TITLE: Alternate Definition of Quotient Category
QUESTION [5 upvotes]: A quotient category is usually (at least from most sources I had seen) defined by something like $\mathcal C/\sim$ for some equivalence relation $\sim$, for example in this post. I am wondering if one can define a quotient of categories of the form $\mathcal B/\mathcal A$ for some subcategory $\mathcal B\subseteq \mathcal A$? I think this may require the categories (both $\mathcal A$ and $\mathcal B$) to be additive (or some other properties?) to replace the operation defined on the equivalence relation, but I haven't found any reference that describes such construction. Any help is appreciated!
Edit: Consider $\mathcal A$ and $\mathcal B$ to be additive categories where $\mathcal A$ is a subcategory of $\mathcal B$. I am wondering if I can construct a "quotient category" $\mathcal B/\mathcal A$ with $F:\mathcal B\to \mathcal B/\mathcal A$ (such that $F\circ \mathbf{inc}: \mathcal A\to \mathcal B/\mathcal A$ is the zero map) satisfying the following universal property: for every category $\mathcal C$ with a functor $G:\mathcal B\to \mathcal C$ that satisfies $G(\mathcal A) =0$, there is a unique functor $\bar G:\mathcal B/\mathcal A\to \mathcal C$ such that $\bar G\circ F =G$.

REPLY [5 votes]: If $\mathcal{A}$ is an additive subcategory of an additive category $\mathcal{B}$, then there is a simple notion of "quotient category" $\mathcal{B}/\mathcal{A}$ that is often used.
It can be regarded as the special case of the quotient category $\mathcal{B}/\!\!\sim$  where, for a pair of maps $f,g:X\to Y$, $f\sim g$ iff $f-g$ factors through an object of $\mathcal{A}$.
This has the universal property that the quotient functor $Q:\mathcal{B}\to\mathcal{B}/\mathcal{A}$ is the universal additive functor from $\mathcal{B}$ to an additive category such that $Q(\mathcal{A})=0$ (i.e., like the universal property stated in the edit to the question, but with "additive" inserted everywhere).
This construction is used quite a bit in the representation theory of finite dimensional algebras. For example, the "stable module category" is the quotient of the module category by the subcategory consisting of projective modules.