Top does not satisfy the axiom of choice I am trying to think about the axiom of choice from a categorical point of view.
I found out in some book that one possible formalization of this axiom is obtained by stating: "a category $\mathcal{C}$ satisfies the axiom of choice if all epics in $\mathcal{C}$ are split".
In this way $\mathbf{Set}$ does satisfy the axiom of choice (and that is fine). The category of groups doesn't though! What about $\mathbf{Top}$, i.e. the category of topological spaces? Does a not split epimorphism exist in $\mathbf{Top}$?

Top indeed does not satisfy the axiom of choice. To show this it suffices to exhibit a fiber bundle without a section, and for example the unit tangent bundle of $S^2$ has this property by the hairy ball theorem. Various other kinds of examples are possible; see, for example, this blog post. 
