# Semidirect products and PROPs

$\newcommand{\p}{\mathcal{P}}$Let G be a group, and let $G_n$ denote the wreath product $G^n \rtimes \Sigma_n$.

There seems to be a notion of a PROP $\p$ where the role of the symmetric groups $\Sigma_n$ is instead played by the groups $G_n$. An example would be $G=SO(N)$ and $\p$ the PROP associated to the framed little N-disc operad. Here $\p(1,n)$ has an action of $G_1^{op} \times G_n$ -- the copy of $G_1^{op}$ rotates the entire disc "counterclockwise" (i.e. an element $g \in SO(N)$ acts via $g^{-1}$), and the action of $G_n$ on $\p(1,n)$ is the evident one. The gluing maps are then suitably equivariant under this simultaneous action of $G$ on the input/output legs, so to speak.

Is there a standard name for this kind of PROP/operad? Cf. how one calls an operad where $\Sigma_n$ has been replaced by $B_n$ a braided operad. I would also be happy to hear of any paper where this kind of gadget has been defined and/or studied.

Addendum, Jan 23 2013. Sorry if it is poor form to bump an inactive question only to advertise your own work, but I just ran across this old question. When I had thought some more about this I realized eventually that what I described above is really a special case of the notion of a colored PROP/operad, except the collection of colors do not form a set but a category. In this case there is only one color but its automorphism group is $G$, that is, the collection of colors is exactly the one-object category corresponding to the group $G$.

The notion of a PROP/operad which is colored by a category is defined in my paper http://arxiv.org/abs/1205.0420 . This is actually somewhat more general than what Salvatore and Wahl define, even when the category in question is a group.

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