From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated B-class, Top-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
B Class
Top Importance
 Field: Algebra
One of the 500 most frequently viewed mathematics articles.

Pseudo-philosophical gobbledygook[edit]

Therefore knowing what counts as an isomorphism is as good as knowing what we mean by structure, of a given kind.

I'm removing this until someone can rephase it. It sounds like pseudo-philosophical gobbledygook. In what sense is the one knowledge "as good as" the other? Who is this mysterious "we"? Is "structure" meant in a technical sense? If so, what's the definition? --Ryguasu 21:42 26 Jun 2003 (UTC)

There isn't a universal definition of structure that covers all mathematical structures that are discussed.

One way round this is to say 'we may not know what structure means in the abstract, but we can be given information about when two structures are the same'. It's like saying, faced with some unfamiliar type of money, this coin has the same value as this note, this pile can be exchanged for that one. Without trying to say what 'money' (or the value it represents ) is.

A simple example from arithmetic: we can write numbers in binary or base 10 notation. For the most part we don't care, since the results of calculations will be the same, after conversion. A represention of 'eigh't as 100 is 'as good' as 8.

A purely mathematical example would be a metric space X, which gives rise to a topological space in a standard way. If we are prepared to consider another topological space Y that is related to X by a homeomorphism as suitable for our purposes, that tells us that the open set structure is all we care about. If on the other hand we insist that Y be another metric space and the homeomorphism actually an isometry, that says we actually care about the metric.

That probably isn't the usual case, in fact: a constant multiplier in the metric is like changing your basic unit of measurement, say from metre to kilometre, and so yet another idea of 'isomorphism' can be brought it as a 'similarity'. That tells you that the structure that matters is the ratio of distances, e.g. similar rather than just congruent triangles in plane geometry.

Charles Matthews 07:58 29 Jun 2003 (UTC)

In the definition of the isomorphism, it is said that the functions f and f-1 should be bijective homomorphisms. But homomorphism is, as I remember, a group mapping, so I believe it ought to be morphism in that section, and that one should wait with the homomorphism until the definition of group isomorphism. Anyone here who could say whether I'm wrong about this? Mikez 10:38, 29 Jan 2004 (UTC)

Anyone who follows the homomorphism link gets an idea of the morphism concept immediately. I've added an informal note at that point, too.

Charles Matthews 10:48, 29 Jan 2004 (UTC)

Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e. structure-preserving mappings.

It seems like this sort of a statement should have a bit more context. Of course, this is a true statement in all of the familiar categories, but it's also a bit misleading in the sense that the important aspect of an isomorphism is that it is a morphism with an inverse. That this implies bijectivity in the usual algebraic categories is really a theorem, not the definition.

Personally I'd leave it like it is, but also expand the mention of the categorical definition (near the end) to be explicit. Most mathematicians would give a definition like this one unless specifically asked for the category theory definition and the same is true for most books that are not category theory books. --Zero 10:18, 25 Feb 2005 (UTC)

You may want to add a link to the First Isomorphism Theorem.

Merge with Cryptomorphism?[edit]

See Talk:Cryptomorphism. Qwertyus 21:04, 1 August 2006 (UTC)

No Way!!! Isomorphism is an extremely broad and widely used term familiar to every mathematician whereas cryptomorphism is highly specific to a particular context and is NOT in widespread use. In my opinion it would be a mistake to merge the two. Someone seeking information on isomorphisms doiesn't need to be told about cryptomorphisms. Hawthorn.

--Army1987 21:56, 24 November 2006 (UTC)

Lead cube versus wood cube[edit]

User:Army1987 removed the following physical analogy example –

A solid cube made of wood and a solid cube made of lead are both solid cubes; although their matter differs, their geometric structures are isomorphic.

– supplying this comment: "→Physical analogies - No, they don't. The lead cube contains less atoms, and two sets with different cardinalities can't be isomorphic."

Although the removed example was arguably not the best illustration of isomorphic structures (and ignoring the fact that there's no reason the cubes couldn't contain the same number of atoms), I don't think that the cardinality argument holds water. The isomorphism being illustrated was clearly the cube-ness of the objects, rather than their physical properties; and two cubes can certainly be viewed as isomorphic structures. (It is after all a basis of topology.) At any rate, non-mathematical use of the term isomorphism is imprecise; thus in biology, isomorphs are organisms with analogous structures – obviously failing a "count of atoms" test.

