TITLE: Why chain homotopy when there is no topology in the background?
QUESTION [21 upvotes]: Given two morphisms between chain complexes  $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow D_{n+1}$ such that $f_n-g_n= \partial_D \psi_n+\psi_{n-1}\partial_C$. I can motivate this definition only when the chain complexes are associated to some topological space. For example if $C_\bullet$ and $D_\bullet$ are simplicial chain complexes, then for a simplex $\sigma$, a homotopy between $f(\sigma)$ and $g(\sigma)$ is something like $\psi(\sigma)\approx\sigma\times [0,1]$, whose boundary is $f(\sigma)-g(\sigma)-\psi(\partial\sigma)$.
But chain homotopy also features in contexts where there is no topological space lurking in the background. Examples are chain homotopy of complexes of graphs (e.g. Conant-Schneiderman-Teichner) or chain homotopy of complexes in Khovanov homology. In such contexts, the motivation outlined in the previous paragraph makes no sense, with "the boundary of a cylinder being the top, bottom, and sides", because there's no such thing as a "cylinder". Thus, surely, the "cylinder motivation" isn't the most fundamental reason that chain homotopy is "the right" relation to study on chain complexes. It's an embarassing question, but:

What is the fundamental (algebraic) reason that chain homotopy is relevant when studying chain complexes?

REPLY [5 votes]: This was a comment, but I guess it was not so clear, although it is hiding in some of the above answers:
There is an object in $Ch^+(R)$ that behaves like an interval. Think of the simplicial interval, and you get a chain complex $I_*$ over $R$ with $I_0=Re_0 \oplus Re_1$, $I_1=Rf_0$ and all other groups 0, with differential $d(f_0)=e_1−e_0$. This plays the role of the unit interval. Now a chain homotopy of two chain maps $f,g:A_∗ \to B_∗$ really is $H:A_∗ \otimes I_* \to B_*$ where the tensor product takes place in R chain complexes. I am not sure how well this object behaves with respect to the Dold-Kan maps, but it certainly is the normalized chains on $\Delta^1$. You could similarly look at maps of $R$ complexes from $I_*$ into $Hom_R(A_*,B_*)$.
With this in hand you can form cylinder objects and cofiber sequences in the "obvious" way, and things start looking a bit more geometric.