Talk:Isomorphism

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.