Information om | Engelska ordet FUNCTORS

Exempel på hur man kan använda FUNCTORS i en mening

• It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category.
• Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.
• In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories.
• Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory.
• The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful.
• Categorically, the group ring construction is left adjoint to "group of units"; the following functors are an adjoint pair:.
• Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.
• This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" G and the natural isomorphisms between FG, GF and the identity functors.
• The coequalizer of these two functors is the monoid of natural numbers under addition, considered as a one-object category.
• One may also start with a contravariant left-exact functor F; the resulting right-derived functors are then also contravariant.
• Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set).
• Roughly speaking, the pullback mechanism (using precomposition) turns several constructions in differential geometry into contravariant functors.
• The categorical concept of limit-preserving and limit-reflecting functors is in complete harmony with order theory, since orders can be considered as small categories defined as poset categories with defined additional structure.
• Here, the forgetful functor from commutative algebras to vector spaces or modules (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication).
• In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving continuous maps to the category of sets and functions.
• Lawvere's Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advanced categorical logic especially using his 1963 discovery that existential and universal quantifiers can be characterized as special cases of adjoint functors.
• Beck's monadicity theorem or the Beck tripleability theorem (1964/2003), by Jonathan Mock Beck, on monadic functors in category theory.
• It was found that the connectives of pre-categorical logic were more clearly understood using the concept of adjoint functor, and that the quantifiers were also best understood using adjoint functors.
• In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level.
• Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties.

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