4 problems

Let $\Sigma: \mathscr{T}op_* \to \mathscr{T}op_*$ denote the usual reduced suspension functor, and let $F: {\mathscr{T}op^{\mathsf{f}}_*} \to {\mathscr{S}p}$ be a functor whose der…

Work at a prime $p$, and consider the connecting maps in the Taylor tower evaluated at ${\mathbb C}^0$: … A contracting homotopy for the complex of spectra ${\overline{M}}(k)$ woul…

Let $P$ be an operad of spectra, and let $\mathcal{C}$ be the category of $P$-algebras. Write $I_{\mathcal{C}}$ for the identity functor on $\mathcal{C}$, and let…

Let $\mathcal{C}$ and $\mathcal{D}$ be categories appropriate for doing calculus, with stabilizations $\mathsf{Spec}(\mathcal{C})$ and $\mathsf{Spec}(\mathcal{D})$. For a functor…