Equivariant map - Revision history

MphLee at 09:36, 12 November 2025

2025-11-12T09:36:51Z

MphLee at 08:34, 12 November 2025

2025-11-12T08:34:11Z

MphLee: Created page with "Let \(M\) be a monoid a,d \(X,Y\) two sets. A function \(\phi:X\to Y\) from the set \(X\) tot he set \(Y\) is defined to be '''\(M\)-equivariant''' if all the following condit..."

2025-11-12T08:31:31Z

Created page with "Let \(M\) be a monoid a,d \(X,Y\) two sets. A function \(\phi:X\to Y\) from the set \(X\) tot he set \(Y\) is defined to be '''\(M\)-equivariant''' if all the following condit..."

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Let \(M\) be a monoid a,d \(X,Y\) two sets. A function \(\phi:X\to Y\) from the set \(X\) tot he set \(Y\) is defined to be '''\(M\)-equivariant''' if all the following conditions holds

<ol style="list-style-type:lower-roman">
<li> exists an \(M\)-action \(f:M\times X \to X\) on the set \(X\);
<li> exists an \(M\)-action \(g:M\times Y \to Y\) on the set \(Y\);
<li> (equivariance) for all \(t\in M\), for all \(x\in X\) the equation \(\phi(f(t,x))=g(t,\phi(x))\) holds.
</ol>

We denote by \(M{\rm Set}((X,f),(Y,g) )\) the set of \(M\)-equivariant maps from the M-Sets \((X,f)\) and \((Y,g)\). It is a subset of the set \(Y^X\) of all the set-theoretic function from \(X\) to \(Y\).

==Particular cases==
Some special cases are [[Abel function]], [[Schröder coordinates]], [[Böttcher coordinate]], some [[primitive recursive functions]] and [[linear transformations]] of [[vecvtor spaces]].
The concept is very flexible since every function \(\phi:x\to Y\) can bee seen as \(M\)-equivariant with respect to the [[constant action]], i.e. the action of ''doing-nothing'', on the sets \(X\) and \(Y\).

==General case==
Equivariant maps are a special kind of [[natural transofrmations]] \(\phi:X\to Y\) between two [[functors]] \(X,Y: {\bf B}M \to {\rm Set}\) going from the monoid \(M\), seen as a single-object category, to the category of sets or other categories.

@@ Line 1: / Line 1: @@
-Let \(M\) be a monoid a,d \(X,Y\) two sets. A function \(\phi:X\to Y\) from the set \(X\) to the set \(Y\) is defined to be '''\(M\)-equivariant''' if all the following conditions hold
+Let \(M\) be a monoid and \(X,Y\) sets. A function \(\phi:X\to Y\) from the set \(X\) to the set \(Y\) is defined to be '''\(M\)-equivariant''' if all the following conditions hold
 <ol style="list-style-type:lower-roman">
-<li> exists an \(M\)-action \(f:M\times X \to X\) on the set \(X\);
+<li> exists a left \(M\)-action \(f:M\times X \to X\) on the set \(X\);
-<li> exists an \(M\)-action \(g:M\times Y \to Y\) on the set \(Y\);
+<li> exists a left \(M\)-action \(g:M\times Y \to Y\) on the set \(Y\);
 <li> (equivariance) for all \(t\in M\), for all \(x\in X\), the equation \(\phi(f(t,x))=g(t,\phi(x))\) holds.
 </ol>
 We denote by \(M{\rm Set}((X,f),(Y,g) )\) the set of \(M\)-equivariant maps from the M-Sets \((X,f)\) and \((Y,g)\). It is a subset of the set \(Y^X\) of all the set-theoretic functions from \(X\) to \(Y\).
 ==Particular cases==
@@ Line 15: / Line 17: @@
 ==General case==
 Equivariant maps are a special kind of [[natural transofrmations]] \(\phi:X\to Y\) between two [[functors]] \(X,Y: {\bf B}M \to {\rm Set}\) going from the monoid \(M\), seen as a single-object category, to the category of sets or other categories.