  Suppose one has a map of split short exact sequences in a category of
modules, or more generally, in any abelian category. Do the short exact
sequences split compatibly, i.e., does there exist a splitting of each short
exact sequence which commutes with the map of short exact sequences? The answer
is sometimes yes and sometimes no. We define and prove basic properties of an
obstruction group to the existence of compatible splittings.

Representation Theory;K-Theory and Homology