I believe that the term isomorphism finds a good deal of general non-mathematical (and thus non-rigorous) use. I will put a comment to that effect in the "physical analogies" section. Trevor Hanson 19:41, 12 November 2006 (UTC)

This article is about mathematical isomorphisms, see also Isomorphism (disambiguation).
Indeed. (Though the section on physical analogies stretches the discussion to fringe areas, by providing intuitive examples for the lay reader. This seems reasonable, since the term isomorphism links directly to this article, rather than the disambiguation page. But if you think my addition to the analogies section is off-topic, do feel free to remove it.) The distinction you make between isomorphisms between infinitely divisible Platonic solids and isomorphisms between the structures and atoms of physical objects is of course valid, and not at all 'unworthy'. I don't think that the original 'lead cube versus wood cube' example was intended to imply anything more than isomorphic shapes; but you're correct that it was ambiguous. Trevor Hanson 23:27, 12 November 2006 (UTC)
Two cubes can be put in biunivocal correspondence with each other, there is a function which nds each point of one to one and only one point of the other. If the function chosen is an isometry, it preserves distances between two points of one cube and the corresponding two points of the other. This is an isomorphism. (There are bijections from a cube and a line, from a cube and the powerset of N, etc., but these can't be obviously isomorphisms.)
I agree that the theoretical shape of a cube made of wood is isomorphic to that of a cube made of lead, but their theoretical shapes are just approximations. At a microscopic level they have very different structures. Yes, this does sound like a subtle distinction unworthy being done, and I myself would believe it is, if I didn't know that some people were fooled into believing that a gold ball could be duplicated via the Banach-Tarski paradox... --Army1987 20:30, 12 November 2006 (UTC)

Another problem in that section is this one: "A list of customer names and a list of telephone numbers, where each entry on either list corresponds to a single entry on the other list. Note that the lists exhibit the isomorphism, rather than the actual customers and telephones (which may or may not be isomorphic)." As well as it being written badly, I can't see that it is anything more than two sets and a bijection between them. If there is no structure in one set that is carried onto to structure in the other set, why bother calling it an isomorphism and why is it a useful example? McKay 03:57, 14 November 2006 (UTC)

They are isomorphic, for example, with respect with the measure d(x,x) = 0 and d(x,y) = 1 ∀ xy. Too bad that any two set with the same cardinality are. So that is not a very instructive example. However, it should be noted that, althought is is not a good example of isomorphism, there is a bijection (that sending every name to the phone number of the person with that name) being chosen among the n! possible ones. In contrast, there are as many quark flavours as many Rammstein members, but there is no bijection between them which is more 'important' or 'useful' or 'logical' than the other 719 possible ones. --Army1987 21:56, 24 November 2006 (UTC)

Consider changing main 'isomorphism' target to disambig page?[edit]

The latest revert suggests that perhaps the main entry for isomorphism should be the disambiguation page rather than the term in mathematics, and that this article should be renamed isomorphism (mathematics). Comments? Objections? That would seem the more normal Wikipedia model. Trevor Hanson 19:01, 24 November 2006 (UTC)

Almost all articles in "What links here" seem to be about maths, so I'm not sure this is the right thing to do... --Army1987 21:44, 24 November 2006 (UTC)
Hmmm, yes of course you are right. I wish I'd looked at "What links here" first. :) Many wrong-headed edits to this article (such as yesterday's involving Gestalt psychology, as well as recent edits of mine) seem to be good-intentioned attempts to deal with the facts that a) isomorphism, used without context, links here, and b) the term is widely used outside maths, with different meanings; and this seems potentially confusing, especially to non-English speakers. But as you point out, the term's richest technical usage, and no doubt its origin, is mathematical. We do always seem torn between two views of Wikipedia: as a union of specialized repositories, for use by experts; versus a trans-specialty gateway, for use by outsiders. And of course, it must be both things. I've been on the other side of this argument elsewhere. I guess I'd better leave this particular article to mathematicians. Trevor Hanson 05:24, 25 November 2006 (UTC)

Gestalt psychology[edit]

I was the person who added the Gestalt Isomorphism entry to the applications section of this page and was disappointed to see it removed by the author (whose right to do so I so not dispute). The Gestalt Psychology article includes a link to this Isomorphism article, but anyone following it would not really be well-enough informed as to the meaning Kohler and Wertheimer meant to convey by their choice (perhaps more accurately, the translator's choice, as the original works were in German)of this term for their idea. So the question is, where to put the Gestalt meaning of isomorphism? Perhaps in the Gestalt Psychology article, in place of the link, but then the author of that article might do what the author of the isomorphism article did. Mark Cole.

