I've been doing some homological algebra and some work on showing some extensions are equivalent, and a thought just came to me, which is that I didn't know how to write down what the analogue of the trivial extension for higher Ext (i.e. n>1). I felt the obvious thing was to write down two direct sums in the middle, but if we take our maps to be inclusion and projection as in the short exact sequence in the n=2 case, our sequence isn't exact and not an extension (at least not in general).
So I was wondering what the analogue for a split extension is, as I have some interest in higher Ext groups and those extensions in my project?