40 problems

Let $G$ be a compact Lie group, let $H<G$ be a closed subgroup, and let $N_H^G$ be the point-set relative norm indexed on universes $U_H$ and $U$. Suppose $X'$ and $X$ satisfy the…

Model-structure conjecture. For a wide range of $A_{\infty}$-operads ${\cal C}$, the category of $\widehat{{\cal C}}$-precats admits a closed model structure with these cofibration…

Let $Y$ be a small category and let $\mathbf{M}$ be a left Quillen presheaf on $Y$. Suppose that $\mathbf{M}$ satisfies hypothesis (o), introduced in Section 19, which in particula…

Let $PrCat$ be the model category of Segal precategories, let $QCat$ be the model category of quasi-categories, and let $\phi:PrCat\to QCat$ and $\psi:QCat\to PrCat$ be the adjoint…

The Generalized Pushout Conjecture. The following statements hold: first, the map $p:X\to X_+$ is a weak equivalence; second, given a connected diagram of controlled theories $J$,…

Let $G$ be a finite group, let $(\operatorname{sOp}^{\operatorname{col}})^G$ be the category of $G$-objects in simplicial coloured operads, and let $G$-$\infty$-$\operatorname{Op}$…

Let $\mathfrak{K}_{n}$ and $\mathfrak{M}_{n}$ be the model categories of marked Kan complexes and marked complicial sets, respectively, and let $(i_{\Delta},-_{\Delta})$ be the adj…

Non-generation conjecture. The set $R_{1}(\widetilde{J})$ is not a generating set of acyclic cofibrations for $\mathcal{SS}\mathrm{ets}_{1}$. The question is motivated by the fact…

Let $s\mathscr{S}\mathrm{et}$ be the category of simplicial sets, $s\mathscr{S}\mathrm{et}^+$ the category of marked simplicial sets, and $s\mathscr{S}^+$ the category of simplicia…

Let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ be the indicated maps of finite topological spaces, and let $\operatorname{Top}$ denote the category of topological spa…

Let $a54$ be a SOGAT, and let $a54$ be equipped with weakly stable identity types satisfying function extensionality and saturation, which is the condition called external univalen…

Take ${\mathbf{Top}}$ to be the category of $\Delta$-Hausdorff $\Delta$-generated spaces. Left properness conjecture. The q-model structures of ${\rm{\mathbf{{\mathcal{G}}dTop}}}$…

For an integer $m$, let $\mathbf{sSet}^m$ denote the category of stratified simplicial sets and let $\infty\text{-}\mathbf{Cat}^m$ denote the category of marked strict $\infty$-cat…

Let $P$ be a finite set of maps between finite topological spaces. For a string of letters $l$ and $r$, let $(P)^{sl}$, $(P)^{slr}$, $(P)^{srl}$, and $(P)^{sr}$ denote the iterated…

Let $M$ be the five-point space with two open and three closed points, let $\Lambda$ be the three-point space with one open and two closed points, and let…

Let $M$ be the five-point space with two open and three closed points, let $\Lambda$ be the three-point space with one open and two closed points, and let…

Let $\operatorname{SpHorn}$ be the set of special horn inclusions. A map of simplicial sets is bijective on $0$-simplices when it induces a bijection on its sets of $0$-simplices.…

Let $\mathsf{cSet}^+$ denote the category of marked cubical sets and let $\mathsf{PreComp}$ denote the category of marked simplicial sets (pre-complicial sets). Let…

Topological–simplicial correspondence conjecture. The left adjoint functors are either on top or on the left, and the vertical solid arrows are Quillen equivalences.

Cofibrancy preservation conjecture. These pullback functors preserve cofibrations between cofibrant objects. This generalizes the Interval Cofibrancy Theorem, which is the special…

For simplicial sets $A$ and $B$, let $\mathbf{Cyl}(A,B)$ be the category of cylinders from $A$ to $B$. Thus an object is a simplicial set $X$ equipped with a map $X\to\Delta[1]$ wh…

Let $n$ be a nonnegative integer, let $n\mathcal{C}\!\mathit{at}$ denote the category of strict $n$-categories, and let … be the natural $n$-nerve, with…

Let $n\geq 0$. Write $\mathit{s}\mathcal{S}\!\mathit{et}^{t\Delta^{\operatorname{op}}}$ for the category of prestratified simplicial spaces, equipped with the cartesian model struc…

Classification conjecture. There are bijections

The hammock-space conjecture. The space $\widehat{A}$ is weakly equivalent to $\emph{Bhaut} A$: