Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

$\newcommand{\p}{\mathcal{P}}$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 how one calls an operad where $\Sigma_n$ has been replaced by $B_n$ a braided operad.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.

share|improve this question

1 Answer 1

Dear Dan,

You can find this notion defined in the paper "Framed discs operads and the equivariant recognition principle" by Nathalie Wahl and Paolo Salvatore: http://arxiv.org/abs/math/0106242 . .

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.