definition of chain homotopy

Question

We can define the notion chain homotopy in the category of chain of $R-$modules. But do we have any definition of chain homotopy in any abelian category?
Analogously I can define the notion of chain map in Abelian category (map between two chain making the ladder commute) but it seems if I define chain homotopy in the same manner as I do for chain of $R-$modules then it is difficult to check if two chain map are homotopic then they induce the same map in homology.

Answer 1

As implied in the question and noted in the comments, the module version of the definition of chain homotopy extends without change to any abelian category. To show that if $f$ and $g$ are chain homotopic, then they induce the same map on homology, we first have to define the induced map. Since homology is the quotient $\ker d / \textrm{im}~ d$, the map induced by $f$ is defined by restricting to $\ker d$ and the noting that modding out by $\textrm{im}~d$ results in a well-defined map. If $f-g = sd + ds$, then when restricting to $\ker d$ we get $f-g=ds$. This means that the image of $f-g$ is in $\textrm{im}~d$, and so is zero in homology.