In order theory the term Galois connection (due to Ore 44, who spelled it “connexion”, the French spelling) can mean both: “adjunction between posets” and “dual adjunction between posets”; the former notion is sometimes called “monotone Galois connection” and the latter “antitone Galois connection”. In this article the term “Galois connection” shall mean “dual adjunction between posets”.
The term Galois correspondence is also in use. For some authors it is synonymous to “Galois connection”, others reserve it for its restriction to its fixed points, where it becomes an adjoint equivalence.
The example that gives the concept its name is the relation between subgroups and subfields in Galois theory (see below), but adjunctions between posets, hence Galois connections, appear also in many other and entirely different contexts, see further below.
Given posets and , a Galois connection between and is a pair of order-reversing functions and such that and for all , .
A Galois correspondence is a Galois connection which is an adjoint equivalence (so and for all , ).
Any Galois connection , induces a Galois correspondence between and , given by the composites and .
For any of the form , we have and also where the inequality follows from and antitonicity of . Hence for all . Similarly for all .
The Galois theory normally taught in graduate-level algebra courses (and based on the work of Évariste Galois) involves a Galois connection between the intermediate fields of a Galois extension and the subgroups of the corresponding Galois group.
Frequently Galois connections between collections of subsets (power sets) arise where is “the set of all standing in some relation to every ” and dually is “the set of all standing in some relation to every .”
Examples of this class of Galois connections include the following
(Zariski topology) The closed subsets in the Zariski topology on affine space or on the set of maximal ideals of a polynomial ring, which may be understood as the fixed points of a Galois connection between polynomials and affine space/maximal ideal. This is discussed at Zariski topology – In terms of Galois connections.
(orthogonality classes) Given a category , then on the poset of sub-classes of morphisms the operations of forming left and right classes with orthogonality lifting property constitute a Galois connection.
In fact all Galois connections between power sets arise this way, see below.
We now spell out in detail the Galois connections induced from a relation:
(Galois connection induced from a relation)
Consider two sets and a relation
Define two functions between their power sets , as follows. (In the following we write to abbreviate the formula .)
Define
by
Define
by
The construction in def. has the following properties:
and are contravariant order-preserving in that
if , then ;
if , then
The adjunction law holds:
which we denote by writing
both as well as take unions to intersections.
Regarding the first point: the larger is, the more conditions that are placed on in order to belong to , and so the smaller will be.
Regarding the second point: This is because both these conditions are equivalent to the condition .
Regarding the third point: Observe that in a poset such as , we have that iff for all , iff (this is the Yoneda lemma applied to posets). It follows that
and we conclude by the Yoneda lemma.
(closure operators from Galois connection)
Given a Galois connection as in def. , consider the composites
and
These satisfy:
For all then .
For all then .
is idempotent and covariant.
and
For all then .
For all then .
is idempotent and covariant.
This is summarized by saying that and are closure operators (idempotent monads).
The first statement is immediate from the adjunction law (prop. ).
Regarding the second statement: This holds because applied to sets of the form , we see . But applying the contravariant map to the inclusion , we also have .
This directly implies that the function . is idempotent, hence the third statement.
The argument for is directly analogous.
(closed elements)
Given a Galois connection induced from a relation as in def. , then
is called closed if ;
the closure of is
and similarly
is called closed if ;
the closure of is .
It follows from the properties of closure operators, hence form prop. :
(fixed points of a Galois connection)
Given a Galois connection induced from a relation as in def. , then
the closed elements of are precisely those in the image of ;
the closed elements of are precisely those in the image of .
We says these are the fixed points of the Galois connection. Therefore the restriction of the Galois connection
to these fixed points yields an equivalence
now called a Galois correspondence.
Given a Galois connection induced from a relation as in def. , then the sets of closed elements according to def. are closed under forming intersections.
If is a collection of elements closed under the operator , then by the first item in prop. it is automatic that , so it suffices to prove the reverse inclusion. But since for all and is covariant and is closed, we have for all , and follows.
Every Galois connection between full power sets,
is of the form in def. above: there is some binary relation from to such that
Indeed, define by stipulating that is true if and only if . Because is a left adjoint, it takes colimits in (in this case, unions) to colimits in , which are intersections in . Since every in is a union of singletons , this gives
which is another way of writing the formula for given above. We observe that
if and only if
(now viewing extensionally in terms of subsets). This last symmetrical expression in and means
which means we have a Galois connection between and under this definition; since is uniquely determined by the presence of a Galois connection with , we conclude that all Galois connections between power sets arise in this way, via a relation between and .
The concept is due to
Introduction is in
See also
Last revised on April 17, 2024 at 16:49:37. See the history of this page for a list of all contributions to it.