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Longer titles found: Isomorphism-closed subcategory (view), Isomorphism (Gestalt psychology) (view), Isomorphism (crystallography) (view), Isomorphism (disambiguation) (view), Isomorphism (sociology) (view), Isomorphism extension theorem (view), Isomorphism of categories (view), Isomorphism problem (view), Isomorphism problem of Coxeter groups (view), Isomorphism theorems (view), Graph isomorphism (view), Group isomorphism (view), Order isomorphism (view), Graph isomorphism problem (view), Cantor's isomorphism theorem (view), Subgraph isomorphism problem (view), Musical isomorphism (view), Norm residue isomorphism theorem (view), Antiisomorphism (view), Group isomorphism problem (view), Quasi-isomorphism (view), Exceptional isomorphism (view), Ornstein isomorphism theorem (view), Eichler–Shimura isomorphism (view), Uniform isomorphism (view), Satake isomorphism (view), Induced subgraph isomorphism problem (view), Borel isomorphism (view), Myhill isomorphism theorem (view), Harish-Chandra isomorphism (view), Potential isomorphism (view), Computable isomorphism (view), Fractional graph isomorphism (view), Almgren's isomorphism theorem (view), Choi–Jamiołkowski isomorphism (view), Duflo isomorphism (view)

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alternate case: isomorphism

Curry–Howard correspondence (6,375 words) [view diff] exact match in snippet view article find links to article

programs and mathematical proofs. It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types

inverse f−1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of

topological vector space isomorphism (abbreviated TVS isomorphism), also called a topological vector isomorphism or an isomorphism in the category of TVSs

B} be a real vector bundle of rank n. Then there is an isomorphism called a Thom isomorphism Φ : H k ( B ; Z 2 ) → H ~ k + n ( T ( E ) ; Z 2 ) , {\displaystyle

evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily not the

that for commutative C*-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching

associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural

(without explicit type annotations) is undecidable. Under the Curry–Howard isomorphism, System F corresponds to the fragment of second-order intuitionistic

of projection onto the sublattice [a, b], a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation

{\displaystyle H} ) is a H {\displaystyle H} -bundle Q {\displaystyle Q} and an isomorphism ϕ Q : Q × H G → P {\displaystyle \phi _{Q}\colon Q\times _{H}G\to P}

= λ · 1. This gives an isomorphism from A to C. The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach

to be any finite abelian extension of F, this law gives a canonical isomorphism θ L / F : C F / N L / F ( C L ) → Gal ⁡ ( L / F ) , {\displaystyle \theta

statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map θ : C K / N L / K ( C L ) → Gal ⁡ ( L /

{\displaystyle \pi :Y\to X} . This map has the property that it is an isomorphism on U = X − Sing ( X ) {\displaystyle U=X-{\text{Sing}}(X)} and the fiber

Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras

theory, the Riemann-Hilbert correspondence provides a complex analytic isomorphism between two of the three natural algebraic structures on the moduli spaces

In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions

The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such

{O}}_{X_{0}})\\\cong &{\mathcal {O}}_{X_{0}}\end{aligned}}} The last isomorphism comes from the isomorphism I / I 2 ≅ I ⊗ O A n O X 0 {\displaystyle {\mathcal {I}}/{\mathcal

representation of F is a pair (A, Φ) where Φ : Hom(A,–) → F is a natural isomorphism. A contravariant functor G from C to Set is the same thing as a functor

closed oriented manifold of dimension n. Then Poincaré duality gives an isomorphism HiX ≅ Hn−iX. As a result, a closed oriented submanifold S of codimension

2-category), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou. A 2-category C consists of:

{\displaystyle X.} A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a linear homeomorphism. F.

