Sheaf (mathematics)

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In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a>, <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a>, <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>) attached to the <a href="/facts/Open_set/QfTCIqMu">open sets</a> of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> and defined locally with regard to them. For example, for each open set, the data could be the ring of <a href="/facts/Continuous_function/sKbl02pB">continuous</a> <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets <a href="/facts/Cover_(topology)/2VUdelA4">covering</a> the original open set (intuitively, every datum is the sum of its constituent data).
The field of mathematics that studies sheaves is called sheaf theory.
Sheaves are understood conceptually as general and abstract <a href="/facts/Mathematical_object/Juhyow9M">objects</a>. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as <a href="/facts/Sheaf_of_rings/JyeaTMtX">sheaves of rings</a>, for example, depending on the type of data assigned to the open sets.
There are also <a href="/facts/Map_(mathematics)/FABF1l2f">maps</a> (or <a href="/facts/Morphism/Kdpuyz3S">morphisms</a>) from one sheaf to another; sheaves (of a specific type, such as sheaves of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a>) with their morphisms on a fixed topological space form a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a>. On the other hand, to each <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> there is associated both a <a href="/facts/Direct_image_functor/z3mckxvp">direct image functor</a>, taking sheaves and their morphisms on the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> to sheaves and morphisms on the <a href="/facts/Codomain/MGc43nwQ">codomain</a>, and an <a href="/facts/Inverse_image_functor/wDG4ARH0">inverse image functor</a> operating in the opposite direction. These <a href="/facts/Functor/l2u3rIBE">functors</a>, and certain variants of them, are essential parts of sheaf theory.
Due to their general nature and versatility, sheaves have several applications in topology and especially in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic</a> and <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>. First, geometric structures such as that of a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> or a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as <a href="/facts/Vector_bundles/cXACAa1I">vector bundles</a> or <a href="/facts/Divisor_(algebraic_geometry)/V0QDypeo">divisors</a> are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general <a href="/facts/Sheaf_cohomology/TrcpmD6k">cohomology theory</a>, which encompasses also the "usual" topological cohomology theories such as <a href="/facts/Singular_cohomology/J7FJzTNh">singular cohomology</a>. Especially in algebraic geometry and the theory of <a href="/facts/Complex_manifold/DlvPnpzm">complex manifolds</a>, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of <a href="/facts/D-module/l1y6evYe">D-modules</a>, which provide applications to the theory of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>. In addition, generalisations of sheaves to more general settings than topological spaces, such as the notion of a sheaf on a category with respect to some <a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a>, have provided applications to <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a> and to <a href="/facts/Number_theory/zhoa7zrc">number theory</a>.

Sheaf (mathematics) open-in-new

Sheaf (mathematics)