Suppose I have a chain complexes
$$\dots \longrightarrow C_2 \stackrel{\partial_2}\longrightarrow C_1 \stackrel{\partial_1}\longrightarrow C_0\stackrel{\partial_0}\longrightarrow 0$$
$$\dots \longrightarrow D_2 \stackrel{\partial_2}\longrightarrow D_1 \stackrel{\partial_1}\longrightarrow D_0\stackrel{\partial_0}\longrightarrow 0$$
of abelian groups. When $f,g: C \to D$ are chain maps these are said to be chain homotopic when there is a collection of group homomorphisms $h_n:C_n \to D_{n+1}$ such that $f_n -g_n = \partial_{n+1} h_n + h_{n-1}\partial_n$ for all $n$.
How should I interpret this for $n=0$? How do we define $h_{-1}$? Do we just consider it to be $0$?
Background question: I'm reading Hatcher's book "Algebraic topology".
Best Answer
The problem is that Hatcher does not properly define the concepts of chain maps and chain homotopies. He introduces them "en passant" when considering induced maps $f_\sharp :C_n(X) \to C_n(Y)$ and prism operators $P : C_n(X) \to C_{n+1}(Y)$ and leaves the technical details of a general definition to the reader.
It is clear that a chain map $\mathbf f : \mathbf C \to \mathbf D$ between chain complexes $\mathbf C, \mathbf D$ is a collection of group homomorphisms $f_n : C_n \to D_n$, $n \ge 0$, such that $\partial_{D,n} f_{n} = f_{n-1} \partial_{C,n}$ for all $n \ge 0$.
According to Hatcher on p. 113, a chain homotopy $\mathbf P : \mathbf C \to \mathbf D$ between chain maps $\mathbf f, \mathbf g : \mathbf C \to \mathbf D$ is a collection of group homomorphisms $P_n:C_n \to D_{n+1}$ such that $f_n - g_n = \partial_{n+1} P_n + P_{n-1}\partial_n$ for all $n$; but he is not specific about $n$. Clearly we need it for all $n \ge 0$ and this requires $P_{-1} : C_{-1} \to D_0$. Though not explicitly stated, it should be clear that Hatcher understands $C_{-1} = 0$ since this is the range of $\partial_0$. And since the zero map is the unique map $0 \to D_0$, we automatically have $P_{-1} = 0 : 0 \to D_0$.
Thus, formally, $\mathbf P$ is a collection of group homomorphisms $P_n:C_n \to D_{n+1}$, $n \ge -1$, such that $f_n - g_n = \partial_{n+1} P_n + P_{n-1}\partial_n$ for all $n \ge 0$.
A more general approach would be to define a chain complex as a collection of abelian groups $C_n$ and group homomorphisms $\partial_n : C_n \to C_{n-1}$, $n \in \mathbb Z$, such that $\partial_n \partial_{n+1} = 0$ for all $n$. It should now be obvious how to define chain maps and chain homotopies for such general chain complexes. Then we can say that the chain complexes occuring in simplicial and singular homology are "nonnegative" which means that $C_n = 0$ for $n < 0$.