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...

particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations...

by the Dolbeault–Grothendieck lemma. The construction of a scheme structure on (representable functor version of) the Picard group, the Picard scheme...

Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation...

section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the...

type by letting A be a non-constant sheaf of abelian groups on S. For a subgroup scheme H of a group scheme G, the functor that takes an S-scheme T to...

studying schemes purely as functors on the category of algebras. To a scheme and even a stack one may associate an étale topos, an fppf topos, or a Nisnevich...

is a group functor: a functor from the category of commutative rings over k to the category of groups. Even better, it is represented by a scheme (since...

an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. C P ∞...

determined by this functor of points. The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the categorical...

associated moduli functor P Z n : Sch → Sets {\displaystyle \mathbf {P} _{\mathbb {Z} }^{n}:{\text{Sch}}\to {\text{Sets}}} sends a scheme X {\displaystyle...

)} , and X → B {\displaystyle X\to B} is a finite type map of schemes, their Hilbert functor is represented by an algebraic space. One of the motivating...

Spec(R). The scheme X is determined up to isomorphism by the functor S ↦ X(S); this is the philosophy of identifying a scheme with its functor of points...

(covariant) functor D : J → C. The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram...

context of schemes. Then a presheaf on X {\displaystyle X} is a contravariant functor from O ( X ) {\displaystyle O(X)} to the category of sets, and a sheaf...

both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating...

is called the quot functor which has a natural stratification into a disjoint union of subfunctors, each of which is represented by a projective S {\displaystyle...

this case with proper (compact) support; they are bundled up into a single functor by means of the derived category formulation of homological algebra...

mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety...

takes A to the set X(A) of A-points of X. We then say that this functor is representable by the scheme X. The affine Grassmannian is a functor from k-algebras...

a function). In some languages, particularly C++, function objects are often called functors (not related to the functional programming concept). A typical...

to set them up as a representable functor question, then apply a criterion that singles out the representable functors for schemes. When this programmatic...

Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically...

abstractly it is the Yoneda functor represented by x {\displaystyle x} in the category of schemes over X {\displaystyle X} . The functor F {\displaystyle F} is...

using functors on the category Art k {\displaystyle {\text{Art}}_{k}} of local Artin algebras over a field. A pre-deformation functor is defined as a functor...

is also called a simplicial sheaf on S m S {\displaystyle Sm_{S}} . Step 1c: fibre functors. For any smooth S {\displaystyle S} -scheme X {\displaystyle...

formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral...

if X {\displaystyle X} is a scheme in ( S c h / S ) {\displaystyle (Sch/S)} , then it determines the contravariant functor h = Hom ⁡ ( − , X ) {\displaystyle...

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent...

Affine scheme Monad (category theory) Comonad Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor...