Chain homotopy

Definition
Given two chain complexes $$A$$ and $$B$$, and chain maps $$f,g:A \to B$$, an algebraic homotopy or chain homotopy between $$f$$ and $$g$$ is an expression of $$f-g$$ as $$dk+kd$$ where $$k$$ is a collection of homomorphisms from $$A_n$$ to $$B_{n+1}$$ for every $$n$$.

Equivalently, two homomorphisms between chain complexes are in algebraic homotopy if they lie in the same coset of the group of homomorphisms of the form $$dk + kd$$.

If a chain homotopy exists between $$f$$ and $$g$$ we say that $$f,g$$ are chain-homotopic chain maps.

Facts
If $$f$$ and $$g$$ are two homotopic maps between topological spaces, then the induced maps between the singular complexes are in algebraic homotopy.