In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories...

(mathematics) Equivalence relation Equivalence class Equivalence of categories, in category theory Equivalent infinitesimal Identity Matrix equivalence in linear...

Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of equivalence of categories; roughly...

weak equivalence may refer to: Weak equivalence of categories Weak equivalence (homotopy theory) Weak equivalence (formal languages) Weak equivalence principle...

space X♭ over K♭. The tilting equivalence is a theorem that the tilting functor (-)♭ induces an equivalence of categories between perfectoid spaces over...

situation is called equivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous...

mathematics, when the elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally...

analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by Jacob Lurie (2009). Quasi-categories are certain...

Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a previous equivalence of categories between cubical omega-groupoids and...

specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two...

homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'...

{\operatorname {Hom} }}(X,V)} is an equivalence of categories for each ∞-category V, where ho means the homotopy category of an ∞-category, f ∗ : Hom _ ( Y , V ) ≃...

useful information. Because of this, one often studies a ring by studying the category of modules over that ring. Morita equivalence takes this viewpoint to...

equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a {\displaystyle...

although a category may have many distinct skeletons, any two skeletons are isomorphic as categories, so up to isomorphism of categories, the skeleton of a category...

restriction of the above canonical functor to an appropriate subcategory will be an equivalence of categories. In the following we will describe the role of injective...

the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations. Model categories were defined by Quillen...

categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of...

map F X , Y {\displaystyle F_{X,Y}} is a weak equivalence. Full subcategory Equivalence of categories Mac Lane (1971), p. 15 Jacobson (2009), p. 22 Mac...

general, adjunctions are not equivalences—they relate categories of different natures. The monad theory matters as part of the effort to capture what it...

Negrepontis also deduces Gelfand duality, i.e., the equivalence of categories between the opposite category of compact Hausdorff spaces and commutative C*-algebras...

transformation Equivalence of categories Subcategory Faithful functor Full functor Forgetful functor Yoneda lemma Representable functor Functor category Adjoint...

is an equivalence of categories, where Hom _ ( − , − ) {\displaystyle {\underline {\operatorname {Hom} }}(-,-)} denotes (roughly) the category of natural...

{\boldsymbol {G-Set}}:p\mapsto p^{-1}(x)} is an equivalence of categories.: 68–70  An important practical application of covering spaces occurs in charts on SO(3)...

The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form...

classification of Lie groups and the representation theory of Lie groups. For finite-dimensional representations, there is an equivalence of categories between...

defines an equivalence of categories between π1(X, p) and the fundamental groupoid of X. More precisely, this exhibits π1(X, p) as a skeleton of the fundamental...

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 )...

categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see § Categorical considerations. A topological...

equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement...