An isomorphism maps a mathematical structure to a mathematical structure in a way that preserves some part of the structure. It must also be possible to reverse an isomorphism with an inverse mapping.[1]
“Two mathematical structures are isomorphic if an isomorphism exists between them…
“An automorphism is an isomorphism from a structure to itself”
“In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:
- “An isometry[2] is an isomorphism of metric spaces.
- “A homeomorphism is an isomorphism of topological spaces.
- “A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.
- “A permutation is an automorphism of a set.
- “In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.
“In certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on 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.” “In mathematical jargon, one says that two objects are the same up to an isomorphism.”
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Information from “Isomorphism,” Wikipedia, retrieved 11/7/2020
[1] What is the Gist of an “Inverse Function”?
[2] What is the Gist of “Isometry” (Mathematics)?
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Disclaimer:
I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).
