"equivalence of categories" meaning in English

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        {
          "text": "There is an equivalence of categories between the category of simply typed lambda calculi and the category of cartesian closed categories; this was shown by Lambek and Scott.",
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          "ref": "©2000, Karen E. Smith with Lauri Kahanpää, Pekka Kekäläinen, and William Traves, edited by S. Axler, F.W. Gehring, and K.A. Ribet, An Invitation to Algebraic Geometry (Universitext), New York: Springer, →ISBN, →OCLC, §2.5, page 24:",
          "text": "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 #x5C;mathbb#x7B;C#x7D;-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 #x5C;mathbb#x7B;C#x7D;-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 #x5C;mathbb#x7B;C#x7D;-algebras.",
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        "An adjunction whose unit and counit are both natural isomorphisms."
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        "(category theory) An adjunction whose unit and counit are both natural isomorphisms."
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          "text": "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 #x5C;mathbb#x7B;C#x7D;-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 #x5C;mathbb#x7B;C#x7D;-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 #x5C;mathbb#x7B;C#x7D;-algebras.",
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