Talk:Isomorphism

Near Useless Introduction (Jan 2020)
I don't usually stop to make Talk section comments on articles, but this has got to be one of the worst written intros I've ever read on this site. I understand that Wikipedia is meant to be comprehensive, but the level of technical vocabulary and field expertise needed to make sense of the intro is ridiculous. Every term in the intro is defined by another technical synonym or antonym, themselves almost circularly defined using the term "isomorphism". So not only do you not know what isomorphism means, you can't figure it out without understanding it's opposite and equivalents. Decoding this mess is like solving a simultaneous algebraic equation.

The intro thus becomes incomprehensible to anyone but experts, and experts are the last people who benefit from Wikipedia articles. What's the point of an article that a layman has to study to understand? You don't even need to dumb down the content, there are really good incremental intros on chemistry and quantum physics pages that gradually ramp up in complexity to create a comprehensive article. A non expert usually delves a good way into the page before losing understanding. I've timestamped this comment because it appears this article has been haplessly confusing for near a decade without improvement. — Preceding unsigned comment added by 2001:D08:C2:13A9:D9B4:BB43:8AE6:1563 (talk) 16:09, 4 January 2020 (UTC)


 * I was getting all set to give the usual defensive response but in fact the intro really is terrible. --JBL (talk) 16:40, 4 January 2020 (UTC)


 * I agree. This intro seems to be an exemplar of the maxim of why you should never allow a category theorist to write an introduction. When (not if) we get around to fixing this intro, I am sure that there will be some objections along the line that category theory is the right mathematical setting for this topic. I agree with that statement, but it is not germane to the right way to present the topic in an encyclopedic setting. I've been thinking about how to fix this without ruffling too many feathers, but have not yet come up with a satisfactory solution. --Bill Cherowitzo (talk) 19:34, 5 January 2020 (UTC)

I have written a new version. I am quite sure that it is not worse than the preceding one, and that it is easier to improve further. Please, feel free to edit it.

Also, most of the article would require to be rewritten in the same spirit. D.Lazard (talk) 14:11, 7 January 2020 (UTC)


 * Thanks, D.Lazard, your edits were an enormous improvement. --JBL (talk) 18:19, 8 January 2020 (UTC)


 * As part of the rewrite, the definition of canonical isomorphism seems to have changed. In the current version, "canonical" seems to synonymous with "unique".  After some searching, I have not been able to locate a place where canonical is defined in this way, but I am not an expert.  Perhaps a reference would help? Will Orrick (talk) 15:38, 15 April 2020 (UTC)
 * It is difficult to find a reference, as "canonical" is jargon that is rarely explicitly defined. I have clarified the sentence, and added an example where there is no uniqueness.
 * IMO, when one says that an isomorphism is canonical, there is always some kind of uniqueness. For example, in the case of the group isomophism between $$A/\ker(f)$$ and $$\operatorname{im}(f)$$ there may be many isomorphisms between the two groups, but there is only one that is compatible with the the (canonical) homorphisms from $$A$$ to these groups. More precisely there is a unique isomorphism between the pairs $$(B, A\to B)$$ formed by a group and an homomorphism from $$A$$ to this group.
 * This can surely be sourced in some books of category theory, but I am unable to explain it here at the right level of technicality. So, I have left an imprecise formulation.D.Lazard (talk) 17:00, 15 April 2020 (UTC)
 * Thanks for the edits. The new wording gives a clearer picture of how the term seems to be used in practice. Will Orrick (talk) 02:17, 16 April 2020 (UTC)

Categorical view
It was me that just updated the "categorical" section. I agree about too many words to describe simple things. An isomorphism is an invertible morphism, that's it. Vlad Patryshev (talk) 00:03, 1 July 2020 (UTC)

Surjective bijective
Surjective & bijective thus Isomorphic ? .... 0mtwb9gd5wx (talk) 19:40, 24 August 2021 (UTC)
 * See . D.Lazard (talk) 08:25, 25 August 2021 (UTC)


 * In set theory an isomorphism between posets explicitly is required to translate the comparison of inputs and outputs, i.e. for isomorphism f:(X,&leq;X)→(Y,&leq;Y) we have a&leq;Xb implies f(a)&leq;Yf(b). Should more precise conditions for being an isomorphism like this one be listed in this article? C7XWiki (talk) 07:14, 7 September 2022 (UTC)
 * Have you read the article? Order isomorphisms between posets are explicitly mentioned in the section Isomorphism, for example. JBL (talk) 17:18, 7 September 2022 (UTC)

Isomorphism
Replace 108 by 144 Len loker (talk) 16:49, 26 March 2024 (UTC)


 * Where? D.Lazard (talk) 17:16, 26 March 2024 (UTC)

Equality versus Isomorphism
I removed some of the misleading or confused parts of the equality versus isomorphism part. For instance the discussion of natural isomorphism was confusing as it made it seem as if whether an isomorphism was natural or not was somehow a feature of only the two structures and was the suggestion that category theory doesn't produce equal objects (it's a less useful notion but it exists in some formalizations).

However, I suggest cutting more or almost everything to just include a brief paragraph about this point. If one wants to go into this level of detail perhaps a new page where one can give the appropriate context (are we talking about this from a foundations of math perspective, working mathematican's perspective a philosophical perspective) which isn't very clear. I'm not ready to just do this without more feedback as someone obviously put a lot of effort into this and I don't want to just yank it if I'm getting something wrong. Peter M. Gerdes (talk) 02:46, 4 July 2024 (UTC)


 * General speaking, my thought is that the article should just stick to the discussion of an isomorphism (a morphism that happens to have an inverse), and not get too deep into the philosophical matter of equality vs isomorphism. Especially tricky is a matter of a canonical isomorphism. It is often the case that a canonical isomorphism is treated as an equality (e.g., $$X \times Y$$ are canonically isomorphic to $$Y \times X$$ but we treat them as the same). This is a delicate hard-to-source matter. A better place for a discussion like that might be in canonical map or a place that discusses equality in mathematics, including a new article suggested above. —- Taku (talk) 06:01, 11 July 2024 (UTC)


 * As a related matter, I have proposed the merger of isomorphism class into the article. I think that is a type of materials that should be emphasized in this article; i.e., an isomorphism is an equivalence relation. —— Taku (talk)