TITLE: What is the name of the following categorical property?
QUESTION [6 upvotes]: 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?).

REPLY [7 votes]: "every bijective morphism determines an isomorphism"
I think you mean that the forgetful functor reflects invertibility.
Let $\bf A$ be the "category of objects with structure" and their structure-preserving maps (homomorphisms), $\bf S$ the category of carriers (maybe sets and functions) and $U:{\bf A}\to{\bf S}$ the "forgetful" functor between them.  In fact, just let $U:{\bf A}\to{\bf S}$ be any functor you like.
Now let $f:X\to Y$ be any morphism of $\bf A$.  You are saying that, whenever $U f:U X\to U Y$ is an isomorphism (such as a bijection) then $f$ was already an isomorphism.
The forgetful functor from any category of algebras has this property, as more generally does the right adjoint of any monadic adjunction.   However, the underlying set functor from the usual category of topological spaces does not, because there are many different topologies that can be put on a set.