Equivariant map

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

1. exists an \(M\)-action \(f:M\times X \to X\) on the set \(X\);
2. exists an \(M\)-action \(g:M\times Y \to Y\) on the set \(Y\);
3. (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.