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). Let $ \mathfrak A $ be an Abelian category. A full subcategory $ {\mathfrak A ^ \prime } $ of $ \mathfrak A $ 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

$$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 , $$

$ B \in \mathop{\rm Ob} {\mathfrak A ^ \prime } $ if and only if $ A , C \in \mathop{\rm Ob} {\mathfrak A ^ \prime } $. The quotient category $ \mathfrak A / {\mathfrak A ^ \prime } $ is constructed in the following way. Let $ ( R , \mu ] $ be a subobject of the direct sum $ A \oplus B ( \pi _ {1} , \pi _ {2} ) $, where $ \pi _ {1} $ and $ \pi _ {2} $ are projections, and suppose that the square

$$ \begin{array}{rcl} R &\rightarrow ^ { {\pi _ 2} \mu } & B \\ size - 3 {\pi _ {1} \mu } \downarrow &{} &\downarrow size - 3 \beta \\ A & \mathop \rightarrow \limits _ \alpha & C \\ \end{array} $$

is a pushout. The subobject $ ( R , \mu ] $ is called an $ {\mathfrak A ^ \prime } $- subobject if $ \mathop{\rm Coker} \pi _ {1} \mu , \mathop{\rm Ker} \beta \in \mathop{\rm Ob} {\mathfrak A ^ \prime } $. Two $ {\mathfrak A ^ \prime } $- subobjects are equivalent if they contain an $ {\mathfrak A ^ \prime } $- subobject. By definition, the set $ H _ {\mathfrak A / {\mathfrak A ^ \prime } } ( A , B ) $ consists of equivalence classes of $ {\mathfrak A ^ \prime } $- subobjects of the direct sum $ A \oplus B $. Ordinary composition of binary relations in an Abelian category is compatible with the equivalence introduced, which makes it possible to define the quotient category $ \mathfrak A / {\mathfrak A ^ \prime } $. This quotient category turns out to be an Abelian category. An exact functor $ T : \mathfrak A \rightarrow \mathfrak A / {\mathfrak A ^ \prime } $ can be defined by associating with each morphism $ \alpha : A \rightarrow B $ its graph in $ A \oplus B $. A thick subcategory $ {\mathfrak A ^ \prime } $ is called a localizing subcategory if the functor $ T $ has a full and faithful right adjoint $ S : \mathfrak A / {\mathfrak A ^ \prime } \rightarrow \mathfrak A $. 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 [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

[1]	I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) MR0236236 Zbl 0197.29205
[2]	P. Gabriel, "Des catégories abéliennes" Bull. Soc. Math. France , 90 (1962) pp. 323–448 MR0232821 Zbl 0201.35602
[3]	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 Zbl 0197.29301

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. [a1]). A thick subcategory $ \mathfrak A ^ \prime $ is a localizing subcategory if and only if: 1) every object of $ \mathfrak A $ has a largest subobject in $ \mathfrak A ^ \prime $; and 2) given an object $ A $ for which this greatest subobject is $ 0 $, there exists a monomorphism $ A \rightarrow B $, where $ B $ has the property that each morphism $ C \rightarrow B $ in the quotient category $ \mathfrak A / {\mathfrak A ^ \prime } $ derives from a unique morphism $ C \rightarrow B $ in $ \mathfrak A $( see [a2]). The quotient category $ \mathfrak A / {\mathfrak A ^ \prime } $ may also be defined as a category of fractions (cf. [a3]), in which one formally adjoins inverses for these morphisms in $ \mathfrak A $ which are "isomorphisms modulo A'" in the sense that their kernels and cokernels both belong to $ \mathfrak A ^ \prime $. The class of all isomorphisms modulo $ \mathfrak A ^ \prime $ admits both a calculus of left fractions and a calculus of right fractions; this corresponds to the fact that the canonical functor $ T : \mathfrak A \rightarrow \mathfrak A / {\mathfrak A ^ \prime } $ 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 [a4], [a5].

In the context of non-Abelian categories, a localization of a category $ C $ is generally taken to mean a functor $ T : C \rightarrow D $ which is exact (i.e. preserves finite limits and colimits) and has a full and faithful right adjoint $ S $; equivalently, the localization of $ C $ may be identified with the (full, reflective) subcategories of $ C $ 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 $ [ C ^ {op} , \mathop{\rm Set} ] $ in terms of Grothendieck topologies on $ C $[a6]; more generally, the localizations of an arbitrary (elementary) topos $ E $ are classified by Lawvere–Tierney topologies in $ E $[a7]. (See also [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 [a9] and [a10]. [a11] studies the ordered set of localizations of a given category; it turns out that under reasonable hypotheses this set is a complete lattice satisfying an infinite distributive law.

References

[a1]	J.-.P. Serre, "Groupes d'homotopie et classes de groupes abéliens" Ann. of Math. , 58 : 2 (1953) pp. 258–294 MR0059548 Zbl 0052.19303
[a2]	N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973) MR0340375 Zbl 0271.18006
[a3]	P. Gabriel, M. Zisman, "Categories of fractions and homotopy theory" , Springer (1967) MR0353301
[a4]	J.S. Golan, "Localization of noncommutative rings" , M. Dekker (1975) MR0366961 Zbl 0302.16002
[a5]	J.S. Golan, "Torsion theories" , Longman (1986) MR0880019 MR0848356 Zbl 0657.16017 Zbl 0587.16017
[a6]	M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos (SGA 4, vol. I) , Lect. notes in math. , 269 , Springer (1972) MR1080173 MR1080172 MR0717602 MR0717586 MR0505104 MR0505101
[a7]	P.T. Johnstone, "Topos theory" , Acad. Press (1977) MR0470019 Zbl 0368.18001
[a8]	M. Adelman, P.T. Johnstone, "Serre classes for toposes" Bull. Austral. Math. Soc. , 25 (1982) pp. 103–115 MR0651424 Zbl 0459.18005
[a9]	F. Borceux, G. van den Bossche, "Algebra in a localic topos with applications to ring theory" , Lect. notes in math. , 1038 , Springer (1983) MR0724431 Zbl 0522.18001
[a10]	F. Borceux, B. Veit, "On the left exactness of orthogonal reflections" J. Pure Appl. Alg. , 49 (1987) pp. 33–42 MR0920514 Zbl 0632.18002
[a11]	F. Borceux, G.M. Kelly, "On locales of localizations" J. Pure Appl. Alg. , 46 (1987) pp. 1–34 MR0894389 Zbl 0614.18005