Split epi and split mono without iso I'm asking for examples of interesting categories in which there exist non-isomorphic objects $X$ and $Y$, a split monomorphism $f : X \to Y$, and a split epimorphism $g : X \to Y$. Spelled out, there should exist maps $f : X \leftrightarrow Y : f'$ such that $f'f = \mathrm{id}_{X}$ and maps $g : X \leftrightarrow Y : g'$ such that $gg' = \mathrm{id}_{Y}$ such that there is no pair of maps $h : X \leftrightarrow Y : h'$ satisfying $h'h = \mathrm{id}_{X}$ and $hh' = \mathrm{id}_{Y}$.
My professor found a seemingly relevant exercise from Rowen's "Graduate Algebra: Noncommutative view" suggesting this may occur in R-Mod, but I haven't got the book on hand and remember having trouble understanding the exercise anyways. Additionally, he more specifically asked if this can happen in Top.
 A: The situation can also be described by two non-isomorphic objects $X,Y$ which admit split monomorphisms $X \to Y$ and $Y \to X$ in both directions.
In fact, this happens in the category of abelian groups. See here and here for rather complicated examples. It was already asked for more easy examples here.
A: In topological terms, you are asking for two non-homeomorphic spaces, each 
of which is homeomorphic to a retract of the other.
You can easily find examples of such spaces by stringing together an infinite chain of copies of two non-homeomorphic spaces. Taking a circle and a segment as an example, we get subspaces of
the plane shaped like o--o--o--o--o--o--o--.... and --o--o--o--o--o--o--o....
Each of these spaces can be obtained from the other by mapping the first
link in the chain to its connecting point, which is a retraction.
Note that the fact that the chains are not homeomorphic cannot be deduced from the fact that the segment and the circle are not homeomorphic, but
we can easily see that every point of the first space is a local cut point,
which is not true for the second one.
Countless examples can be constructed in similar ways. For a compact
metrizable example you might consider the one point compactification of
the spaces above.
