Define the fact that $Hom_{C}(X,Y)$ is bijective for category $C.$

Question

In completion to my question here:

Proving that $f^{-1} \in \operatorname{Hom}(Y,X).$

I know that: If $X,Y$ are groups and $f\in \operatorname{Hom}(X,Y)$ is bijective, then $f^{-1} \in \operatorname{Hom}(Y,X).$

My question is:

Why this statement is not correct in other categories than groups? could anyone help me answer this question, please?

In helping me answering my previous question @Tsemo in the previous question asked me that question:

how we define the fact that $Hom_{C}(X,Y)$ is bijective for category $C,$could you please help me in answering that question? I have no clear definition in my mind.

EDIT:

My confusion arises from the definition of Isomorphism that my professor gave to us, he said:

$f\in \operatorname{Hom}(X,Y)$ is as isomorphism if it is bijective and $f^{-1} \in \operatorname{Hom}(Y,X).$

He added that: in the category of groups $f\in \operatorname{Hom}(X,Y)$ is as isomorphism if it is bijective only.

This what confuses me, because I used to know that Isomorphism means homomorphism and bijection.

Answer 1

In a category $\mathcal C$, a morphism $f:X\to Y$ is called an isomorphism if there is a morphism $g:Y\to X$ such that $g\circ f = 1_X$ and $f\circ g = 1_Y$.

If your category is concrete, i.e.

• objects are sets with extra stuff (e.g. groups, spaces),
• morphisms are stuff-preserving functions (e.g. homomorphisms, continuous functions),

then, unwrapping the definition, a morphism $f:X\to Y$ is an isomorphism iff it has an inverse ($f^{-1}:Y\to X$) preserving the extra stuff ($f\in\hom(Y,X)$, i.e. not just a bijection).

Your teacher proved the following:

Fact. In the category of groups and homomorphisms between them, a (homo)morphism $f:X\to Y$ is an isomorphism iff its a bijection. In other words, the set-theoretic inverse $f^{-1}:Y\to X$ is already a homomorphism.

This does not occur in general:

Fact. In the category of spaces and continuous functions, there are bijections which are not homeomorphisms. For instance, if $X$ has nonequivalent topologies $\mathcal O_1, \mathcal O_2$ then the identity $id:(X,\mathcal O_1)\to (X,\mathcal O_2)$ is a bijection, but not an homeomorphism. Example.

Fact. In the category of smooth manifodls and smooth functions, there are bijections which are not diffeomorphisms. John Douma's $f(x) = x^3$ is a counterexample.