Equivariant map
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
- exists a left \(M\)-action \(f:M\times X \to X\) on the set \(X\);
- exists a left \(M\)-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 \(Y\).
Notation From now we use the following abuse of notation: we use \(X\) to denote the left action \((X,f)\), i.e. the set equipped with the action, the infix notation for the action \(tx:=f(t,x)\) and denote by \(M{\rm Set}(X,Y )\) the set \(M{\rm Set}((X,f),(Y,g) )\).
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.
Properties
Let \(\phi\) be \(M\)-equivariant, \(x_0,x_1 \in X\) and \(\lambda\in {\rm ht}_X(x_0,x_1):=\{\lambda\in M\,|\, \lambda x_0=x_1\}\). If \(\phi(x_0)=\phi(x_1)=y\) then \(\lambda\) is a period of \(y\in Y\).
proof. This follows from the Grothendieck construction \(\int:M{\rm Set}\to {\rm Cat}/_{{\bf B}M }\), being a functor: each \(\phi\in M{\rm Set}(X,Y )\) is lifted to a functor \(\int\phi\in {\rm Cat}/_{{\bf B}M}(\int X,\int Y )\). Since the height-sets \({\rm ht}_X(x_0,x_1)\) in \(X\) are in bijection with the homsets in \(\int X\), i.e. \({\rm ht}_X(x_0,x_1)\cong \int X(x_0,x_1)\) the functor \(\int\phi:\int X\to \int Y\) induces a family of set-maps \({\rm ht}_X(x_0,x_1)\to {\rm ht}_Y(\phi(x_0),\phi(x_1))\). By assumption \(\phi(x_0)=\phi(x_1)=y\) we induce an inclusion map $${\rm ht}_X(x_0,x_1)\subseteq {\rm per}_Y(y) $$