nLab quotient category

Idea

The term quotient category has more than one meaning.

1. Sometimes it refers to the Serre quotient category by a thick subcategory. In this case it is not the category that is a quotient, but its objects are quotients. This is an instance of a localization functor, often a strict localization functor.

2. Other times, it refers to an actual quotient of some kind in Cat. There are different possible meanings of this, but one of the more useful is the quotient of the homsets of a category by equivalence relations compatible with composition.

References

• Saunders MacLane, §II.8 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]