Definition & Betydelse | Engelska ordet ISOMORPHISMS

Definition av ISOMORPHISMS

1. böjningsform av isomorphism

Exempel på hur man kan använda ISOMORPHISMS i en mening

• The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms.
• Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space.
• The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.
• It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms).
• It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.
• The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used for injections, surjections, and bijections, as well as the cofibrations, fibrations, and weak equivalences in a model category.
• In category theory, if two objects X and Y are isomorphic, then the isomorphisms between them, Iso(X,Y), form a torsor for the automorphism group of X, Aut(X), and likewise for Aut(Y); a choice of isomorphism between the objects gives rise to an isomorphism between these groups and identifies the torsor with these two groups, giving the torsor a group structure (as it has now a base point).
• This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" G and the natural isomorphisms between FG, GF and the identity functors.
• The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
• Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties.
• In mathematics, a skeleton of a category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms.
• These isomorphisms are difficultly accounted by theories reasoning from the differences among national economies and cultural traditions, however, they are sensible outcomes if nation-states are enactments of the world cultural order.
• The forgetful functor from topological spaces to sets is not monadic as it does not reflect isomorphisms: continuous bijections between (non-compact or non-Hausdorff) topological spaces need not be homeomorphisms.
• More generally, every Lie group G is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of G on G (every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in G).
• Minkowski's question mark function can be used to determine the order isomorphisms between the quadratic algebraic numbers and the rational numbers, and between the rationals and the dyadic rationals.
• Thus a skew lattice may be viewed as a coset atlas of rectangular skew lattices placed on the vertices of a lattice and coset bijections between them, the latter seen as partial isomorphisms.
• In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
• K contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts.
• As suggested by the name, the further requirement that the category be skeletal is often assumed for the definition of "posetal"; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered class satisfies antisymmetry and hence, if a set, is a poset.
• In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (a commutative semigroup of idempotents).

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