Questions tagged [galois-connections]

The tag has no usage guidance.

36 questions
97 views

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 ...
57 views

Equivalence between Category of Covers and $\pi_1(X)$ Sets

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 38): In order to show the category equivalence claimed in Thm 2....
48 views

What makes a good mathematical theory?

I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when ...
56 views

Infimum and Supremum (of sets) - Formal Concept Analysis

I am taking a course of Introduction to Formal Concept Analysis and I have an uncertainty about the definition of supremum (least comum superconcept) and infimum (greatest comum subconcept) of formal ...
54 views

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. ...
39 views

Image and Preimage - Proof of Galois Connection

Here is a problem from my Graduate Abstract Algebra course. I'm not quite sure how to go about part d at all, though the rest of the parts were easily proved using some basic machinery I already knew. ...
51 views

Does $f[A] \cap B\subseteq f[A\cap f^{-1}[B]]$ generalize beyond sets?

If $f:X\to Y$, and $A\subseteq X$, $B\subseteq Y$, then the equation $f[A] \cap B\subseteq f[A\cap f^{-1}[B]]$ holds. Indeed, let $y\in f[A]\cap B$, then $y=f(x)$ for some $x\in A$; since $f(x)\in B$ ...
53 views

17 views

Conditions for embedding to be part of Galois connection?

I am working though 7 sketches in compositionality and have almost reached the end of chapter 1, which is very much concerned with Galois Connections. One of the questions on the subject that is not ...
92 views

71 views

Under what conditions $c = \gamma(\alpha(c))$ for a Galois connection?

I have the basic definition of Galois connection. Let $(C,\leq)$ and $(A,\sqsubseteq)$ be partial orders and $\alpha: C \rightarrow A$, $\gamma: A \rightarrow C$ monotonic functions. They form a ...
67 views

Reference request: the category of adjunctions between posets as categories that induce a partiuclar monad

I am interested in the category $A$ of adjunctions that induce a monad $c : C \to C$ where $C$ is a poset. (The description of $A$ is in a previous math.se post.) For a general $C$, of course, $A$ ...
Show $F(U) = K((x^q -x)^{q-1})$.
Let $K$ be a finite field with $q$ elements. Show that if $U$ is the subgroup of $Aut(K(x)/K)$ which consists of all mappings $\sigma$ of the form $(\sigma \theta)(x) = \theta(ax+b)$ with $a \neq 0$ ...