(which is the strong dual of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important

In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not

above, a, l, and r are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively. Some examples and non-examples

In graph theory and theoretical computer science, a maximum common subgraph may mean either: Maximum common induced subgraph, a graph that is an induced

In graph theory and theoretical computer science, a maximum common subgraph may mean either: Maximum common induced subgraph, a graph that is an induced

{\displaystyle {\bar {\partial }}_{E}} . In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or ∂ ¯ {\displaystyle

is invertible in the ground field K, there exists a canonical linear isomorphism between ⋀V and Cl(V, Q). That is, they are naturally isomorphic as vector

presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the

{\mathcal {O}}_{X}(D)} defines a monoid isomorphism from the Weil divisor class group of X to the monoid of isomorphism classes of rank-one reflexive sheaves

_{2}\right);{\mathcal {D}}^{\prime }\left(\Omega _{1}\right)\right)} is an isomorphism of TVSs (where L b ( C c ∞ ( Ω 2 ) ; D ′ ( Ω 1 ) ) {\displaystyle

sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve. The isomorphism obtained in this way will in general

is formalized by the simplicial approximation theorem. A simplicial isomorphism is a bijective simplicial map such that both it and its inverse are simplicial

If u : X → Y {\displaystyle u:X\to Y} is a (surjective) vector space isomorphism then so is the transpose t u : Y ′ → X ′ . {\displaystyle {}^{t}u:Y^{\prime

{\displaystyle \psi _{\mathcal {B}}:V\rightarrow \mathbb {R} ^{n}} which is an isomorphism of inner product spaces: to make this more explicit we can write ψ B

bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless E {\displaystyle E} is equipped with an inner product. This is

consists of the following data: A distinguished section o : M → E. A linear isomorphism of vector bundles θ : TM → o*VE from the tangent bundle of M to the pullback

isotopic are, in fact, equal. Isomorphism is also an equivalence relation and its equivalence classes are called isomorphism classes. An alternate representation

relationship between the additive and multiplicative notations is the group isomorphism φ : ( [ 0 , 1 ] , + ) → ( S 1 , ⋅ ) φ ( x ) = x e 2 π i θ {\displaystyle

In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal

D ( α ) = [ M ] ∩ α {\displaystyle D(\alpha )=[M]\cap \alpha } is an isomorphism for all k. This is the Poincaré duality. In particular, every closed

{F}}(U)\to {\mathcal {F}}(V),\,f\otimes s\mapsto f\cdot s|_{V}} is an isomorphism. For each open affine subscheme U = Spec ⁡ A {\displaystyle U=\operatorname

{\displaystyle G} and Q {\displaystyle Q} to be abelian groups, then the set of isomorphism classes of extensions of Q {\displaystyle Q} by a given (abelian) group

→ T f ( x ) Y {\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y} is a linear isomorphism for all points x ∈ X {\displaystyle x\in X} . This implies that X {\displaystyle

for each vector space, it is worth to leave this isomorphism implicit, and to work up to an isomorphism. As several bases of the same vector space are considered

related to mathematical logic and proof theory via the Curry–Howard isomorphism and they can be considered as the internal language of certain classes

{\displaystyle \phi _{x}:x\times L\rightarrow p^{-1}(x)\,} is a Lie algebra isomorphism. Any Lie algebra bundle is a weak Lie algebra bundle, but the converse

is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of X {\displaystyle X} and Y {\displaystyle Y} )

employed the concept in her original formulation of the three Noether isomorphism theorems. A group with operators ( G , Ω ) {\displaystyle (G,\Omega )}

{\mathrm {T} _{X/Y}}_{p}\longrightarrow \mathbb {R} } that induces an isomorphism of sheaves T X / Y ≃ ( I Y / I Y   2 ) ∨ {\displaystyle \mathrm {T} _{X/Y}\simeq

always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention

representation or Gelfand isomorphism for a commutative C*-algebra with unit A {\displaystyle A} is an isometric *-isomorphism from A {\displaystyle A}

intuitive idea is that ω(X) provides a vertical component of X, using the isomorphism of the fibers of π with H to identify vertical vectors with elements

whereas the octonions have a positive-definite signature (8,0). Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional

algebraic vector bundle E on X and an integer i, there is a natural isomorphism: H i ( X , E ) ≅ H n − i ( X , K X ⊗ E ∗ ) ∗ {\displaystyle H^{i}(X,E)\cong

rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank

Novikov ring associated the group of covering transformations. This isomorphism intertwines the quantum cup product structure on the cohomology of M

