equivalence of categories

English

Noun

equivalence of categories (plural equivalence of categoriess)

(category theory) An adjunction whose unit and counit are both natural isomorphisms.
- ©2000, Karen E. Smith; Lauri Kahanpää; Pekka Kekäläinen; William Traves, S. Axler; F.W. Gehring; K.A. Ribet, editors, An Invitation to Algebraic Geometry (Universitext), New York: Springer, →ISBN, OCLC 643228412, §2.5, page 24:
  The defining feature of algebraic geometry is the remarkable fact that not only does the geometry determine the algebra, but conversely, the algebra determines the geometry. That is, given any finitely generated ${\\displaystyle \\mathbb {C} }$ -algebra R without nilpotent elements, there exists an affine algebraic variety V, uniquely defined up to isomorphism, such that R is isomorphic to the coordinate ring of V. Moreover, any homomorphism between such ${\\displaystyle \\mathbb {C} }$ -algebras uniquely defines a morphism of the corresponding varieties. In fancy language, there is an equivalence of categories between the category of affine algebraic varieties and finitely generated, reduced ${\\displaystyle \\mathbb {C} }$ -algebras.

Hypernyms

adjunction

单词	equivalence of categories
释义	equivalence of categories English Noun equivalence of categories (plural equivalence of categoriess) (category theory) An adjunction whose unit and counit are both natural isomorphisms. ©2000, Karen E. Smith; Lauri Kahanpää; Pekka Kekäläinen; William Traves, S. Axler; F.W. Gehring; K.A. Ribet, editors, An Invitation to Algebraic Geometry (Universitext), New York: Springer, →ISBN, OCLC 643228412, §2.5, page 24: The defining feature of algebraic geometry is the remarkable fact that not only does the geometry determine the algebra, but conversely, the algebra determines the geometry. That is, given any finitely generated ${\\displaystyle \\mathbb {C} }$ -algebra R without nilpotent elements, there exists an affine algebraic variety V, uniquely defined up to isomorphism, such that R is isomorphic to the coordinate ring of V. Moreover, any homomorphism between such ${\\displaystyle \\mathbb {C} }$ -algebras uniquely defines a morphism of the corresponding varieties. In fancy language, there is an *equivalence of categories* between the category of affine algebraic varieties and finitely generated, reduced ${\\displaystyle \\mathbb {C} }$ -algebras. Hypernyms adjunction
随便看	ᩉ᩠ᩃᩋᩯ᩠ᨶ ᩉ᩠ᩃᩡᩋᩯ᩠ᨶ ᩉ᩠ᩅᩫ ᩉᩢ᩠ᨶ ᩉᩥᨲ ᩉᩩᨾ ᩉᩫ᩠ᩅ ᩉᩮᩣᨲᩥ ᩉᩮᩣᨶ᩠ᨲ ᩉᩯ᩵ᩉᩯ᩠ᨶ ᩊ ᩋ ᩋᨠᩁᨱ ᩋᨠᩣᩃᩥᨠ ᩋᨠᩥᨬ᩠ᨧᨶ ᩋᨬ᩠ᨬ ᩋᨬ᩠ᨬᨲᩁ ᩋᨭᩛ ᩋᨭᩛᨦ᩠ᨣᩥᨠ ᩋᨭᩛᩥ ᩋᨲ᩠ᨲᨶ᩺ ᩋᨲ᩠ᨳᩥ ᩋᨲᩥᨠ᩠ᨠᨾᨲᩥ ᩋᨲᩦᨲ ᩋᨳ