Constant sheafIn mathematics, the constant sheaf on a topological space associated to a set is a sheaf of sets on whose stalks are all equal to . It is denoted by or . The constant presheaf with value is the presheaf that assigns to each non-empty open subset of the value , and all of whose restriction maps are the identity map . The constant sheaf associated to is the sheafification of the constant presheaf associated to . This sheaf identifies with the sheaf of locally constant -valued functions on .
Čech cohomologyIn mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Let X be a topological space, and let be an open cover of X. Let denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover consisting of sufficiently small open sets, the resulting simplicial complex should be a good combinatorial model for the space X.
Morphism of algebraic varietiesIn algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties.
Leray spectral sequenceIn mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. Let be a continuous map of topological spaces, which in particular gives a functor from sheaves of abelian groups on to sheaves of abelian groups on .
Invertible sheafIn mathematics, an invertible sheaf is a sheaf on a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Let (X, OX) be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of tensor product of OX-modules. The identity element for this operation is OX itself.
Filtered categoryIn , filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category, which will be recalled below. A is filtered when it is not empty, for every two objects and in there exists an object and two arrows and in , for every two parallel arrows in , there exists an object and an arrow such that . A filtered colimit is a colimit of a functor where is a filtered category.
Projective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.
Elliptic geometryElliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
ClockA clock or chronometer is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month, and the year. Devices operating on several physical processes have been used over the millennia. Some predecessors to the modern clock may be considered "clocks" that are based on movement in nature: A sundial shows the time by displaying the position of a shadow on a flat surface.
Clock of the Long NowThe Clock of the Long Now, also called the 10,000-year clock, is a mechanical clock under construction that is designed to keep time for 10,000 years. It is being built by the Long Now Foundation. A two-meter prototype is on display at the Science Museum in London. , two more prototypes are on display at The Long Now Museum & Store at Fort Mason Center in San Francisco. The project was conceived by Danny Hillis in 1986. The first prototype of the clock began working on December 31, 1999, just in time to display the transition to the year 2000.
Arthur VerhaegenArthur Théodore Verhaegen (31 August 1847 in Brussels – 5 September 1917 in Elsene) was a Gothic Revival Belgian architect and a politician of the Catholic Party, one of the founders of Belgian Christian democracy. He was a grandson of the politician and lawyer Pierre-Théodore Verhaegen. Verhaegen began his career as a civil engineer at Charleroi. In 1872, after his marriage to Claire Lammens, he moved to Ghent and dedicated himself to arts and crafts in the Gothic Revival style.
Injective objectIn mathematics, especially in the field of , the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of . The dual notion is that of a projective object. An in a is said to be injective if for every monomorphism and every morphism there exists a morphism extending to , i.e. such that . That is, every morphism factors through every monomorphism . The morphism in the above definition is not required to be uniquely determined by and .