Gelfand–Naimark theorem, there exists a Hilbert space H and an isometric *-isomorphism from A to a norm-closed subalgebra of the algebra B(H) of continuous

t_{X}:\operatorname {H} ^{n}(X,\omega _{X})\to k} that induces a natural isomorphism of vector spaces Hom X ⁡ ( F , ω X ) ≃ H n ⁡ ( X , F ) ∗ , φ ↦ t X ∘

(electrical circuits), regarding isomorphism of electrical circuits Duality (mechanical engineering), regarding isomorphism of some mechanical laws AdS/CFT

measure space X, L∞(X) is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism. In fact we can state this more precisely as follows:

For a finite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate.

between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from

minimal DFA is unique up to unique isomorphism. That is, for any minimal DFA acceptor, there exists exactly one isomorphism from it to the following one: Let

Zeeman, gives conditions for a morphism of spectral sequences to be an isomorphism. Comparison theorem—Let E p , q r , ′ E p , q r {\displaystyle E_{p,q}^{r}

{A}{I}}\right)^{p}} , that can easily be proven to be an isomorphism. Since A/I is a field, f' is an isomorphism between finite dimensional vector spaces, so n

U ) ] G {\displaystyle \pi ^{\#}:k[U]\to k[\pi ^{-1}(U)]^{G}} is an isomorphism. (Here, k is the base field.) The notion appears in geometric invariant

In mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann

A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound

In mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann

"anabelian question" has been formulated as How much information about the isomorphism class of the variety X is contained in the knowledge of the étale fundamental

be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/cc in

{A}{I}}\right)^{p}} , that can easily be proven to be an isomorphism. Since A/I is a field, f' is an isomorphism between finite dimensional vector spaces, so n

classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of)

in the study of algebraic varieties. Let (X, OX) be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of

U ) ] G {\displaystyle \pi ^{\#}:k[U]\to k[\pi ^{-1}(U)]^{G}} is an isomorphism. (Here, k is the base field.) The notion appears in geometric invariant

In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are

space of a vector bundle V over X, then the Gysin maps are just the Thom isomorphism. Then, using the splitting principle, it suffices to check the theorem

smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving

{\displaystyle (Q,\ell )} to Q {\displaystyle Q} . This morphism is an isomorphism on the open subset of all points ( Q , ℓ ) ∈ X {\displaystyle (Q,\ell

Pic ⁡ ( X ) {\displaystyle \operatorname {Pic} (X)} of X, the group of isomorphism classes of line bundles on X. 3.  In general, O X ( D ) {\displaystyle

resemblance or physical isomorphism approach to how information enters the brain and is stored and deployed. This isomorphism between brain and world

(L^{\times })^{n}\right)/(K^{\times })^{n}.} In this case there is an isomorphism Δ ≅ Hom c ⁡ ( Gal ⁡ ( L / K ) , μ n ) {\displaystyle \Delta \cong \operatorname

(2)}\cdots v_{\sigma (n)}.} Then, one has the theorem that this map is an isomorphism of K-modules. Still more generally and naturally, one can consider U(L)

open set in Rn. f#: O|f(U) → f∗ (OM|U) is an isomorphism of sheaves. The localization of f# is an isomorphism of local rings f#f(p) : Of(p) → OM,p. There

c} nor c 0 {\displaystyle c_{0}} is reflexive. In the first case, the isomorphism of ℓ 1 {\displaystyle \ell ^{1}} with c ∗ {\displaystyle c^{*}} is given

nature". The term "isogeny" was introduced by Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny. Let

