Representable functor information

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

Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable. Let...

to the tensor product functor – ⊗ {\displaystyle \otimes } R M: Ab → Mod-R. Ext functor Functor category Representable functor Also commonly denoted Cop...

of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and...

− , X ) {\displaystyle {\text{Hom}}(-,X)} be the contravariant representable functor from C {\displaystyle C} to Set {\displaystyle {\textbf {Set}}}...

Representability in mathematics can refer to the existence of a representable functor in category theory Birch's theorem about the representability of...

Subcategory Faithful functor Full functor Forgetful functor Yoneda lemma Representable functor Functor category Adjoint functors Galois connection Pontryagin...

Grassmannian can be constructed as a scheme by expressing it as a representable functor. Let E {\displaystyle {\mathcal {E}}} be a quasi-coherent sheaf...

X ∈ A {\displaystyle X\in {\mathcal {A}}} to the contravariant representable functor: Y ( h ∙ ) : A → [ A o p , V ] {\displaystyle Y\;(h^{\bullet }):{\mathcal...

covariant functor F X : C → Set {\displaystyle F_{X}:C\to {\textbf {Set}}} . This functor is called representable (more generally, a representable functor is...

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 homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations...

Suppose that this inclusion morphism G → F is representable by open immersions, i.e., for any representable functor Hom(−, X) and any morphism Hom(−, X) → F...

_{tr}(X)\cong \mathbb {Z} \oplus \mathbb {Z} _{tr}(X,x)} . There is a representable functor associated to the pointed scheme G m = ( A 1 − { 0 } , 1 ) {\displaystyle...

Dolbeault-Grothendieck lemma. The construction of a scheme structure on (representable functor version of) the Picard group, the Picard scheme, is an important...

that this functor is representable by the scheme X. The affine Grassmannian is a functor from k-algebras to sets which is not itself representable, but which...

most famous basic results of category theory; it describes representable functors in functor categories. Duality: Every statement, theorem, or definition...

introduced) this is a question of whether a certain functor is representable: the contravariant functor from the homotopy category to the category of sets...

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

{X}}_{\operatorname {Spec} (k)}} is representable as an algebraic space. Another important equivalence of having a representable diagonal is the technical condition...