Isomorphism is a very general concept that appears in several areas of mathematics. The word derives from the Greek iso, meaning "equal," and morphosis, meaning "to form" or "to shape."

Formally, an isomorphism in a category is a morphism with a morphism satisfying and , where and denote the identity morphisms of and (The Stacks Project Authors 2026). Informally, an isomorphism is a map that preserves sets and relations among elements. " is isomorphic to " is written . Unfortunately, this symbol is also used to denote geometric congruence.

An isomorphism from a set of elements onto itself is called an automorphism.

See also

Automorphism, Ax-Kochen Isomorphism Theorem, Graph Isomorphism, Homeomorphism, Isomorphic Graphs, Isomorphic Groups, Morphism Explore this topic in the MathWorld classroom

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References

The Stacks Project Authors. "Definitions." §4.2 in The Stacks Project. Definition 4.2.4, Tag 0017, 2026. https://stacks.math.columbia.edu/tag/0017.

Referenced on Wolfram|Alpha

Isomorphism

Cite this as:

Weisstein, Eric W. "Isomorphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Isomorphism.html

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