# 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 . 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.

How to Cite This Entry:
Localization in categories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Localization_in_categories&oldid=47688
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article