Isomorphism in 3 minutes

An Isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them.

Isomorphism between the types means that we can convert T  →  U and then U → T lossless.

For example Int to String conversion is sort of isomorphism, but not all the possible values of String can be converted to Int. For example when we try to convert “Assa” into Int we get an Exception. If we want to use identity element for example 0 for any non-number String - Int2String won’t be Isomorphic anymore.

The real isomorphic mapping from String to Int can be done via co-algebra (list’s co-algebra):

You’ve probably generated the tons of Unit tests for  JSON → String and String → JSON isomorphism prove.

Singleton types are always isomorphic to itself, for example type Unit has only one set’s member Unit or () and it’s always isomorphic to itself. Scala compiler has a special flag -Ywarn-value-discard that checks method returns type may break isomorphism rule (make sense only for non side-effect calls).

Currying functions are isomorphic to each other:

Function1 in Scala doesn’t have the curry method and that is why curry isn’t isomorphic for all the Scala’s functions - but it could be if curry from Function1 returned Function1[T, Function1[T1, U]].


Isomorphism as applied to web development means been able to render pages on the both server and client sides. It’s used in context of NodeJS ecosystem because of been able to reuse libraries/frameworks in backend and frontend.