is a sheaf ''X'' on the category of simplicial commutative rings which admits an open affine covering \. From the locally ringed space point-of-view, 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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
