In algebraic topology, a presheaf of spectra on a
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 to ...
''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 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 ch ...
. A theorem of Jardine says that such presheaves form a
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. ...
, 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, 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
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Algebraic topology
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