nLab
quotient category

Quotient category is an alternative name (used especially in 1960s and 1970s) for a result of a strict localization functor, that is, given a category $C$ and a class of morphisms $\Sigma \subset C$ the category ${\Sigma }^{-1}C$ equipped with a functor $Q_{\Sigma }:C\to {\Sigma }^{-1}C$ sending all morphisms in $\Sigma$ to isos and which has a strict universal property, that is, for every other functor $F:C\to A$ inverting all morphisms in $\Sigma$, there is a factorization $F=\tilde{F}\circ Q$.

Note that large part of a topos community and category community calls by localization those (non-strict) localization functors, for which the localization functor is left exact and admits a right adjoint. These people often use quotient functor when removing the “admits right adjoint” and “left exact” conditions..

Following the extensions of early work of Serre, the term (Serre’s) quotient category is especially used when the input is a thick subcategory $T$ of an abelian category $A$, instead of the class $\Sigma$. A nonempty full subcategory of an abelian category is thick in strong sense if it is closed under subquotients and extensions. Then one defines $T/A$ to have the same objects as $T$ and

$$(A/T)(X,Y):=\mathrm{colim}A(X\prime ,Y/Y\prime )$$(A/T)(X,Y) := colim A(X',Y/Y')

where the colimit runs through all subobjects $X\prime \subset X$, $Y\prime \subset Y$ such that $X/X\prime \in \mathrm{Ob}T$, $Y\prime \in \mathrm{Ob}T$. The quotient functor $Q:A\to A/T$ is obvious.

A thick subcategory (here always in strong sense) is said to be localizing if and $Q$ admits a right adjoint $A/T\to A$, often called the section functor. Every coreflective thick subcategory admits a section functor, and the converse holds if $A$ has injective envelopes. A thick subcategory $T\subset A$ is a coreflective iff $(T,F)$ is a torsion theory where

$$F:=\{X\in \mathrm{Ob}A\mid A(T,X)=0\}$$F := \{X\in Ob A\,|\,A(T,X) = 0\}

Yaron: The above definition of a quotient category appears to be different from that of CWM (p. 51 of the second edition), where a quotient category is obtained by identifying arrows. Perhaps there is a need for disambiguation?

Toby: See discussion on the Forum.