Terminology: Isomorphism to mean bijection

Question

All my textbooks have never mentioned "isomorphism" to mean bijection, and they explicitly specified "bijection" or "bijective functions". Today, I faced this person who said they were taught "isomorphism" to mean bijection, and it is common usage to call a bijection an isomorphism.

Is this true, does mathematical literature use "isomorphism (on Set)" to exactly mean bijection, without no extra constraints? Should I be prepared to face such term in the wild (in non-Category Theory context)?

Here by isomorphism I mean isomorphism on Set category. As you know, isomorphism coincides with bijection within the Set category. So I wonder if "isomorphism" is often used interchangeably with "bijection".

It's one of those context-dependent things, I think. But since the natural intention of an isomorphism is to establish two items have the "same structure" and can be treated identically in whatever sense is relevant, it seems to naturally necessitate the use of a bijection, if just to ensure that the items in question actually have the same number of elements. — PrincessEev, Sep 17 '22 at 07:26
@PrincessEev Sorry, it seems like I worded my question confusingly. I meant using the term "isomorphism on Set" to imply bijective function. Thus, not using the term "bijection" or "bijective" at all. — Abastro, Sep 17 '22 at 07:31

Martin Argerami · Accepted Answer · 2022-09-17T08:01:27.570

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Is this true, does mathematical literature use "isomorphism" (perhaps with some qualifiers) to mean bijection? Should I be prepared to face such term in the wild (in non-CT context)?

As for isomorphism, outside of category theory the standard use of the word is to mean "bijective homomorphism". The only cases where some people may use the word in a context where it is not bijective, is when someone says "an isomorphism from A into B"; this intends to mean that the function is an injective homomorphism and that it can be considered an isomorphism if one restricts the codomain to be the function's image.

edited Sep 17 '22 at 08:01

answered Sep 17 '22 at 07:32

Martin Argerami

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CT= category theory? – peek-a-boo Sep 17 '22 at 07:35
Yes, the question has been edited now. – Martin Argerami Sep 17 '22 at 08:00
I see, then my question becomes: Is the term "isomorphism" used when there is no structure, where any function would be homomorphism? That is, do we use "Isomorphism on Set category" in place of "bijection" in common mathematics literature? I am sorry if I am not being clear. – Abastro Sep 17 '22 at 08:40
That would be less common, though not unheard of. Sounds better to say "isomorphic" than it is to say "isocardinal" (which I have never seen). – Martin Argerami Sep 17 '22 at 13:25
I see, thank you! – Abastro Sep 18 '22 at 22:42

Terminology: Isomorphism to mean bijection

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