The 'normal' Wikipedia way is to have the source article (Gestalt Psychology) point to the disambiguated isomorphism (Gestalt psychology) page. I believe this is now how things stand.
Just to clarify: We are all "the author" of this page; I have no special claim to it (less than most in fact; this particular page is really the domain of mathematicians). I did step in, and moved the Gestalt Isomorphism text from this page to a new article: isomorphism (Gestalt psychology). I should point out that:
  • The material added was good, but it was not an application of mathematical isomorphism, any more than are the biological or computer science uses of the term. (This point was driven home to me when one of my earlier edits drew the same reaction.)
  • I moved this text to a new article because I saw that this text had already been added, removed, and added again, and I wanted to avoid a 'revert war' -- three reverts triggers bad things. Since creating a new article was the preferred Wikipedia solution, I decided to step in and do this – rather than just saying "this belongs on a disambiguation page." I'm sorry if this bruised any toes.
  • I have just changed the link in Gestalt psychology to point to that new disambiguated page, so I hope everything is satisfactory now?
  • I recently questioned (above) the decision of having isomorphism link to this article, rather than to the disambiguation page; but the valid point was made that there are innumerable mathematical articles referring to this page in a technical way, and very few articles from other fields. (I still think it could be argued either way...but not by me.)
Again, there is no such thing as "the author of an article" in Wikipedia. There is an "original author" who sets the style in such matters as citation format and whether to say "color" or "colour" (and those get changed, too). But all work is collaborative. Trevor Hanson 20:21, 26 November 2006 (UTC)

As the instigator of all this, I am guilty more of being a Wikipedia novice than of anything else. The only reason I added the Gestalt Psychology note to the Isomorphism article a second time was that I thought I had done something wrong (technically speaking) the first time when it disappeared. I certainly had no intention of starting any sort of "reversion war". The Isomorphism link in the Gestalt psychology article now points to my entry on Gestalt Isomorphism on the related disambiguation page. I am happy and no feelings were bruised along the way. Also, I am grateful for the heads up about global authorship. Mark Cole.

No worries, and I'm glad if things look OK to you now. BTW, in case you were thinking that a 'revert war' was like a childish 'flame war', that isn't the issue. Three reverts triggers automatic consequences; I decided to step in because I thought this might occur essentially by accident. Also: In case you don't know, you can type four tildes (~~~~) after your post to have the system automatically include your name and the time. Trevor Hanson 03:41, 29 November 2006 (UTC)

Thanks for helping me out in this expedition into uncharted waters, Trevor. 16:20, 29 November 2006 (UTC)

Clock analogy[edit]

"The Clock Tower in London (that contains Big Ben) and a wristwatch; although the clocks vary greatly in size, their mechanisms of reckoning time are isomorphic."

Would it be better to make a comparison between:

"The Clock Tower in London (that contains Big Ben) and a digital wristwatch; although the clocks vary in size and type of display, their mechanisms of reckoning time are isomorphic."

Or is this unneccessary, given the following dice example? —Matthew0028 05:29, 30 December 2006 (UTC)

Tic-Tac-Toe Analogy[edit]

When playing Tic-Tac-Toe X usually wins on the fourth move. Finding three numbers that adds up to fifteen is not the same game. There is an analogy for playing the most efficient game, but doesn't cover every possible way in which the game may be played, and won, with more than three moves as is often seen when inexperienced children play it. —The preceding unsigned comment was added by (talk) 21:28, 19 February 2007 (UTC).

I think you misunderstand the 15-game. You aren't limited to just saying 3 numbers. But when you say more, only 3 of them have to add up to 15. The analogy stands. —The preceding unsigned comment was added by (talk) 16:15, 9 May 2007 (UTC).

Matrix examples[edit]

I think this article would benefit from an example that an mxn matrix is isomorphic to an nxm matrix. Isomorphism has special application to linear algebra. Psyadam 20:42, 29 March 2007 (UTC)Adam Henderson, March 29, 2007

Isomorphic "sets"?[edit]


Can you explain this?

