Inspired by [Group Equivariant Convolutional Networks](https://arxiv.org/abs/1602.07576) by Taco Cohen and Max Welling I have been thinking about the following construction and I wonder where else it has been described and studied.


Suppose $G$ is a group and $X$ and $Y$ are sets with a $G$-action. Let $f : X \to Y$, not necessarily $G$-equivariant. Then we can cook up augmentations of $X$, $Y$ and $f$ ($X\_G$, $Y\_G$ and $f\_G$ respectively) such that $f\_G$ *is* $G$-equivariant.


The recipe is to define $X\_G = G \times X$ with $G$-action $g(h, x) = (gh, gx)$ (resp. for $Y\_G$) and $f\_G : X\_G \to Y\_G$ by $f\_G(g, x) = (g, gf(g^{-1}x))$.


Then we have that


* $f\_G(g(h, x)) = f\_G(gh, gx)$
* $= (gh, ghf((gh)^{-1}gx)$
* $= (gh, ghf(h^{-1}g^{-1}gx))$
* $= (gh, ghf(h^{-1}x))$
* $= g(h, hf(h^{-1}x))$
* $= gf\_G(h, x)$


i.e. $f\_G$ is $G$-equivariant.


It seems like this should be a well-known construction but I just can't quite grasp its implications at the moment. Is there somewhere I can read more about it?



What is this concept related to group actions?
In fact $X\_G=G\times X$ with the action $g(h,x)=(gh,gx)$ is $G$-equivariantly isomorphic to $G\times|X|=G\times X$ with the action $g(h,x)=(gh,x)$. The isomorphism $i\_X:X\_G\to G\times|X|$ is given by $i\_X(h,x)=(h,h^{-1}x)$, with the inverse $i\_X^{-1}(h,x')=(h,hx')$.


Now observe that for any set $S$, the $G$-set $G\times S$ with the action $g(h,s)=(gh,s)$ is the free $G$-set on $S$; thus any map $f:S\to Y$ to a $G$-set $Y$ extends uniquely to a $G$-equivariant map $\bar f:G\times S\to Y$ with $\bar f(1,s)=f(s)$, namely $\bar f(h,s)=hf(s)$.


It follows that for any map $f:X\to Y$ between $G$-sets, there is a unique $G$-equivariant map $\tilde f:X\_G\to Y$ with $\tilde f(1,x)=f(x)$, namely, $\tilde f=\bar f\circ i\_X$, i. e. $\tilde f(h,x)=hf(h^{-1}x)$:
$$
X\_G\xrightarrow{i\_X}G\times|X|\xrightarrow{\bar f}Y
$$


In the above notation your $f\_G$ is $(\pi\_G,\tilde f)$, where $\pi\_G:X\_G=G\times X\to G$ is the projection (which is obviously equivariant).


