what exactly is an isomorphism? - Mathematics Stack Exchange An isomorphism picks out certain traits of one object, certain traits of the other, and shows that the two objects are the same in that specific way Two sets are "isomorphic" when there is a $1-1$ mapping between them, so in this case isomorphism means having the same cardinality--the same number of elements
linear algebra - Difference between epimorphism, isomorphism . . . Isomorphism: a homomorphism that is bijective (AKA 1-1 and onto); isomorphic objects are equivalent, but perhaps defined in different ways Endomorphism : a homomorphism from an object to itself Automorphism : a bijective endomorphism (an isomorphism from an object onto itself, essentially just a re-labeling of elements)
What is the difference between homomorphism and isomorphism? Isomorphism is a bijective homomorphism I see that isomorphism is more than homomorphism, but I don't really understand its power When we hear about bijection, the first thing that comes to mind is topological homeomorphism, but here we are talking about algebraic structures, and topological spaces are not algebraic structures
Difference between ≈, ≃, and ≅ - Mathematics Stack Exchange The symbol ≅ is used for isomorphism of objects of a category, and in particular for isomorphism of categories (which are objects of CAT) The symbol ≃ is used for equivalence of categories At least, this is the convention used in this book and by most category theorists, although it is far from universal in mathematics at large
Questions on isomorphism of graphs - Mathematics Stack Exchange Here is the definition of isomorphism of graph: G = (V , E) and G′ = (V′ , E′) are called isomorphic if a bijection f : V → V′ exists such that {x,y}∈E if and only if {f(x), f(y)}∈E’ holds for all x,y ∈ V , x != y Such an f is called an isomorphism of the graphs G and G′
Whats an Isomorphism? - Mathematics Stack Exchange for the reliance on isomorphism in proofs For example, the internal direct sum of subspaces of a vector space is isomorphic to the external direct sum of these subspaces One can prove that the internal direct sum is associative and commutative and then call on isomorphism to say the same applies to the external direct sum
basic difference between canonical isomorphism and isomorphims The isomorphism requires something specific to this vector space in order to define it But we don't require that to define this isomorphism $\phi$ of a vector space with its second dual We can define just from the definition of "dual of a real vector space": $$\forall v \in V, f \in V^*, \phi(v)(f) := f(v)$$ That is why $\phi$ is "natural"
soft question - What is an Isomorphism: Linear algebra - Mathematics . . . An isomorphism is a bijective homomorphism "Structure" can mean many different things, but in the context of linear algebra, almost exclusively means the vectorial structure -- i e all those rules about addition and scalar multiplication
What is a natural isomorphism? - Mathematics Stack Exchange It's also natural in the technical sense: there is a natural transformation $\eta$ from the identity functor to the double-dual functor $(-)^{**}$, and the component $\eta_V : V \to V^{**}$ of $\eta$ at each finite-dimensional vector space is an isomorphism