"In a certain sense, isomorphic sets are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined."

Huh? I didn't think that "sets" could be isomorphic. What, are we supposed to say that the sets A = { 1, 2, 3, 4 } and B = { Roswell UFO, Bill Gates's house, Microsoft, IBM } have some sort of "isomorphism"? That doesn't make any sense. Sets alone do not really have much "structure" to them, you need to set up relationships between elements of the set that create structure, and structure is what isomorphism deals with. But then it is misleading to call the resulting object just a "set". What gives? Is somehow, the Roswell UFO isomorphic to the number 3? Not unless you add some sort of relationship that puts a structure on those sets, but then just saying "sets" gets misleading! mike4ty4 06:16, 7 May 2007 (UTC)

I've changed "sets" to "structures". Sets can be isomorphic (which just means that they have the same cardinality, since the isomorphisms in the category of sets are simply the bijections), but that's not what this sentence is talking about. --Zundark 07:45, 7 May 2007 (UTC)

i think this article is really helpful. isomorphism confused me before, now i get it. thank you

pronunciation of "isomorphism"[edit]

I think it would be quite a good idea to add the IPA pronunciation of "isomorphism" (which is not really easy to find out, believe me). At least someone could write it here. Thanks. 23:20, 3 September 2007 (UTC)

What's this Hofstadter Business?[edit]

I liked G.E.B. as much as the next guy, but what's with the Hofstadter quote at the beginning of the article? Not only does it add nothing to the article, but it's also confusing. "Complex structures"? Stick with the well thought out description already agreed on. Quote removed. Rljacobson (talk) 03:11, 13 February 2008 (UTC)

Physical analogies?[edit]

Hello, I'm concerned that the "physical analogies" make it sound like "isomorphism" is a vague concept. This is misleading: one of the most important facets of abstract maths is that "isomorphism" is a precise thing and definitely not vague.

How can two decks of cards be isomorphic? A deck of cards is not a mathematical set, so what is "a function between decks of cards"? Or is there some "category of decks of cards"?

Similarly, how can two clocks be isomorphic? in what sense is a clock a set? and what is a function between two clocks? when is it structure-preserving? is there a category of clocks?

If "isomorphism" is a vague notion that exists in general language, and if we really need to mention that here, then it should be clear that this has little to do with "abstract algebra". Shall I cut the section? Or move it lower and make it clear that it is not mathematics? Or should we suggest what a "morphism of clocks" could be? Any thoughts? Sam Staton (talk) 15:49, 18 May 2008 (UTC)

No responses, so I cut the section. Sam (talk) 13:19, 6 August 2008 (UTC)


Is "≅" standard notation? I certainly used it a lot in college, as in "Z/6Z ≅ Z/3Z x Z/2Z", and nobody called me on it. The = article says "The symbol “≅” is often used to indicate isomorphic algebraic structures or congruent geometric figures," no citation. (talk) 22:55, 6 January 2009 (UTC)

Yes, that is absolutely standard. Occasionally, people sloppily just write "=" for isomorphism, but "≅" is unambiguous. Jakob.scholbach (talk) 23:31, 6 January 2009 (UTC)

Isomorphism vs. Equality[edit]

The issue of isomorphism vs. equality is fundamental to category theory, and extraordinarily confusing to novices. I’ve found it easiest to understand and to explain by using examples – the abstract statement (equal as objects vs. have a map between them) is terse and opaque – and accordingly I’ve had a stab at explaining the difference as of this revision, giving simple examples (ABC, 123 – Sesame Street level), and elementary motivational algebraic and geometric examples (double dual, Riemann sphere), and the Mazur reference, and mentioning confusing subtleties but relegating them to footnotes (not mentioning them is confusing because these are important distinctions, but mentioning them in the running text is distracting). Hope these help, and feel free to improve!

—Nils von Barth (nbarth) (talk) 07:25, 13 December 2009 (UTC)

Nice work! Couple suggestions[edit]

Decent intro, a motivating Purpose section, practical and abstract examples. "Nice job!" to those who've made this article as clear and helpful as it is.

Some suggestions:

1. I'm, pretty sure "A relation-preserving isomorphism" could use the addition of:

"for all u, v in X".

... could someone add this who's confident that's correct?

