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 point of fact, much more is known in this case, right?).



What is the name of the following categorical property?
The comments on the question point out that it's not really well-posed: the property "bijective" isn't defined for morphisms of an arbitrary category. 


However, for maps between sets, "bijective" means "injective and surjective". A common way to interpret "injective" in an arbitrary category is "monic", and a common way to interpret "surjective" in an arbitrary category is "epic". So we might interpret "bijective" as "monic and epic".


Then JHS's question becomes: is there a name for categories in which every morphism that is both monic and epic is an isomorphism? And the answer is yes: [balanced](http://ncatlab.org/nlab/show/balanced+category).


It's not a particularly inspired choice of name, nor does it seem to be a particularly important concept. But the terminology is quite old and well-established, in its own small way.


Incidentally, you don't *have* to interpret "injective" and "surjective" in the ways suggested. You could, for instance, interpret "surjective" as "regular epic", and indeed that's often a sensible thing to do. But then the question becomes trivial, since any morphism that's both monic and regular epic is automatically an isomorphism.