on distances in graphs and on the color-coding technique for subgraph isomorphism. With Howard Karloff, he is the namesake of the Karloff–Zwick algorithm

on distances in graphs and on the color-coding technique for subgraph isomorphism. With Howard Karloff, he is the namesake of the Karloff–Zwick algorithm

nature". The term "isogeny" was introduced by Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny. Let

nodes, and exactly 206 regular semigroups of order four up to isomorphism and anti-isomorphism. Sloane, N. J. A. (ed.). "Sequence A058763 (Integers which

map S : M × N → X {\displaystyle S:M\times N\to X} is a vector space isomorphism. The addition map is bijective. M ∩ N = { 0 } {\displaystyle M\cap N=\{0\}}

associated L-function). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be

the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism. An alphabet is a finite set. The free monoid on a given alphabet is

⋅ A 5 ≅ 2 I ; {\displaystyle 2\cdot A_{5}\cong 2I;} this isomorphism covers the isomorphism of the icosahedral group with the alternating group A 5 ≅

\cdots .} The multiplication in T(V) is determined by the canonical isomorphism T k V ⊗ T ℓ V → T k + ℓ V {\displaystyle T^{k}V\otimes T^{\ell }V\to

{\text{End}}({\widehat {G}})^{\text{op}}} . More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories

This technique has given PTASs for the following problems: subgraph isomorphism, maximum independent set, minimum vertex cover, minimum dominating set

\diamond )} is an isomorphism, then ⁠ ι ∘ f {\displaystyle \iota \circ f} ⁠, being a composition of antiisomorphism and isomorphism, is an antiisomorphism

is the image by a linear isomorphism of the canonical basis of F n {\displaystyle F^{n}} , and that every linear isomorphism from F n {\displaystyle F^{n}}

isomorphic to R, and under any such isomorphism the exterior product is a perfect pairing. That is, it yields an isomorphism ϕ : V   → ≅   Hom ⁡ ( ∧ n − 1 V

of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with the isomorphism of O G × S X {\displaystyle {\mathcal {O}}_{G\times _{S}X}} -modules

are equal. (Here isomorphic means that there exists a vector bundle isomorphism E → F {\displaystyle E\to F} which covers the identity i d X : X → X

In mathematics, a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f ( x +

\mathbb {Z} )\rightarrow H_{k}(X,\mathbb {Z} )} in singular homology is an isomorphism for k < n − 1 {\displaystyle k<n-1} and is surjective for k = n − 1 {\displaystyle

group. The Schur cover of a perfect group is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up

(categorical) isomorphism; for example, "The tensor product in a weak monoidal category is associative and unital up to a natural isomorphism." vanish To

concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions

the cap product with an element of the kth cohomology group yields an isomorphism to the (n − k)th homology group. The space is essentially one for which

induces a natural isomorphism TM = T*M and therefore an isomorphism Cℓ(TM) = Cℓ(T*M). There is a natural vector bundle isomorphism between the Clifford

Hausdorff, then χ {\textstyle \chi } is an isomorphism. More generally, χ {\textstyle \chi } is an isomorphism whenever the Čech cohomology of all presheaves

induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is an isomorphism of schemes. A semigroup is said to be seminormal

birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open

induces a natural isomorphism TM = T*M and therefore an isomorphism Cℓ(TM) = Cℓ(T*M). There is a natural vector bundle isomorphism between the Clifford

V^{*}} . Choosing an isomorphism of V {\displaystyle V} with V ∗ {\displaystyle V^{*}} therefore determines a (non-canonical) isomorphism between G r k ( V

Hausdorff, then χ {\textstyle \chi } is an isomorphism. More generally, χ {\textstyle \chi } is an isomorphism whenever the Čech cohomology of all presheaves

S)(R)=R} . One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating

generated over Q, Florian Pop proves that an isomorphism of the absolute Galois groups yields an isomorphism of the fields: Theorem. Let K, L be finitely