2. I would love to read somewhere on this page a clear discussion of the phrase "up to isomorphism" which one sometimes sees, including on this page.

3. An clarification of "identify" as used here would also be helpful. Does this mean "provide a mapping between members of two sets? Or is it something stricter, as section "Relation with equality" suggests. (Well actually it says (A,B,C) can't be identified with (1,2,3), but then it says it can, but that doesn't count because you'd be choosing a particular isomorphism. So there's apparently some other issue implied here.) Gwideman (talk) 23:37, 18 December 2009 (UTC)


I had a hard time reading this article. I could hardly even understand what isomorphism is. Yes, I know that the article seems to be mostly written by math nerds (no offence). But unless the article is made simpler to understand for a newbie, there is even no sense for it to exist! Artem Karimov (talk) 20:05, 1 January 2011 (UTC)


The introduction, as it currently reads:

In abstract algebra, an isomorphism[1] is a bijective homomorphism.[2] Two mathematical structures are said to be isomorphic if there is an isomorphism between them. In category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f −1f = idX and f f −1 = idY.[3]

is basically useless to a general audience. Essentially the entire introductory explanation of a isomorphism is 'a bijective homomorphism'. That may be adequate as a technical reference; but not for an encyclopedic article.

There exists a non-technical, jargon-minimized explanation for what an isomorphism is --- and that's what should be in the introduction of the article. All Clues Key (talk) 15:53, 7 September 2012 (UTC)

I have added a paragraph to the lead in an attempt to address this, as this topic is simple enough to be described more accessibly. — Quondum 17:44, 7 September 2012 (UTC)
Thanks Quondum, I think this is a good addition. What about something like:

In essence, an isomorphism is a mapping from one object (i.e. the set elements and operations) to another object in which they are indistinguishable given only a selection of their features. A named isomorphism indicates which features are selected for this purpose. Thus, for example, two objects may be group isomorphic without being ring isomorphic, if < criteria 1,2 ... >, since the latter isomorphism selects the additional structure of the multiplicative operator.

Where, maybe, you (or someone) could add the relevant criteria for the 'group isomorphic' vs. 'ring isomorphic' example. AllCluesKey (talk) 18:20, 7 September 2012 (UTC)
How about being more concrete:

In essence, an isomorphism is a bijective mapping from one object (i.e. the set elements and selected operations) to another object in which they are indistinguishable. A named isomorphism indicates which features are selected for this purpose. Thus, for example, the cyclic group Zn and ring of integers modulo n Z/nZ are group isomorphic without being ring isomorphic, since in the group lacks the multiplicative operator required by a ring isomorphism.

This is not the ideal example: I would have preferred to have had analogous structures for each example object – e.g. a multiplication operator – that breaks the ring isomorphism without being absent. Also, I'm sure far simpler examples might be better (e.g. set isomorphism vs. group isomorphism with the Klein group and the cyclic group of order four, Z4?). — Quondum 18:56, 7 September 2012 (UTC)

Flaw in introduction?[edit]

There is a flaw in the definition in the introduction:

The notion of a category theoretic isomorphism is quoted in [1] and [3] but incorrectly. Although it can be true that an isomorphism is a bijective (homo)morphism, this does not hold in general. (It does hold in [1] I believe, but for a more generally known example, take the class of groups, with grouphomomorphisms as category.) The point is that the 'inverse' that is spoken of needs to be a morphism in the same category.

For a counter example: Take the category of topological spaces with continuous functions as morphisms. Suppose that we have a set X, with two topologies A and B, such that B is strictly contained in A as a set. Then the identity map from X with the topology A to X with B as topology, id: (X,A) -> (X,B), is a bijective function and it is continuous i.e. a homomorphism of topological spaces. It however is not an isomorphism.

