Localization of a category
Encyclopedia
In mathematics
, localization of a category consists of adding to a category
inverse morphism
s for some collection of morphisms, constraining them to become isomorphism
s. This is formally similar to the process of localization of a ring
; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to
homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.
Some significant examples follow.
introduced the idea of working in homotopy theory modulo
some class C of abelian group
s. This meant that groups A and B were treated as isomorphic, if for example A/B lay in C. Later Dennis Sullivan
had the bold idea instead of using the localization of a topological space
, which took effect on the underlying topological space
s.
s over a commutative ring
R, when R has Krull dimension
≥ 2, it can be useful to treat modules M and N as pseudo-isomorphic if M/N has support of codimension at least two. This idea is much used in Iwasawa theory
.
in homological algebra
proceeds by a step of adding inverses of quasi-isomorphism
s.
from an abelian variety
A to another one B is a surjective morphism with finite kernel
. Some theorems on abelian varieties require the idea of abelian variety up to isogeny for their convenient statement. For example, given an abelian subvariety A1 of A, there is another subvariety A2 of A such that
is isogenous to A (Poincaré's theorem: see for example Abelian Varieties by David Mumford
). To call this a direct sum
decomposition, we should work in the category of abelian varieties up to isogeny.
C and some class w of morphisms in the category, there is some question as to whether it is possible to form a localization w-1 C by inverting all the morphisms in w. The typical procedure for constructing the localization might result in a pair of objects with a proper class of morphisms between them. Avoiding such set-theoretic issues is one of the initial reasons for the development of the theory of model categories
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, localization of a category consists of adding to a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
inverse morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s for some collection of morphisms, constraining them to become isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s. This is formally similar to the process of localization of a ring
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.
Some significant examples follow.
Categorical definition
Let A be a category. A localization is an idempotent and coaugmented functor. A coaugmented functor is a pair (L,l) where L:A → A is an endofunctor and l:Id → L is a natural transformation from the identity functor to L (called the coaugmentation). A coaugmented functor is idempotent if, for every X, both maps L(lX),lL(X):L(X) → LL(X) are isomorphisms. It can be proven that in this case, both maps are equal.Serre's C-theory
SerreJean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
introduced the idea of working in homotopy theory modulo
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
some class C of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s. This meant that groups A and B were treated as isomorphic, if for example A/B lay in C. Later Dennis Sullivan
Dennis Sullivan
Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
had the bold idea instead of using the localization of a topological space
Localization of a topological space
In mathematics, well behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in ....
, which took effect on the underlying topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s.
Module theory
In the theory of moduleModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s over a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R, when R has Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
≥ 2, it can be useful to treat modules M and N as pseudo-isomorphic if M/N has support of codimension at least two. This idea is much used in Iwasawa theory
Iwasawa theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...
.
Derived categories
The construction of a derived categoryDerived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
in homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
proceeds by a step of adding inverses of quasi-isomorphism
Quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism is a morphism A → B of chain complexes such that the induced morphisms...
s.
Abelian varieties up to isogeny
An isogenyIsogeny
In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....
from an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
A to another one B is a surjective morphism with finite kernel
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
. Some theorems on abelian varieties require the idea of abelian variety up to isogeny for their convenient statement. For example, given an abelian subvariety A1 of A, there is another subvariety A2 of A such that
- A1 × A2
is isogenous to A (Poincaré's theorem: see for example Abelian Varieties by David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
). To call this a direct sum
Direct sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
decomposition, we should work in the category of abelian varieties up to isogeny.
Set-theoretic issues
In general, given a categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
C and some class w of morphisms in the category, there is some question as to whether it is possible to form a localization w-1 C by inverting all the morphisms in w. The typical procedure for constructing the localization might result in a pair of objects with a proper class of morphisms between them. Avoiding such set-theoretic issues is one of the initial reasons for the development of the theory of model categories
Model category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
.