In Hatcher's Algebraic Topology textbook, he has been referring to "direct sum of exact sequences". As far as I know he's never defined this and I can't find what I'm looking for online.
Without a precise definition I can only guess what he means but I'm not entirely sure.
For example, he refers to taking the direct sum of free resolutions of abelian groups $H$ and $H'$ to obtain a free resolution for $H \oplus H'$.
What exactly does he mean here?
Just take the direct sum term-by term. Explicitly, suppose $$\dots\stackrel{f_3}\to X_2\stackrel{f_2}\to X_1\stackrel{f_1}\to X_0$$ and $$\dots\stackrel{g_3}\to Y_2\stackrel{g_2}\to Y_1\stackrel{g_1}\to Y_0$$ are two sequences of maps of (say) abelian groups. Their direct sum is then the sequence $$\dots\stackrel{h_3}\to X_2\oplus Y_2\stackrel{h_2}\to X_1\oplus Y_1\stackrel{h_1}\to X_0\oplus Y_0$$ where the maps $h_n:X_n\oplus Y_n\to X_{n-1}\oplus Y_{n-1}$ are defined by $h_n(x,y)=(f_n(x),g_n(y))$. It is straightforward to check that if the two original sequences were exact, then so is the direct sum.