Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects?

Examples of categories of that type abound: Gr, Set, ...

An specific example of a category where the constraint doesn't hold is given by Top: a morphism there is a continuous function between topological spaces. Now, it is easy to give here a concrete example of a bijective morphism between [0,1) and $\mathbb{S}^{1}$ that fails to be an isomorphism of topological spaces. In fact, much more is know known in this case, right?

Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects?

Examples of categories of that type abound: Gr, Set, ...

An specific example of a category where the constraint doesn't hold is given by Top: a morphism there is a continuous function between topological spaces. Now, it is easy to give there a concrete example of a bijective morphism between [0,1) and $\mathbb{S}^{1}$ that fails to be an isomorphism (homeomorphism) in the cat..of topological spaces. In fact, much more is know in this case, right?

Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects?

Examples of categories of that type abound: Gr, Set, ...

An specific example of a category where the constraint doesn't hold is given by Top: a morphism there is a continuous function between topological spaces. Now, it is easy to give there a concrete example of a bijective morphism between [0,1) and $\mathbb{S}^{1}$ that fails to be an isomorphism (homeomorphism) in the cat...