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In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.
If a category is equivalent to the opposite (or dual) of another category then one speaks of
a duality of categories, and says that the two categories are dually equivalent.
An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be naturally isomorphic to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.
and 24 Related for: Equivalence of categories information
In category theory, a branch of abstract mathematics, an equivalenceofcategories is a relation between two categories that establishes that these categories...
morphism f: c → d in C, is again isomorphic to C. Equivalenceofcategories Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts...
(mathematics) Equivalence relation Equivalence class Equivalenceofcategories, in category theory Equivalent infinitesimal Identity Matrix equivalence in linear...
mathematics, when the elements of some set S {\displaystyle S} have a notion ofequivalence (formalized as an equivalence relation), then one may naturally...
situation is called equivalenceofcategories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous...
specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form ofequivalence between two...
useful information. Because of this, one often studies a ring by studying the categoryof modules over that ring. Morita equivalence takes this viewpoint to...
restriction of the above canonical functor to an appropriate subcategory will be an equivalenceofcategories. In the following we will describe the role of injective...
space X♭ over K♭. The tilting equivalence is a theorem that the tilting functor (-)♭ induces an equivalenceofcategories between perfectoid spaces over...
equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a {\displaystyle...
categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of...
a system of homotopy categories given by the diagram categories I → M {\displaystyle I\to M} for a category with a class of weak equivalences ( M , W )...
also refer to a category with a monoidal-category action. The categoriesof left and right modules are abelian categories. These categories have enough projectives...
homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'...
map F X , Y {\displaystyle F_{X,Y}} is a weak equivalence. Full subcategory Equivalenceofcategories Mac Lane (1971), p. 15 Jacobson (2009), p. 22 Mac...
representations of quantum groups and Hecke algebras, and on the geometric Langlands program (Satake equivalenceofcategories). He is currently a Professor of Mathematics...
False equivalence A false equivalence or false equivalency is an informal fallacy in which an equivalence is drawn between two subjects based on flawed...
equivalent if they have the same truth value in every model. The logical equivalenceof p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as...
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise...
The equivalence principle is the hypothesis that the observed equivalenceof gravitational and inertial mass is a consequence of nature. The weak form...
product of the slice categories C/X × C/Y to the slice category C/(X + Y) is an equivalenceofcategories for all objects X and Y of C. The categories Set...
stable module category to itself. For certain rings, such as Frobenius algebras, Ω{\displaystyle \Omega } is an equivalenceofcategories. In this case...
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