Questions tagged [galois-connections]
58 questions
11
votes
0 answers
Galois connection arising from discussion of flat module and pure exact sequence.
There is somewhat of symmetry in the definition of flat module and pure short exact sequence which can be made precise as follows.
Let $\mathcal{R}$ be the class of all right $R$-modules, $\mathcal{S}$ be the class of all short exact sequences of…
Zhenhui Ding
- 161
9
votes
4 answers
The Galois connection between topological closure and topological interior
[Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.]
Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and closed sets $\mathcal{C}\subseteq 2^X$. Consider the…
Drew Armstrong
- 1,023
7
votes
2 answers
Galois theory, had it solved any major problems beside its original applications to classical problems?
Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible polygons) ?
I understand that Galois Theory had…
ben
- 191
7
votes
0 answers
Charecterization of Topologies via Galois Connections
Let $X$ and $Y$ be two sets and let $F: \mathcal{P}(X) \to \mathcal{P}(Y)$ and $G: \mathcal{P}(Y) \to \mathcal{P}(X)$ be two set functions that satisfy
$$F(A) \subseteq B \iff A \subseteq G(B). \tag{1}$$
That is, the pair $F$ and $G$ constitute a…
201p
- 877
5
votes
1 answer
Adjoint to multiplication in a GCD lattice
Consider the lattice on the nonzero natural numbers where the meet $a \wedge b $ is defined to be the greatest common divisor of $a$ and $b$, and the join $a \vee b$ is the least common multiple. There's a partial order in that $ a \leq b \equiv a…
Apocalisp
- 308
5
votes
1 answer
Galois connection and order-isomorphisms
Assume that we have a Galois connection formed by two monotone maps $f\colon X\to Y$ and $g\colon Y\to X$.
I want to know whether the following statement is true: if $f$ is bijective, then $f$ is an order-isomorphism, that is, $f(x)\leq f(y)$…
J. Karen
- 337
4
votes
1 answer
Do any familiar adjunctions arise from this construction?
I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide.
For any set $X$, let $\mathscr P(X)$ be the category whose…
Brian Fitzpatrick
- 26,933
4
votes
1 answer
Computing all the Galois groups of reducible polynomials of degree 5 over $\mathbb{Q}$
in my Algebra class, it was given as an exercise to find all possible Galois groups of reducible polynomials of degree 5 over $\mathbb{Q}$ without repeated roots.
Where, for a field $F$, we call the Galois group of a polynomial $f(x)\in F[x]$ the…
F. Salviati
- 383
4
votes
3 answers
Image of sup/inf under Galois connection between lattices
This is Lemma 2.4 in Pete Clark's notes on commutative algebra.
Given two posets $(X,\le)$ and $(Y,\le)$, an (antitone) Galois connection between $X$ and $Y$ is a pair of maps $F:X\to Y$ and $G:Y\to X$ such that $F$ and $G$ are antitone (=…
PatrickR
- 7,165
4
votes
1 answer
Does $\mathcal C=\mathrm{Pol}(\mathrm{Inv}(\mathcal C))$ hold for clones on an infinite set?
I would like to know, whether the following theorem holds also for clones on an infinite set?
Theorem (Geiger; Bodnarcuk, Kaluznin, Kotov, Romov). Let $A$ be a finite set, then
$$\mathrm{Clo}(A)=\mathrm{Pol}(\mathrm{Inv}(\mathbf A)). $$
It is…
user771160
- 987
4
votes
0 answers
Trivial covering of a topological space
We have a connected topological space $X$ and $p:E\to X$ a Galois covering projection with $E$ not necessarily connected, let us call $\text{Aut}(E/X)=G$. Suppose it is given that every homomorphism of $\pi_1(X,x)\to G$ is trivial. Then we have to…
shadow10
- 5,737
4
votes
1 answer
A reference for an explicit statement of the Galois correspondence in a Galois category
The definition of a Galois category was cooked up intentionally to create the general setting where Galois correspondences appear. There are plenty of the resources (e.g. here and here) that go into detail about Galois categories, their properties,…
Santana Afton
- 6,978
4
votes
0 answers
Condition for Being Galois
I am thinking about the following excerpt of Patrick Morandi's "Field and Galois Theory" from chapter 1 section 2, page 21:
Here, $K$ is a field, and $\mathcal{F}(G)$ denotes the subfield $\{ x \in K : \sigma (x) = x\ \forall x \in G \}$ of $K$.…
user900250
4
votes
1 answer
Galois comodules
I tried to figure out why Galois comodules is a generalization of several Galois aspects, but I could not?
I am really interested in Comodule theory, and I am very curious to know the answer for this question. I know Galois theory is a nice…
Dan
- 685
3
votes
0 answers
Is there a way to state the fundamental theorem of Galois theory in a categorical sense?
I will state the fundamental theorem of Galois theory to make things clear. Let $F/K$ be a finite dimensional Galois extension. Let $A$ be the set of all intermediate fields of $F/K$, and let $B$ be the set of all subgroups of…
zxcv
- 1,649