Aug 4, 2021 · 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.

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)

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.

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.

An "isomorphism of sets" (that is, an isomorphism in the category of Sets) is just a bijection. So what ...

Isomorphism (in a narrow/algebraic sense) - a homomorphism which is 1-1 and onto. In other words: a homomorphism which has an inverse. In other words: a homomorphism which has an inverse. However, homEomorphism is a topological term - it is a continuous function, having a continuous inverse.

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.

Jun 16, 2013 · $\begingroup$ The example of geometry is badly chosen. There are many different forms of geometry (Euclidean, projective, Riemannian, differential, to name a few) which are totally incomparable; not only they are not isomorphic, they don't even have the same language, so talking about isomorphism between then is meaningless.

Dec 12, 2014 · If you are talking just about sets, with no structure, the two concepts are identical. Usually the term "isomorphism" is used when there is some additional structure on the set. For example, if the sets are groups, then an isomorphism is a bijection that preserves the operation in the groups: $\varphi(ab) = \varphi(a)\varphi(b)$.

Oct 15, 2018 · To show these are the only isomorphism, you want to use the fact that an isomorphism of a cyclic group is completely determined by where it sends a generator, that a generator must get sent to another generator, and that …