Théorie des catégoriesLa théorie des catégories est l'étude des structures mathématiques et de leurs relations. Ce domaine est né du constat de l'abondance de caractéristiques partagées par diverses classes liées à des structures mathématiques. Les catégories sont utilisées dans la plupart des branches mathématiques et dans certains secteurs de l'informatique théorique et en mathématiques de la physique. Elles forment une notion unificatrice.
Forgetful functorIn mathematics, in the area of , a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure.
Proof (truth)A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
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 .
Functor represented by a schemeIn algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural bijections) the set of all morphisms . The scheme X is then said to represent the functor and that classify geometric objects over S given by F. The best known example is the Hilbert scheme of a scheme X (over some fixed base scheme), which, when it exists, represents a functor sending a scheme S to a flat family of closed subschemes of .
Strong dual spaceIn functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise.
Verdier dualityIn mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.
Injective hullIn mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . A module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective. Here, the base ring is a ring with unity, though possibly non-commutative. An injective module is its own injective hull. The injective hull of an integral domain is its field of fractions .
Preadditive categoryIn mathematics, specifically in , a preadditive category is another name for an Ab-category, i.e., a that is over the , Ab. That is, an Ab-category C is a such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: and where + is the group operation. Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below).
Catégorie des ensemblesEn mathématiques, plus précisément en théorie des catégories, la catégorie des ensembles, notée Set ou Ens, est la catégorie dont les objets sont les ensembles, et dont les morphismes sont les applications d'un ensemble dans un autre. Sa définition est motivée par le fait qu'en théorie des ensembles usuelle, il n'existe pas d'« ensemble de tous les ensembles », car l'existence d'un tel objet résulterait en une contradiction logique : le paradoxe de Russell.
HypergrapheLes hypergraphes sont des objets mathématiques généralisant la notion de graphe. Ils ont été nommés ainsi par Claude Berge dans les années 1960. Les hypergraphes généralisent la notion de graphe non orienté dans le sens où les arêtes ne relient plus un ou deux sommets, mais un nombre quelconque de sommets (compris entre un et le nombre de sommets de l’hypergraphe). Certains théorèmes de la théorie des graphes se généralisent naturellement aux hypergraphes, par exemple le théorème de Ramsey.
