This isn't quite the question you asked, but does address the notion of ''bijective'' morphisms in categories, so I hope you'll forgive this digression.

The examples you've mentioned - **Set**, **Gp**, **Top** - are all *concrete*, meaning they are equipped with a forgetful functor *U* to **Set**. We say a morphism *f* in a concrete category *C* is *injective* if its image *Uf* is injective, i.e., monic in the category **Set**. Dually, *f* is *surjective* if *Uf* is surjective. One usually thinks of concrete categories as "sets with structure", so these definitions coincide with the common use of such terminology: e.g., we call a map of spaces surjective when the underlying map of sets is.

So we have four adjectives to use for arrows in *C*: monic, epic, injective, surjective. It's an easy exercise to see that all injections are monic and all surjections are epic. The converse is not true in general, but finding examples of monos that aren't injective and epis that aren't surjective can be tricky, and here's why.

Often, particularly in ''algebraic'' examples, the functor *U* : *C* → **Set** has a left adjoint *F*. When this is the case, it is an easy exercise to see that every mono must be injective. Dually, if *U* has a right adjoint, then every epi is surjective. So for example, the forgetful functor *U* : **Top** → **Set** has both adjoints, and hence for spaces the notions injective/surjective and monic/epic coincide, at which point Tom's post answers your question.

Here are some examples of concrete categories where these concepts differ, all of which can be found in Francis Borceux's *Handbook of Categorical Algebra* (I think). In the category of divisible abelian groups, the quotient map $\mathbb{Q} \rightarrow \mathbb{Q}/\mathbb{Z}$ is monic, though it's clearly not injective. In the category of monoids, the inclusion $\mathbb{N} \rightarrow \mathbb{Z}$ is epic, though not surjective. In the category of Hausdorff spaces, the epis are continuous functions with dense image, so also need not be surjective.