In algebraic topology, a presheaf of spectra on a
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 po ...
''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 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 ...
. A theorem of Jardine says that such presheaves form a
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 abstra ...
, where ''F'' →''G'' is a weak equivalence if the induced map of
homotopy sheaves 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.
References
External links
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Algebraic topology
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