In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).

An isomorphism is simply a bijective homomorphism.

* Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.

* A group automorphism is a group isomorphism from a group to itself.

In this example it is not sufficient for a morphism to be bijective to be an isomorphism.

In a category with exponentials, using the isomorphism ( in computer science, this is called currying ), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

It is essentially unique ( up to isomorphism ).

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

* automorphism if f is both an endomorphism and an isomorphism.

* f is an isomorphism.

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

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η < sub > X </ sub > is an isomorphism for every object X in C.

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.

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.

One can show that this map is an isomorphism, establishing the equivalence of the two definitions.

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.

The mapping Φ: H → H * defined by Φ ( x ) = φ < sub > x </ sub > is an isometric ( anti -) isomorphism, meaning that:

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms ( e. g., a weak equivalence of spaces passes to an isomorphism of homology groups ), verified that all existing ( co ) homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

In general the situation is more complicated and can be characterized as a rotation in the Poincaré sphere about the axis defined by the propagation modes ( this is a consequence of the isomorphism of SU ( 2 ) with SO ( 3 )).

Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism.

an isomorphism ) between individual objects ( not entire categories ) is referred to as a " natural isomorphism ", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors ; formalizing this intuition was a motivating factor in the development of category theory.

The fundamental theorem on homomorphisms ( or first isomorphism theorem ) is a theorem, again taking various forms, that applies to the quotient algebra defined by the kernel.

The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension ( this is done inside local class field theory ) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field.

Rather than working directly with the sheaves, he defined a group using ( isomorphism classes of ) sheaves as generators, subject to a relation that identifies any extension of two sheaves with their sum.

Applying the Curry – Howard isomorphism once more, inductive families correspond to inductively defined relations.

Given a torus T ( not necessarily maximal ), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G ; then the corresponding Weyl group is called the Weyl group of G ( it depends up to isomorphism on the choice of T ).

The canonical isomorphism is defined as follows:

The concept of a homomorphism and an isomorphism may be defined.

The modern statement of the Poincaré duality theorem is in terms of homology and cohomology: if M is a closed oriented n-manifold, and k is an integer, then there is a canonically defined isomorphism from the k-th cohomology group H < sup > k </ sup >( M ) to the ( n − k ) th homology group H < sub > n − k </ sub >( M ).

Suppose P is an exact category ; associated to P a new category QP is defined, objects of which are those of P and morphisms from M ′ to M ″ are isomorphism classes of diagrams

Alternatively, we could have defined to be the above function from the beginning, realized that is an isomorphism, and defined to be its inverse.

NNOs are defined up to isomorphism.

In mathematics, the derived category D ( C ) of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C. The construction proceeds on the basis that the objects of D ( C ) should be chain complexes in C, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes.

In this approach, graphs are treated as database instances, and rewriting operations as a mechanism for defining queries and views ; therefore, all rewriting is required to yield unique results ( up to isomorphism ), and this is achieved by applying any rewriting rule concurrently throughout the graph, wherever it applies, in such a way that the result is indeed uniquely defined.

The double conjugate is naturally isomorphic to, with the isomorphism defined by

A measure of the closeness of this isomorphism is the " redundancy " of the language, defined as the number of editing operations needed to implement a stand-alone change in requirements.

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

A choice of isomorphism is equivalent to a choice of basis for V ( by looking at the image of the standard basis for R < sup > n </ sup > in V ).

The finite fields are classified by size ; there is exactly one finite field up to isomorphism of size p < sup > k </ sup > for each prime p and positive integer k. Each finite field of size q is the splitting field of the polynomial x < sup > q </ sup > − x, and thus the fixed field of the Frobenius endomorphism which takes x to x < sup > q </ sup >.

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.

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.

If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra over F there is a simply connected Lie group G with as Lie algebra, unique up to isomorphism.

* Suppose G and H are topologically finitely-generated profinite groups which are isomorphic as discrete groups by an isomorphism ι.

If a diagram F: J → C has a limit in C, denoted by lim F, there is a canonical isomorphism

If one is given an object L of C together with a natural isomorphism Φ: Hom (–, L ) → Cone (–, F ), the object L will be a limit of F with the limiting cone given by Φ < sub > L </ sub >( id < sub > L </ sub >).

On the category of finite-dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space ( say, by choosing a basis for every vector space and taking the corresponding isomorphism ), but this will not define a natural transformation.

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 ).