If we take a quotient of a (say, abelian) category by subcategory, this can be understood in a "naive" and "homotopy" sense. Is it written somewhere what a homotopy quotient category is, and whether some analogue of the "connecting homomorphism" exists?

(Trying to rephrase an earlier question)

In topology, a continuous map $f: X \to Y$ has a ``homotopy cofiber" $Cf$ included in a cofiber sequence $$ X \to Y \to Cf \to \Sigma X \to \Sigma Y \to \Sigma CF \to \ldots $$ when $f$ is a cofibration, the cofiber is homotopy equivalent to $X/Y$.

Now, if $Y$ is a category, say, of modules over a ring $R$, and $f$ a functor from another category (e.g. embedding of projective modules into all modules), is there a similar cofiber category and a cofiber sequence?