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If we speak simply about isomorphism, we say that it is a bijective homomorphism. But I have read recently another definition, where they say, it is more general if we say, that it is $\psi$, a bijective homomorphism where $\psi^{-1}$ is a bijective homomorphism too.

They say the reason for this is, that it is not by all algebraic structures the case, that by a bijective homomorphism, the inverse function is necessarily a bijective homomorphism.

Could you name me an example, where it is not the case? Thanks!

user3435407
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    I'll post an answer later, but one (non-algebraic) example is given by continuous bijective maps from one topological space onto another, where the source and target spaces have different topologies. For something more algebraic, we can consider maps of varieties. However it is true that for groups, rings, and algebras all bijective homomorphisms are isomorphisms. – Josh Aug 07 '15 at 18:18
  • Ok, thanks, I will wait for the concrete example! – user3435407 Aug 07 '15 at 18:20
  • Take $f$ from $[0,2\pi)$ to the unit circle in $\Bbb R^2$ defined by $f(t)=(\cos(t),\sin(t))$. –  Aug 07 '15 at 19:08
  • @Zircht: Sorry, how do you mean? f(pi/2) = (0,1) ; f(pi) = (-1,0) ; f(3pi/2) = (0, -1) thus its not a homomorphism as (0,1) + (-1,0) is not equal to (0,-1). Or do I misunderstand something? – user3435407 Aug 07 '15 at 19:47
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    The "homomorphisms" in the category of topological spaces are the continuous functions, and they do not behave like homomorphisms in groups, rings,... See here. –  Aug 07 '15 at 20:36
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    @JoshChen There are examples of continuous bijections between spaces with the same topology that are not homeomorphisms. For example here and here. –  Aug 11 '15 at 05:56

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The generalization is a lot better suited to category theory. There we define that a homomorphism $\phi\colon A\to B$ is an isomorphism if there exists a homomorphism $\psi\colon B\to A$ such that $\psi\circ \phi=\operatorname{id}_A$ and $\phi\circ \psi=\operatorname{id}_B$.

First of all, to investigate bijectivity, $\phi$ needs to be a map between sets in the first place. In general categories, not all objects are sets with additional structure, and homomorphisms maps respecting the structure. The standard example is the category of homotopy classes of topological spaces.

Secondly, in certain categories for which objects are sets with structure it is still not true that a homomorphism that is bijective as a set-map is an isomorphism. The standard example is the category of topological spaces.

Hagen von Eitzen
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  • As I don't know topological spaces, I can't really imagine this yet, but I will check it out. However, if you have a very concrete short example in mind, please add it to your answer! Thanks! – user3435407 Aug 07 '15 at 18:36