We prove the formula
\begin{equation*} \text{cat}_G(X\vee Y)=\max \{\text{cat}_G(X),\text{cat}_G(Y)\} \end{equation*} for the equivariant category of the wedge
$X\vee Y$. As a direct application, we have that the wedge
$\bigvee _{i=1}^m X_i$ is
$G$-contractible if and only if each
$X_i$ is
$G$-contractible, for each
$i=1,\ldots ,m$. One further application is to compute the equivariant category of the quotient
$X/A$, for a
$G$-space
$X$ and an invariant subset
$A$ such that the inclusion
$A\hookrightarrow X$ is
$G$-homotopic to a constant map
$\overline {x_0}\,:\,A\to X$, for some
$x_0\in X^G$. Additionally, we discuss the equivariant and invariant topological complexities for wedges. For instance, as applications of our results, we obtain the following equalities:
\begin{align*} \text{TC}_G(X\vee Y)&=\max \{\text{TC}_G(X),\text{TC}_G(Y),\text{cat}_G(X\times Y)\},\\ \text{TC}^G(X\vee Y)&=\max \{\text{TC}^G(X),\text{TC}^G(Y),_{X\vee Y}\text{cat}_{G\times G}(X\times Y)\}, \end{align*} for
$G$-connected
$G$-CW-complexes
$X$ and
$Y$ under certain conditions.