I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies.
The definitions I'm using are:
Let $f,g : X \to Y$ be continuous functions between topological spaces. A homotopy from $f$ to $g$ is a continuous map $H : X \times [0,1] \to Y$ such that $H(\, \cdot\, , 0) = f$ and $H(\, \cdot\, , 1) = g$.
Let $f_{\bullet}, g_{\bullet} : A_{\bullet} \to B_{\bullet}$ be chain maps between chain complexes $(A, d_A)$ and $(B,d_B)$. A chain homotopy from $f$ to $g$ is a sequence of maps $h_n : A_n \to B_{n+1}$ such that $f_n-g_n=d_Bh_n+h_{n-1}d_A$.
I'm aware of the properties of a chain homotopy and how they are similar to those of a homotopy, but I still find the definition quite opaque and the notion quite hard to picture $-$ it would help me a lot if I could think of a chain homotopy in a similar way to how I think of a homotopy.
Or, to make my question a bit less vague, I'd like to know:
- What is the rationale behind the definition of a chain homotopy?
- Is there a fundamental similarity between a chain homotopy and a homotopy, beyond their further consequences?
(General waffle would also be appreciated; I'd really like to develop a good understanding.)