Like Schreiber does in his post, I would advertise the point of view developed by Sagave and Schlichtkrull in their Adv. Math 2012 paper, and used by us to study topological logarithmic geometry. Each symmetric spectrum X has a graded underlying space that is really a J-shaped diagram Y. Here J is a category with nerve QS^0, so hocolim_J Y maps to QS^0. If X is a commutative symmetric ring spectrum then hocolim_J Y is an E_\infty space over QS^0, i.e., it is graded over the sphere spectrum. Sagave started developing this while a postdoc in Oslo. I noted that the nerve of J was not Z but something with \pi_1 = Z/2.

Like Schreiber does in his post, I would advertise the point of view developed by Sagave and Schlichtkrull in their Adv. Math 2012 paper, and used by us to study topological logarithmic geometry. Each symmetric spectrum $X$ has a graded underlying space that is really a $J$-shaped diagram $Y$. Here $J$ is a category with nerve $QS^0$, so $hocolim_J Y$ maps to $QS^0$. If $X$ is a commutative symmetric ring spectrum then $hocolim_J Y$ is an $E_{\infty}$ space over $QS^0$, i.e., it is graded over the sphere spectrum. Sagave started developing this while a postdoc in Oslo. I noted that the nerve of $J$ was not $Z$ but something with $\pi_1 = Z/2$.

Like Schreiber does in his post, I would advertise the point of view developed by Sagave and Schlichtkrull in their Adv. Math 2012 paper, and used by us to study topological logarithmic geometry. Each symmetric spectrum X has a graded underlying space that is really a J-shaped diagram Y. Here J is a category with nerve QS^0, so hocolim_J Y maps to QS^0. If X is a commutative symmetric ring spectrum then Y is an E_\infty space over QS^0, i.e., it is graded over the sphere spectrum. Sagave started developing this while a postdoc in Oslo. I noted that the nerve of J was not Z but something with \pi_1 = Z/2.

Like Schreiber does in his post, I would advertise the point of view developed by Sagave and Schlichtkrull in their Adv. Math 2012 paper, and used by us to study topological logarithmic geometry. Each symmetric spectrum X has a graded underlying space that is really a J-shaped diagram Y. Here J is a category with nerve QS^0, so hocolim_J Y maps to QS^0. If X is a commutative symmetric ring spectrum then hocolim_J Y is an E_\infty space over QS^0, i.e., it is graded over the sphere spectrum. Sagave started developing this while a postdoc in Oslo. I noted that the nerve of J was not Z but something with \pi_1 = Z/2.