Please help me improve this and my future answers, it is my first post on this kind of page and would like to continue help making the math pages better. :) I will check back regularly here and soon probably have an account too. If someone requires a proof of the assertions made, I will gladly provide them! (talk) 19:36, 20 June 2013 (UTC)

I agree. The given definition is not correct for all types of objects, even in abstract algebra (construed broadly). Best would be to give the category-theoretic definition of isomorphism, and then to say that for many specific kinds of objects arising in abstract algebra (groups, rings, etc.) it is a theorem that bijective homomorphisms are the same as isomorphisms. We could still start with a user-friendly sentence such as "In mathematics, an isomorphism is a map from one object X to another object Y under which the structure on X corresponds exactly to the structure on Y. More precisely, ..." Ebony Jackson (talk) 04:25, 17 November 2013 (UTC)
I've rewritten the lede. It should be correct, but the rest of the article now needs some revision to make it consistent. Ozob (talk) 20:31, 17 November 2013 (UTC)
While I don't object to the improved correctness of the lead, it now lacks any interpretation that does not require a degree of familiarity with the concepts. Isomorphism as a concept of indistinguishability of the structure of two objects. Simply using the category-theoretic term "morphism" already excludes a large audience. Would describing an isomorphism as a map that preserves a chosen type of structure. As an example, I am having difficulty in determining why a bijective homomorphism, applied correctly, is not a correct definition for isomorphism. With the example of topological spaces, a mapping would only qualify as a homomorphism if it preserves the topology: every subset must still have the same topology (which is, after all, the structure to be preserved) after the mapping. A requirement of the mapping itself being continuous does not enter into the definition. I mention this really only as a comment on the accessibility of the text. —Quondum 21:59, 17 November 2013 (UTC)
Using the term "map" is just as, if not more, problematic than the term "morphism". "Morphism" is a precisely defined concept; "map" is often just an informal shorthand for "morphism", but to a reader not familiar with it, it will conjure images of maps, leading to greater confusion. That said, I would prefer to use a less formal, less jargony word for a morphism; I considered "map", "function", and "relationship", but thought they were all lacking. I'm open to other ideas if anyone has them.
Regarding topological spaces, a morphism in the category of topological spaces is a continuous function. Consider the half-open interval [0,1) and the function f(t) = exp(2πit). This is a continuous bijection between [0,1) and the unit circle, so it is a morphism in the category of topological spaces. However, f's inverse, the argument function, is not continuous, so while the argument function is a morphism in the category of sets, it is not a morphism in the category of topological spaces, and therefore f is not an isomorphism in the category of topological spaces. Since an isomorphism in the category of topological spaces is called a homeomorphism, f is not a homeomorphism between [0,1) and the unit circle. (Proving that there is no homeomorphism between the two requires additional effort.) Ozob (talk) 04:12, 18 November 2013 (UTC)
I feel like a pipsqueek arguing with the wisdom of ages of mathematicians, but here goes: demanding continuity of the mapping to qualify as a morphism is a constraint on the selection of the mapping designed to preserve structure, but it is not defined in terms of the preservation of any structure of the objects being mapped. In particular, the structure of continuity (such as boundary points) is not mapped to the same in the other. So to me, with this structure being preserved, an isomorphism being defined as a bijective homomorphism works, just like in every other context. But I've probably got it wrong, this is not the place to educate me, and what counts is what the people who really know say. So let's rather look at adding an introductory sentence to the lead that normal mortals can understand. —Quondum 03:59, 20 November 2013 (UTC)
