Timeline for Weak operad and deloopings

6 events

when toggle format	what		by	license	comment
Dec 15, 2011 at 7:24	vote	accept	Ulrich Pennig
Dec 14, 2011 at 16:02	comment	added	Justin Noel		One could use cofibrant operads, this is a perfectly good way to go, but it is not required, nor does it appear in the reference I quoted. All $E_\infty$ operads produce weakly equivalent deloopings. The comparison merely requires taking the product of operads and using the bar construction.
Dec 14, 2011 at 14:08	history	edited	user19952	CC BY-SA 3.0	added 12 characters in body
Dec 14, 2011 at 14:08	comment	added	user19952		Sure: The $\Sigma$-cofibrant condition is sufficient to make homotopy categories equivalent. Every space $X$ acted on by a $\Sigma$-cofibrant $E_\infty$-operad is weakly to a space $Y$ acted on by the Barratt-Eccles operad (for instance). This does not mean however that $X$ itself is acted by the Barratt-Eccles operad, and cofibrant operad come at this point, in order to get homotopy invariant structures.
Dec 14, 2011 at 13:41	comment	added	Justin Noel		@BF: I appreciate your elaboration. I do want to point out that one does not require that the operad to be cofibrant in May's theory, only $\Sigma$-cofibrant. Although we can always replace an operad by a cofibrant replacement, it is often useful to compare algebras over an operad that naturally acts on them (such as the little n-cubes operad or the Barratt-Eccles operad).
Dec 14, 2011 at 13:23	history	answered	user19952	CC BY-SA 3.0