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It is well know the notion of Equivariant map .

Now, consider the situation: $X$ and $Y$ being $G$- and $H$-sets, resp., and a homomorphism $\varphi: G\to H$.

How is called a map $f:X\to Y$ satisfying $f(gx)=\varphi(g)f(x)$ ?

Sigur
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    It's also called equivariant, or $\varphi$-equivariant to be more precise. – Moishe Kohan Aug 05 '17 at 00:34
  • I would not call $f$ an equivariant since $f$ is equivariant if the domain and the codomain of $f$ are acted upon by the same group $G$. I would call it $\varphi$-$\textit{equivariant}$. – Mee Seong Im Aug 05 '17 at 00:39

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