---- > [!definition] Definition. ([[group-equivariant map]]) > Let $G$ be a [[group]] [[group action|acting]] on a set $S$ via $\cdot$ as well as on a set $Y$ via $\star$. A map $f: S \to Y$ is **$G$-equivariant** if it is compatible with the [[group action]]: $f(g \cdot s)=g \star f(s)$ > for all $g \in G$ and $s \in S$. > When $\star$ is the trivial action, we call $f$ **$G$-invariant**. ([[group-invariant function]]) ^13a1d7 > [!note] Categorical Remark. > [[group action#^equivalence|Recall]] that the actions $\cdot, \star$ of $G$ on $S,Y$ is are each [[covariant functor|functors]] $G \to \mathsf{Set}$. A [[natural transformation]] between them is, by definition, the datum of a single (since $G$ is a one-object [[category]]) set-function $f:S \to Y$ that commutes with the actions, i.e., a $G$-equivariant map. ^note > [!basicexample] > In signal processing, we care a lot about systems which are time-~~in~~equivariant, i.e., functions $H$ of signals $x$ such that if $H\big(x(t)\big)=y(t)$ then $H\big(x(t-\tau)\big)=y(t-\tau)$. A time-shift in the input corresponds to the output, time-shifted by the same amount. ^f83edd ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```