{\displaystyle V/\Lambda } is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For n = 1 this is the classical period lattice

concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin ⁡ ( 3 ) ≅ Sp ⁡ ( 1 ) {\displaystyle \operatorname {Spin} (3)\cong

the above property defines the tensor product uniquely up to a unique isomorphism: any other abelian group and balanced product with the same properties

of a linear map. Such a definition can be given using the canonical isomorphism between the space End(V) of linear maps on V and V ⊗ V*, where V* is

by the previous example, as End(L) is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space X forms an abelian

a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G ≅ Aut ⁡ ( Φ ) {\displaystyle G\cong \operatorname

G ] {\displaystyle [G,G]} . More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup. If G {\displaystyle

constructions can be considered an extension of the Curry–Howard isomorphism. The Curry–Howard isomorphism associates a term in the simply typed lambda calculus

\operatorname {Hom} _{R}(M,L)} is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are

respect to base change: Given a morphism S′ → S, there is a natural isomorphism: X ( p / S ) × S S ′ ≅ ( X × S S ′ ) ( p / S ′ ) . {\displaystyle X^{(p/S)}\times

complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly

  ring isomorphism : A ring homomorphism that is bijective is a ring isomorphism. The inverse of a ring isomorphism is also a ring isomorphism. Two rings

homotopy sheaves π ∗ F → π ∗ G {\displaystyle \pi _{*}F\to \pi _{*}G} is an isomorphism. A sheaf of spectra is then a fibrant/cofibrant object in that category

Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by Steinitz

u, v ∈ V′. The mapping f is called an isomorphism between G and G′. When G″ ⊂ G and there exists an isomorphism between the sub-graph G″ and a graph G′

is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970

isomorphism is not canonical, similarly to the situation that a finite-dimensional vector space is isomorphic to its dual, but giving an isomorphism requires

that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix

scientist. Lubiw, Anna (1981), "Some NP-complete problems similar to graph isomorphism", SIAM Journal on Computing, 10 (1): 11–21, doi:10.1137/0210002, MR 0605600

\operatorname {Hom} _{R}(M,L)} is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are

homotopy sheaves π ∗ F → π ∗ G {\displaystyle \pi _{*}F\to \pi _{*}G} is an isomorphism. A sheaf of spectra is then a fibrant/cofibrant object in that category

-homomorphism, then φ {\displaystyle \varphi } is a Freiman s {\displaystyle s} -isomorphism. If φ {\displaystyle \varphi } is a Freiman s {\displaystyle s} -homomorphism

gap> i:=IsomorphismPermGroup(G); # Find an isomorphism from G to a group of permutations. <action isomorphism> gap> Image(i,G); # Generators for the image

theorem has been generalised in various ways, for example by the Almgren isomorphism theorem. There are several other theorems constituting relations between

{\displaystyle A_{R}(R)\to A_{K}(K)} is an isomorphism. If a Néron model exists then it is unique up to unique isomorphism. In terms of sheaves, any scheme A

is up to isomorphism only one such surface, given by blowing up the projective plane in 2 distinct points. Degree 8: they have 2 isomorphism types. One

generators. A particularly simple case of the word problem for groups and the isomorphism problem for groups asks if a finitely presented group is the trivial

{\displaystyle T_{0}} is a topological isomorphism. It follows that T 0 ′ {\displaystyle T_{0}'} is an isomorphism and then im ⁡ ( T ′ ) = ker ⁡ ( T ) ⊥

