View source for Localization in categories

A construction associated with special radical subcategories; it first appeared in Abelian categories in the description of the so-called Grothendieck categories in terms of categories of modules over rings (cf. [[Grothendieck category|Grothendieck category]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602901.png" /> be an [[Abelian category|Abelian category]]. A full subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602902.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602903.png" /> is said to be thick if it contains all subobjects and quotient objects of its objects and is closed with respect to extension, that is, in an exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602904.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602905.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602906.png" />. The quotient category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602907.png" /> is constructed in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602908.png" /> be a subobject of the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l0602909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029011.png" /> are projections, and suppose that the square

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029012.png" /></td> </tr></table>

is a pushout. The subobject <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029013.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029014.png" />-subobject if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029015.png" />. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029016.png" />-subobjects are equivalent if they contain an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029017.png" />-subobject. By definition, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029018.png" /> consists of equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029019.png" />-subobjects of the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029020.png" />. Ordinary composition of binary relations in an Abelian category is compatible with the equivalence introduced, which makes it possible to define the quotient category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029021.png" />. This quotient category turns out to be an Abelian category. An exact functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029022.png" /> can be defined by associating with each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029023.png" /> its graph in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029024.png" />. A thick subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029025.png" /> is called a localizing subcategory if the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029026.png" /> has a full and faithful right adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029027.png" />. A localizing subcategory is always the subcategory of all radical objects for some hereditary radical.

In the category of Abelian groups the subcategory of all torsion groups is a localizing subcategory. The quotient category of any category of modules with respect to a localizing subcategory is a Grothendieck category. Conversely, any Grothendieck category is equivalent to a quotient category of a suitable category of modules.

The concept of a localizing subcategory can also be defined for non-Abelian categories [[#References|[3]]]. However, in the non-Abelian case there usually are few such subcategories. For example, in the category of associative rings there are only the two trivial localizing subcategories, namely the whole category and the full subcategory of it that contains only trivial rings.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bucur,   A. Deleanu,   "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Gabriel,   "Des catégories abéliennes"  ''Bull. Soc. Math. France'' , '''90'''  (1962)  pp. 323–448</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.G. Shul'geifer,   "Localizations and strongly hereditary strict radicals in categories"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1969)  pp. 299–331  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 271–301</TD></TR></table>



====Comments====
The term  "dense subcategory"  is sometimes used in place of  "thick subcategory" ; but  "dense subcategorydense subcategory"  has another, conflicting, meaning. The term  "Serre class18E40Serre class"  is also used for this concept, particularly by algebraic topologists (cf. [[#References|[a1]]]). A thick subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029028.png" /> is a localizing subcategory if and only if: 1) every object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029029.png" /> has a largest subobject in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029030.png" />; and 2) given an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029031.png" /> for which this greatest subobject is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029032.png" />, there exists a monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029034.png" /> has the property that each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029035.png" /> in the quotient category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029036.png" /> derives from a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029038.png" /> (see [[#References|[a2]]]). The quotient category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029039.png" /> may also be defined as a category of fractions (cf. [[#References|[a3]]]), in which one formally adjoins inverses for these morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029040.png" /> which are  "isomorphisms modulo A'"  in the sense that their kernels and cokernels both belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029041.png" />. The class of all isomorphisms modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029042.png" /> admits both a calculus of left fractions and a calculus of right fractions; this corresponds to the fact that the canonical functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029043.png" /> is exact.

Localizations of module categories have been extensively used in non-commutative ring theory and in the attempts to develop a  "non-commutative algebraic geometry" ; see [[#References|[a4]]], [[#References|[a5]]].

In the context of non-Abelian categories, a localization of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029044.png" /> is generally taken to mean a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029045.png" /> which is exact (i.e. preserves finite limits and colimits) and has a full and faithful right adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029046.png" />; equivalently, the localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029047.png" /> may be identified with the (full, reflective) subcategories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029048.png" /> which are the images of these right adjoints. Such localizations cannot be classified by localizing subcategories, as in the Abelian case, but various techniques have been developed for handling them in many particular cases of interest. For example, the  "little Giraud theoremlittle Giraud theorem"  classifies localizations of a functor category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029049.png" /> in terms of Grothendieck topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029050.png" /> [[#References|[a6]]]; more generally, the localizations of an arbitrary (elementary) [[Topos|topos]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029051.png" /> are classified by Lawvere–Tierney topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060290/l06029052.png" /> [[#References|[a7]]]. (See also [[#References|[a8]]] for a topos-theoretic analogue of the notion of Serre class.) For localizations of algebraic categories (and more generally of locally presentable categories), see [[#References|[a9]]] and [[#References|[a10]]]. [[#References|[a11]]] studies the ordered set of localizations of a given category; it turns out that under reasonable hypotheses this set is a [[Complete lattice|complete lattice]] satisfying an infinite distributive law.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-.P. Serre,   "Groupes d'homotopie et classes de groupes abéliens"  ''Ann. of Math.'' , '''58''' :  2  (1953)  pp. 258–294</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Popescu,   "Abelian categories with applications to rings and modules" , Acad. Press  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Gabriel,   M. Zisman,   "Categories of fractions and homotopy theory" , Springer  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.S. Golan,   "Localization of noncommutative rings" , M. Dekker  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.S. Golan,   "Torsion theories" , Longman  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Artin (ed.)  A. Grothendieck (ed.)  J.-L. Verdier (ed.) , ''Théorie des topos (SGA 4, vol. I)'' , ''Lect. notes in math.'' , '''269''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.T. Johnstone,   "Topos theory" , Acad. Press  (1977)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Adelman,   P.T. Johnstone,   "Serre classes for toposes"  ''Bull. Austral. Math. Soc.'' , '''25'''  (1982)  pp. 103–115</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  F. Borceux,   G. van den Bossche,   "Algebra in a localic topos with applications to ring theory" , ''Lect. notes in math.'' , '''1038''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  F. Borceux,   B. Veit,   "On the left exactness of orthogonal reflections"  ''J. Pure Appl. Alg.'' , '''49'''  (1987)  pp. 33–42</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  F. Borceux,   G.M. Kelly,   "On locales of localizations"  ''J. Pure Appl. Alg.'' , '''46'''  (1987)  pp. 1–34</TD></TR></table>