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I am currently working on a project (joint with di Fraia and Ramzi) that constructs a generalization of the question you have in mind. In particular, the following is true: If $M$ is an $E_2$-monoid i …

Given a (for simplicity connective) spectrum $E$ and a pointed CW-space $X$ there is "the" (homological) Atiyah-Hirzebruch spectral sequence $$E_{pq}^2 = \tilde{H}_p( X, \pi_q(E)) \Rightarrow \pi_{p+ …

Suppose $E$ is a connective spectrum, then there exists a natural map in the stable homotopy category $\mathcal{SHC}$, $E \rightarrow P_0 E$, called the $0$-th Postnikov truncation, which is character …

I will answer my own question, in hope that it is helpful to someone. Given a functor $X:D \rightarrow Spc$ of $\infty$-categories, we can take the unstraightening of $X$ (the appropriate generalizati …

It is the classifying category for the left action of $C$ on its product $C \times C$. Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric m …

Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, b …

Consider an $E_n$-monoid X. We can deloop $X$ to an $\infty$-category $\mathbf{B}X$. There's a natural functor $X^\circlearrowleft : \mathbf{B}X \rightarrow \text{Spc}$ given by the left action of $X$ …