I may be misunderstanding your comment, but let me try to say in an imprecise way what I think the issue is. Very roughly (and imprecisely), the structure being preserved by a continuous map is not boundary points, but "nearness". Roughly, continuity means that if two input points are close to each other, then their two outputs are close to each other. It does not mean, however, that the two input points are close IF AND ONLY IF their two outputs are close; e.g., it could happen that a continuous map also sends far away points to the same place. (Analogously, a group homomorphism from Z to Z is allowed to send everything to 0.) If you want the IF AND ONLY IF statement, then you need the map and its inverse both to be continuous. The bijection in Ozob's counterexample does map nearby points to nearby points, but the inverse bijection sends nearby points on the circle on either side of the "cut" to points in [0,1) whose distance is large, nearly 1.
As for the lede: it is important to provide the precise definition that Ozob provided, but perhaps this could come later, in a section marked "Definition", while the lede would explain only intuitively what kind of thing an isomorphism is? Ebony Jackson (talk) 07:17, 20 November 2013 (UTC)
It's helpful to see what other languages do. In Italian, they use the following Douglas Hofstadter quote:
“The word ‘isomorphism’ applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where ‘corresponding’ means that the two parts play similar roles in their respective structures.”
In French: In mathematics, an isomorphism between two structured sets is a bijective function which preserves the structure and whose inverse preserves the structure. — Preceding unsigned comment added by Brirush (talkcontribs) Brirush (talk) 14:36, 20 November 2013 (UTC)
I would like very much to have a more elementary description of an isomorphism in the lede than what I put there. I don't really like the Hofstadter quote, though I think it comes close. I don't like how the French version restricts to the use of sets (there are plenty of isomorphisms in non-concrete categories, like the homotopy category of CW-complexes).
Maybe I'm being too restrictive in insisting that whatever text we have be an English distillation of the category-theoretic definition? Perhaps we should look for something less like a definition than a description. Ozob (talk) 15:32, 20 November 2013 (UTC)
@Ebony: Thank you for the clarification. I take your point about what structure is preserved, so I can see why a homomorphism in one direction is not necessarily a homomorphism in the other direction, even if it is bijective. This would lead us to define an isomorphism as a mapping in which both the mapping and its inverse are homomorphisms. This ties in closely with Brirush's input.
A suggestion: replace "an isomorphism is a morphism that admits a two-sided inverse" with "an isomorphism is a homomorphism of which the inverse is also a homomorphism". Would this work? I think this'll be more generally understandable, as long as it is accurate.
Hofstadter's description is a bit fuzzy, even for plain English. In "plain English" I'd say essentially the same as I'd say for a homomorphism (being a structure-preserving map), adding that it must apply in both directions, finishing with that this means that in terms of the chosen structure (relationships between elements within an object), mapped elements and relationships are indistinguishable. —Quondum 16:01, 20 November 2013 (UTC)
I've tried again on the lede. There's no mention of categories or morphisms now until the second paragraph (and the second paragraph could easily be moved elsewhere). What do you think of this version? Ozob (talk) 16:30, 20 November 2013 (UTC)
I like it a lot betterBrirush (talk) 16:37, 20 November 2013 (UTC)
Yes, now the explanation in the lead is readily readable. We probably need a definition section now for the more formal treatment, which can be added whenever we find the time. One thing about "relationship": it sounds as though what is being described is isomorphic structures (i.e. two structures for which there exists an isomorphism between them) rather than an isomorphism (a map between two structures). —Quondum 06:36, 21 November 2013 (UTC)

