So I am a little confused with the first line of the setup of the Mayer-Vietoris sequence as in Hatcher page 149 (page 158 of the PDF). The setup is as follows. Let $X$ be a space and let $A$ and $B$ be subspaces whose interiors cover $X$. Denote by $C_{n}(A + B)$ the free group generated by singular $n$-simplices. Then Hatcher claims there is an exact sequence $$ 0 \longrightarrow C_{n}(A \cap B) \longrightarrow C_{n}(A) \oplus C_{n}(B) \longrightarrow C_{n}(A + B) \longrightarrow 0 , $$ where the injective map takes a $\sigma \mapsto (\sigma , -\sigma)$ and the surjective map takes $(\psi, \phi) \mapsto \psi + \phi$.
My question may be a stupid one, but aren't $C_{n}(A) \oplus C_{n}(B) $ and $C_{n}(A + B)$ always isomorphic? Indeed $C_{n}(A) \oplus C_{n}(B) $ is just a direct sum of copies of $\mathbb{Z}$ indexed by $n$-simplices living in $A$ and then direct summed with another direct sum of copies of $n$-simplices living in $B$. That makes one direct sum of copies of $\mathbb{Z}$ indexed by $n$-simplices living in either $A$ or $B$. But this is precisely the definition of $C_{n}(A + B)$ as well.
Is writing them differently just a notational convenience, or are there actually situations where they won't be the same? My background tends to be more categorical so I really like to think of isomorphic things as "the same" where possible.
What about the simplices which are both in $A$ and in $B$ ? They appear twice in the definition of $C_n(A)\oplus C_n(B)$. First as a summand in $C_n(A)$ but also as a summand in $C_n(B)$.
The point is, if $\sigma$ is such a simplex $(\sigma,-\sigma)$ is not $0$ in $C_n(A)\oplus C_n(B)$. But clearly, it maps to $0$ via the map $C_n(A)\oplus C_n(B)\rightarrow C_n(A+B)$ (which is $(\psi,\phi)\mapsto \psi + \phi)$ by the way).
You can now easily check that the short exact sequence is indeed exact.