In fact, when A is a commutative unital C *- algebra, the Gelfand representation is then an isometric *- isomorphism between A and C ( Δ ( A )).

Sometimes two quite different constructions yield " the same " result ; this is expressed by a natural isomorphism between the two functors.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.

The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.

A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts:

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

If there exists an isomorphism between two groups, then the groups are called isomorphic.

This has the important consequence that two objects are completely indistinguishable as far as the structure in question is concerned, if there is an isomorphism between them.

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function.

The reason for this is that the existence of a path between two points allows one to identify loops at one with loops at the other ; however, unless is abelian this isomorphism is non-unique.

Furthermore, the classification of covering spaces makes strict reference to particular subgroups of, specifically distinguishing between isomorphic but conjugate subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.

Two mathematical structures are said to be isomorphic if there is an isomorphism between them.

In essence, two objects are isomorphic if they are indistinguishable given only a selection of their features, and the isomorphism is the mapping of the set elements and the selected operations between the objects.

There is an isomorphism from X to Y if the bijective function happens to produce results, that sets up a correspondence between the operator and the operator.

* Group isomorphism, an isomorphism between groups

* Ring isomorphism, an isomorphism between rings.

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

In graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the " edge structure " in the sense that there is an edge from vertex u to vertex v in G if and only if there is an edge from ƒ ( u ) to ƒ ( v ) in H. See graph isomorphism.

It provides a way of viewing institutions outside of the traditional views of economics by explaining why so many businesses end up having the same organizational structure ( isomorphism ) even though they evolved in different ways, and how institutions shape the behavior of individual members.

As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization G < sup > ab </ sup > up to canonical isomorphism, whereas the explicit construction G → G / shows existence.

An isomorphism class is a collection of mathematical objects isomorphic to each other.

Thus, for example, two objects may be group isomorphic without being ring isomorphic, since the latter isomorphism selects the additional structure of the multiplicative operator.

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property that is preserved by an isomorphism and that is true of one of the objects, is also true of the other.

Equality is when two objects are exactly the same, and everything that's true about one object is true about the other, while an isomorphism implies everything that's true about a designated part of one object's structure is true about the other's.

* Universal properties define objects uniquely up to isomorphism.

If this object is a class of transformations ( such as " isomorphism " or " permutation "), it implies the equivalence of objects one of which is the image of the other under such a transformation.

Starting from finite-dimensional vector spaces ( as objects ) and the dual functor, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from " all linear maps " to " linear maps that respect this structure ".

Explicitly, for each vector space, require that it come with the data of an isomorphism to its dual, In other words, take as objects vector spaces with a nondegenerate bilinear form This defines an infranatural isomorphism ( isomorphism for each object ).

One then restricts the maps to only those maps that commute with these isomorphism ( restricts to the naturalizer of η ), in other words, restrict to the maps that do not change the bilinear form: The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual ( each space has an isomorphism to its dual, and the maps in the category are required to commute ).

Conversely, a particular map between particular objects may be called an unnatural isomorphism ( or " this isomorphism is not natural ") if the map cannot be extended to a natural transformation on the entire category.

Thus, one way to show two objects of C are distinct ( up to isomorphism ) is to show that their images under f are distinct ( i. e. not isomorphic ).

Define C < sub > 1 </ sub > as the full subcategory of C consisting of those objects X of C for which ε < sub > X </ sub > is an isomorphism, and define D < sub > 1 </ sub > as the full subcategory of D consisting of those objects Y of D for which η < sub > Y </ sub > is an isomorphism.

Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative " up to " an isomorphism.

However, related categories ( with additional structure and restrictions on the maps ) do have a natural isomorphism, as described below.

For any ( full ) embedding F: B → C the image of F is a ( full ) subcategory S of C and F induces a isomorphism of categories between B and S.

More categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see categorical considerations.

Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry – Howard isomorphism and they can be considered as the internal language of classes of categories, e. g. the simply typed lambda calculus is the language of Cartesian closed categories ( CCCs ).

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.

A functor F: C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets.

Another isomorphism of categories arises in the theory of Boolean algebras: the category of Boolean algebras is isomorphic to the category of Boolean rings.

The word " noncanonically " prevents one from concluding that supermanifolds are simply glorified vector bundles ; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories.

) Also, 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 is unique.

The importance of skeletons comes from the fact that they are ( up to isomorphism of categories ), canonical representatives of the equivalence classes of categories under the equivalence relation of equivalence of categories.

This means that for every pair of categories defined below, there is an isomorphism of categories, for which corresponding objects have the same underlying set and corresponding morphisms are identical as set functions.

In the case where the two monoidal products coincide and the distributivities are taken from the associativity isomorphism of the single monoidal structure, one obtains autonomous categories.