I'm late to the discussion but I just want to say one talks about whether two categories are isomorphic or not: they are isomorphic if there is a functor F from one to the other that admits an inverse. Clearly, the adjectives like "bijective" don't quite apply to this situation. -- Taku (talk) 16:42, 20 November 2013 (UTC)

If the categories are small (having sets of objects and morphisms) then an isomorphism of categories is a bijection on object sets and morphism sets. But isomorphism of categories is a useless notion; natural equivalence of categories is the right concept, and naturally equivalent small categories need not admit a bijection between their object sets. (Consider, for example, a one object category with only the identity morphism and a two object category whose objects are isomorphic and admit no morphisms other than the isomorphisms.) Ozob (talk) 03:42, 21 November 2013 (UTC)
I agree that an isomorphism of categories is rather uninteresting. But my point was that a "bijection", by which I mean "inject", "surjective", is not a concept that applies to a functor (it makes sense on the levels of objects or hom-set). A functor is an equivalence if and only if it is fully faithful and is essentially surjective; so maybe you can say it's an analog of a bijection (I mean "fully faithful" means bijective on hom-sets after all.) But defining an "equivalence" that way is not right, I think, conceptually speaking. -- Taku (talk) 13:15, 21 November 2013 (UTC)
I see! You and I were making different points. I agree with you entirely on this.
However, I disagree with some of the changes you've made to the page. In order:
  • I don't think the case of homomorphisms should be distinguished in the lede as different or more important than any other kind of morphism. While the word "isomorphism" does come from algebra, the concept applies equally well to non-algebraic situations, and I think we are misleading our readers if we suggest otherwise.
  • The words "homomorphism", "function", and "mapping" are all mathematical jargon. The first two have formal definitions, and they don't apply in all cases of interest. While "mapping" feels loose and informal, it's just as much jargon as "homomorphism" and "function", and because it lacks a formal definition, it's less intelligible to the uninitiated (they will probably think of a map).
  • It's true that an isomorphism in a concrete category must be a bijection on underlying sets; but it's possible for the objects of a category to admit something we might call an underlying set without being a concrete category, and then an isomorphism might not be bijective. Consider, for example, the homotopy category of CW-complexes. The objects of this category are CW-complexes, and the morphisms are homotopy classes of continuous functions. This category is not concretizable (a theorem of Freyd), but each object admits an underlying set, because each CW-complex is a topological space with an underlying set. An isomorphism in this category is not necessarily bijective (for example, the inclusion of a 0-cell into a 1-cell is a homotopy equivalence, but not a bijection).
  • Isomorphisms are formalized using category theory. Not can be.
  • I don't know what you mean by "the development of category theory itself requires the notion of "isomorphism" to begin with".
I've made a third try at the lede. Pertinent differences from my last try are:
  • I didn't much like "relationship", and since there are objections, I'm trying "relation". Unfortunately "relation" has a technical meaning, but it's also a plain English language word, so hopefully this won't mislead anyone.
  • Isomorphisms are described as being relations that can be inverted.
  • There's a longer discussion of their relation (ha ha) with bijections.
Ozob (talk) 15:39, 21 November 2013 (UTC)
I like "relation" better, but I don't think any jagon-free description can be given without compromising clarity: it's like to describe African elephant without using the word elephant; that is, without familiarity with homomorphisms or functions, an isomorphism wouldn't make much sense. My general position is that the idealism "wikipedia articles are for everyone" doesn't apply to math articles: that idealism is not compatible with the reality :) Anyway, this is more of matter of the style and I don't insist on my version. The old lede was wrong and that has been fixed, at least. (By the way, I like the "CW-complex" example very much; it's nice to see that in the lede.) -- Taku (talk) 23:07, 21 November 2013 (UTC)
I think that the lead is moving in the direction of an understandable overview, though the category-theoretic definition at the end may still need work. What seems to be coming out is that an isomorphism is (as per Ozob) not merely a special type of homomorphism, such as "a homomorphism of which the inverse is a homomorphism" as I tried to define it, and that it does not have a rigorous definition without category theory (neither of which I am comfortable with, or understand properly). My point here is that the article needs a "definition" section, in which a more rigorous definition should be given. —Quondum 14:30, 22 November 2013 (UTC)
Sort of coming back to my point, what is an "isomorphism" then? There is a clean-cut category-theoretic definition (invertible morphism), which requires no further elaboration. If you can stomach the category of cats (i.e., if you believe in the universe), then this even applies to categories. But again this is the matter of the style. -- Taku (talk) 14:40, 22 November 2013 (UTC)
I agree that the article ought to have a definition section. Probably the second paragraph of the lede would fit better in a separate definition section. Perhaps it could be amplified to explain that often, isomorphisms of algebraic objects are defined to be bijective homomorphisms, but that this and the definition in terms of invertibility are equivalent for such objects (i.e., in the presence of one, the other is a theorem). Also, it would be helpful to have some history here, though I don't know whether anyone has studied the history of the concept of an isomorphism. Certainly the definition in terms of categories was not the original one (maybe van der Waerden's Algebra would be useful here?). Ozob (talk) 04:05, 23 November 2013 (UTC)

I just come to this article discussion. Although I know rather well what is an isomorphism, I cannot understand the new lede: The lack of precision of the formulation make impossible to know if what is described is what all mathematicians call isomorphism. If a mathematician cannot fully understand the lead, what the layman can understand? IMO, nothing. I strongly suggest to come back to the style of the previous lede and just correct the blatant errors and omissions. This could be something like:

In mathematics, an isomorphism is a homomorphism (or more generally a morphism) that admits an inverse. The interest of isomorphisms lies on the fact that two isomorphic objects may not be distinguished by using only the properties which are used to define morphisms, and may thus be identified as long as one consider only these properties and their consequences.
For most algebraic structures, an homomorphism is a an isomorphism if and only if it is bijective. If the source and the target of the morphism are the same object, an isomorphism is called an automorphism. In topology, when the morphisms are continuous functions, isomorphisms are also called homeomorphisms. In mathematical analysis, when the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms.
The notion of isomorphism has been formalized in category theory ...

Note also that the link to the other names of "isomorphism" partially answers to the question about history: Outside algebra, "isomorphisms" seem to have received other names, before the introduction of category theory. D.Lazard (talk) 10:49, 4 December 2013 (UTC)

I want to be on the record that I support Lazard's version. In fact, I have already said "any jagon-free description can be given without compromising clarity", and this is exactly the problem addressed by him. -- Taku (talk) 21:35, 5 December 2013 (UTC)