: X × E G → X {\displaystyle \pi \colon X\times EG\to X} induces an isomorphism of prorings π ∗ : K G ∗ ( X ) I ^ → K ∗ ( ( X × E G ) / G ) . {\displaystyle

minimal polynomial, and any isomorphism of fields F and F' that maps polynomial p to p' can be extended to an isomorphism of the splitting fields of p

module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers are in some sense dual to injective hulls, and are related

universal property of NNOs, meaning they are defined up to canonical isomorphism. If the arrow u as defined above merely has to exist, that is, uniqueness

module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers are in some sense dual to injective hulls, and are related

general, this can be motivated by viewing each τ as representing an isomorphism class of elliptic curves. Every elliptic curve E over C is a complex

they are not given by equalities), but rather are satisfied up to an isomorphism of the next level. An example in topology is the composition of paths

reformulate the Fuchsian uniformization of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the surface)

follows: A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset. The total transform of a normal fundamental point

G(R^{\oplus n})} is an isomorphism for each natural number n, then θ : F ( M ) → G ( M ) {\displaystyle \theta :F(M)\to G(M)} is an isomorphism for any finitely-presented

is (roughly speaking) a direct sum of the spaces of sections of all isomorphism classes of line bundles. Cox rings were introduced by Hu & Keel (2000)

decomposition of the regular representation contains a representative of every isomorphism class of irreducible linear representations of G over K. You can say

finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules:

{N}}\;\mapsto \;(x{\bmod {n}}_{1},\,\ldots ,\,x{\bmod {n}}_{k})} defines a ring isomorphism Z / N Z ≅ Z / n 1 Z × ⋯ × Z / n k Z {\displaystyle \mathbb {Z} /N\mathbb

algebra P ( A ) {\displaystyle P(A)} of primitive elements of A to A is an isomorphism. Here we say A is connected if A 0 {\displaystyle A_{0}} is the field

and N are simply connected. Then W is C-isomorphic to M × [0, 1]. The isomorphism can be chosen to be the identity on M × {0}. This means that the homotopy

equivalence is analogous to the classical Fourier transform that gives an isomorphism between tempered distributions on a finite-dimensional real vector space

zero, so the differential d 2 0 , 1 {\displaystyle d_{2}^{0,1}} is an isomorphism. Given a complex n-dimensional projective variety X there is a canonical

representations. It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology

predual. The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with

equivalence relation by u ≡ v {\displaystyle u\equiv v} if there exists an isomorphism ϕ : S → T {\displaystyle \phi :S\to T} with u = v ∘ ϕ {\displaystyle

with Galois group G. The local reciprocity law describes a canonical isomorphism θ v : K v × / N L v / K v ( L v × ) → G ab , {\displaystyle \theta _{v}:K_{v}^{\times

arbitrary finite-dimensional C*-algebra A takes the following form, up to isomorphism: ⊕ k M n k , {\displaystyle \oplus _{k}M_{n_{k}},} where Mi denotes the

classification theory to general countable p-groups: their isomorphism class is determined by the isomorphism classes of the Ulm factors and the p-divisible part

y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.} An isomorphism of Lie algebras is a bijective homomorphism. As with normal subgroups

is that V should, when restricted to Xi, give back Vi, up to a bundle isomorphism. The data needed is then this: on each overlap X i j , {\displaystyle

For each prime p and positive integer n there are exactly two (up to isomorphism) extraspecial groups of order p1+2n. Extraspecial groups often occur

not an isomorphism factors through B. A is indecomposable and any homomorphism from A to an indecomposable module that is not an isomorphism factors

complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending

deformation theory because deformation functors are classified by quasi-isomorphism classes of L ∞ {\displaystyle L_{\infty }} -algebras. This was later

n}^{m}\otimes H_{j,k}^{n}\otimes V_{m}} The associativity isomorphism induces a vector space isomorphism Φ i , j k , m : ⨁ ℓ H i , j ℓ ⊗ H ℓ , k m → ⨁ n H i

field F p y {\textstyle F^{\mathrm {py} }} containing it, unique up to isomorphism, called its Pythagorean closure. The Hilbert field is the minimal ordered

groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra

X_{n}]\twoheadrightarrow A} is surjective; thus, by applying the first isomorphism theorem, A ≃ K [ X 1 , … , X n ] / k e r ( ϕ a ) {\displaystyle A\simeq