Intertwining for semidirect product operads

Abstract

This paper shows that the semi-direct product construction for G-operads and the levelwise Borel construction for G-cooperads are intertwined by the topological operadic bar construction. En route we give a generalization of the bar construction of M. Ching from reduced to certain non-reduced topological operads. arXiv:1801.00970v2 [math.AT]

Koszul duality for topological E_n-operads

“…where the operad structure on the right-hand side is identified by Westerland [37] as the homotopy fixed point spectra of the S 1 -action on the stable little 2-discs. Drummond-Cole has shown in [17] that the topological operad M 0, * +1 bears a close relationship with the framed little 2-discs operad f E 2 , and Ward's work on bar construction for non-reduced operads [36], together with Theorem 1.1, ties all these constructions together.…”

“…Work of Ward [36,Sec. 4] outlines how the main theorem of this paper may be use to promote Getzler's duality result to an equivalence between operads of spectra…”

Koszul duality for topological E n ‐operads

“…where the operad structure on the right-hand side is identified by Westerland [37] as the homotopy fixed point spectra of the 𝑆 1 -action on the stable little 2-discs. Drummond-Cole has shown in [17] that the topological operad  0, * +1 bears a close relationship with the framed little 2-discs operad 𝑓𝐸 2 , and Ward's work on bar construction for non-reduced operads [36], together with Theorem 1.1, ties all these constructions together. Much of the first author's interest in the present project comes from its connections to Goodwillie calculus.…”

“…Work of Ward [36, section 4] outlines how the main theorem of this paper may be use to promote Getzler's duality result to an equivalence between operads of spectra $$\begin{equation*} K(\Sigma ^\infty _+\overline{\mathcal {M}}_{0,*+1}) \simeq \Sigma \Sigma ^\infty _+ \mathcal {M}_{0,*+1}, \end{equation*}$$where the operad structure on the right‐hand side is identified by Westerland [37] as the homotopy fixed point spectra of the $$S^1$$‐action on the stable little 2‐discs. Drummond–Cole has shown in [17] that the topological operad $$\overline{\mathcal {M}}_{0,*+1}$$ bears a close relationship with the framed little 2‐discs operad $$fE_2$$, and Ward's work on bar construction for non‐reduced operads [36], together with Theorem 1.1, ties all these constructions together.…”