Baer sum

To add two extensions of $M$ by $N$, take their direct sum, form a pullback in a category along the diagonal $M\to M\oplus M$, and form a pushout in a category along addition $N\oplus N\to N$. This produces another extension of $M$ by $N$ and defines addition in ${\mathrm{Ext}}_R^1(M,N)$. In a quiver representation, vertexwise splittings express the two extensions by off-diagonal arrow matrices, and the Baer sum adds those matrices modulo the extension complex of quiver representations coboundaries.