Difference between revisions of "Equivariant map"
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| − | Let \(M\) be a monoid a,d \(X,Y\) two sets. A function \(\phi:X\to Y\) from the set \(X\) | + | 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 |
<ol style="list-style-type:lower-roman"> | <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 \(f:M\times X \to X\) on the set \(X\); | ||
<li> exists an \(M\)-action \(g:M\times Y \to Y\) on the set \(Y\); | <li> exists 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. | + | <li> (equivariance) for all \(t\in M\), for all \(x\in X\), the equation \(\phi(f(t,x))=g(t,\phi(x))\) holds. |
</ol> | </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 | + | 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== | ==Particular cases== | ||
| − | Some special cases are [[Abel function]], [[Schröder coordinates]], [[Böttcher coordinate]], some [[primitive recursive functions]] and [[linear transformations]] of [[ | + | Some special cases are [[Abel function]], [[Schröder coordinates]], [[Böttcher coordinate]], some [[primitive recursive functions]] and [[linear transformations]] of [[vector 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 \(Y\). | + | The concept is very flexible since every function \(\phi:x\to Y\) can always 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== | ==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. | 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. | ||
Revision as of 08:34, 12 November 2025
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
- exists an \(M\)-action \(f:M\times X \to X\) on the set \(X\);
- exists an \(M\)-action \(g:M\times Y \to Y\) on the set \(Y\);
- (equivariance) for all \(t\in M\), for all \(x\in X\), the equation \(\phi(f(t,x))=g(t,\phi(x))\) holds.
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
Some special cases are Abel function, Schröder coordinates, Böttcher coordinate, some primitive recursive functions and linear transformations of vector spaces. The concept is very flexible since every function \(\phi:x\to Y\) can always 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.