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 and a class of morphisms the category equipped with a functor sending all morphisms in to isos and which has a strict universal property, that is, for every other functor inverting all morphisms in , there is a factorization .
Note that large part of a topos community and category community calls by localization those (possibly 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 an early work of Serre by Grothendieck and Gabriel, the term Serre quotient category or simply a quotient category is especially used when the input is a thick subcategory of an abelian category , instead of the class . A nonempty full subcategory of an abelian category is thick in strong sense if it is closed under subquotients and extensions. Then one defines to have the same objects as and
where the colimit runs through all subobjects , such that , . The quotient functor is obvious.
A thick subcategory (here always in strong sense) is said to be localizing if and admits a right adjoint , often called the section functor. Every coreflective thick subcategory admits a section functor, and the converse holds if has injective envelopes. A thick subcategory is a coreflective iff is a torsion theory where
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.