In Mike Hill's paper "On the Algebras over Equivariant Little Disks" ( https://arxiv.org/abs/1709.02005) in section 2.2 there is an information, that algebras over $\mathcal{D}(\sigma)$, that is little disks operad in sign representation of $C_2$ come equipped with a canonical isomorphism $$ i^*_eX\to i^*_eX^{op}. $$ Could somebody explain this isomorphism to me? I have a feeling, that "$X^{op}$" means here the space $X$ after the action of non-trivial element of $C_2$. Then this isomorphism would be induced by this action. Is that correct?
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1$\begingroup$ I believe that this can be parenthesized as $(i_e^* X)^{op}$. In other words, once you have taken $X$ and forgotten the group action to get a space $Y = i_e^* X$, the space $Y$ is an $E_1$-space (in particular, having a binary multiplication) with an isomorphism $Y \to Y^{op}$ (given by acting by the generator of $C_2$, if I understand correctly). $\endgroup$– Tyler LawsonCommented May 3, 2018 at 17:06
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$\begingroup$ Thanks, but I still have another problem - what is $Y^{op}$? $\endgroup$– Igor SikoraCommented May 3, 2018 at 17:15
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2$\begingroup$ It has the same underlying space as Y but with the multiplication in the opposite order. $\endgroup$– Tyler LawsonCommented May 3, 2018 at 19:36
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