# Category: Functors

Analytic functor
No description available.
Presheaf with transfers
In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additi
Schur functor
In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. Th
Presheaf (category theory)
In category theory, a branch of mathematics, a presheaf on a category is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion
Profunctor
In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.
Smooth functor
In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense that it
Gabriel–Popescu theorem
In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu. It characterizes certain abelian categories (the G
Span (category theory)
In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of
Stone functor
In mathematics, the Stone functor is a functor S: Topop → Bool, where Top is the category of topological spaces and Bool is the category of Boolean algebras and Boolean homomorphisms. It assigns to ea
Forgetful functor
In mathematics, in the area of category theory, 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 ou
Free functor
No description available.
Monoidal category action
In algebra, an action of a monoidal category S on a category X is a functor such that there are natural isomorphisms and and those natural isomorphism satisfy the coherence conditions analogous to tho
Pseudo-functor
In mathematics, a pseudofunctor F is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that and do not hold as exact equalities but only up to cohe
Calculus of functors
In algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it general
End (category theory)
In category theory, an end of a functor is a universal extranatural transformation from an object e of X to S. More explicitly, this is a pair , where e is an object of X and is an extranatural transf
Subfunctor
In category theory, a branch of mathematics, a subfunctor is a special type of functor that is an analogue of a subset.
Ind-completion
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category C. The objects in this ind-completed category, denoted Ind(C), are known as
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphism
Bivariant theory
In mathematics, a bivariant theory was introduced by Fulton and MacPherson, in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring
Zuckerman functor
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernst
In mathematics, a preradical is a subfunctor of the identity functor in the category of left modules over a ring with identity. The class of all preradicals over R-mod is denoted by R-pr. There is a n
Diagram (category theory)
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that
Full and faithful functors
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functo
Final functor
In category theory, the notion of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category. A functor is called final if, for any se
Dinatural transformation
In category theory, a branch of mathematics, a dinatural transformation between two functors written is a function that to every object of associates an arrow of and satisfies the following coherence
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathe
Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) a
Simplicial presheaf
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant f
Amnestic functor
In the mathematical field of category theory, an amnestic functor F : A → B is a functor for which an A-isomorphism ƒ is an identity whenever Fƒ is an identity. An example of a functor which is not am
Functor category
In category theory, a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another ob
Hom functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and h
Effaceable functor
In mathematics, an effaceable functor is an additive functor F between abelian categories C and D for which, for each object A in C, there exists a monomorphism , for some M, such that . Similarly, a
Essentially surjective functor
In mathematics, specifically in category theory, a functor is essentially surjective (or dense) if each object of is isomorphic to an object of the form for some object of . Any functor that is part o
Dominant functor
In category theory, an abstract branch of mathematics, a dominant functor is a functor F : C → D in which every object of D is a retract of an object of the form F(x) for some object X of C.
2-functor
In mathematics, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-fu
Polynomial functor
In algebra, a polynomial functor is an endofunctor on the category of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers and the exterior po
Simplicial set
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories.
Exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be direc