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We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobenius algebra and the restriction at $R$ of its Nakayama automorphism is equal to the Nakayama automorphism of $R$. As applications, we obtain several characterizations for an algebra extension associated to a wreath to be Frobenius, respectively separable.

On Frobenius and separable algebra extensions in monoidal categories: applications to wreaths