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 CC and a class of morphisms ΣC\Sigma\subset C the category Σ 1C\Sigma^{-1} C equipped with a functor Q Σ:CΣ 1CQ_\Sigma: C\to \Sigma^{-1} C sending all morphisms in Σ\Sigma to isos and which has a strict universal property, that is, for every other functor F:CAF: C\to A inverting all morphisms in Σ\Sigma, there is a factorization F=F˜QF = \tilde{F}\circ Q.

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 TT of an abelian category AA, instead of the class Σ\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 A/TA/T to have the same objects as TT and

(A/T)(X,Y):=colimA(X,Y/Y) (A/T)(X,Y) := colim A(X',Y/Y')

where the colimit runs through all subobjects XXX'\subset X, YYY'\subset Y such that X/XObTX/X' \in Ob T, YObTY'\in Ob T. The quotient functor Q:AA/TQ: A\to A/T is obvious.

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

F:={XObA|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.

Revised on March 7, 2014 05:23:41 by Zoran Škoda (