16 problems

Let $C$ be the category appearing above, let $L{\mathsf{Sp}}$ be the corresponding localized category of spectra, and write $\partial^{BB}_\ast$ for the Goodwillie derivative funct…

Let $F,G\colon \mathcal{T}_{}\to\mathcal{T}_{}$ be homotopy functors such that $F(\ast)=\ast$, and let $FG$ denote their composite. Write $\partial_*F$ and $\partial_*G$ for their…

Bimodule chain-rule conjecture. For functors

Mixed chain-rule conjecture. There is an equivalence of right $K(\mathsf{OEpi})$-modules

Lin–Nishida conjecture. The diagrams

Let $F\colon \operatorname{\mathit{s}\EuScript{V}}\to\mathit{s}\EuScript{L}$ be a reduced analytic functor, and let $\bar{\partial}^{\star}F$ denote its symmetric sequence of coder…

Let $\mathbb{T}\mathrm{opos}_\infty^{op} \subseteq {\mathbb{C}\mathrm{at}_\infty^{\mathrm{diff}}}$ be the subcategory consisting of $\infty$-toposes and left exact colimit-preservi…

Let $X$ be a space and let $n>2$. Define $LS_2(X)\leq n-2$ when $X$ has Lusternik–Schnirelmann category at most $n$ and admits an open cover $U_1,\ldots,U_n$ such that each map…

Let $n$ be a nonnegative integer, let $mathcal{P}^n$ be a theory in the sense of Lawvere, and let its objects be the free homotopy conilpotent groups of class $n$, namely products…

Dual Goodwillie calculus conjecture. There should be a cooperad $\partial_*(\Sigma^\infty_\mathcal{C}\Omega^\infty_\mathcal{C})$ and an equivalence of operads

Let $E_r^{k,t}(S^1)$ denote the pages of the Goodwillie spectral sequence (GSS) for $S^1$. Goodwillie–Whitehead conjecture, weak form. The GSS for $S^1$ collapses at the $E_2$-page…

For each $k$, let … be the map in Kuhn's Kahn–Priddy sequence, and let … be the $k$-invariant map of the Goodwillie tower for $S^1$. Goodwillie–Whitehead conjecture, strong form. ……

Goodwillie–Whitehead conjecture. The GSS for $S^1$ collapses at $E_2$. This is a calculus analogue of the $2$-primary Whitehead conjecture and would explain the convergence of the…

Iterated bar–Goodwillie approximation conjecture. The iterated bar spectral sequence should be isomorphic, from the $E^2$-stage and up to $\Bbbk$-duality, to the $H_*(-,\Bbbk)$-ver…

Iterated bar–Goodwillie spectral sequence conjecture. These spectral sequences should agree from the $E^2$-stage onward.

For a prime $p$, let $P_{p^n}(\operatorname{id})$ denote the $p^n$-th stage of the Goodwillie tower of the identity functor, let $S^1$ be the circle, and write $(-)_{(p)}$ for loca…