101 problems

Let $n e 1$ and assume the canonical model structure exists on $operatorname{Mod}(IC{n+1})$. Let $0leq k<n$, let $X:IC{n+2}toSet$ be a model, and let $f:D^kto UX$ be a map. Suppose…

Local compactness conjecture. Locally compact fully characterises the connected cofibrant objects: a connected object $\mathbb{X}$ is cofibrant if and only if it is locally compact…

Let the category of one-colored simplicial (wheeled) properads carry the model structure lifted from that on $\mathtt{sSet}^{\Sigma^{\operatorname{op}}\times \Sigma}$, and let 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…

Dendroidal sets form a category equipped with a notion of inner Kan complex; for dendroidal sets $X$, $Y$, and any dendroidal inner Kan complex $K$, write…

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 compact Lie group. Write $\mathcal{A}(G)$ for an abelian category and let $\mathrm{DG}\text{-}\mathcal{A}(G)$ denote the category of differential graded objects of…

Let $\mathsf{CDGA}^1$ be the full subcategory of connected commutative differential graded algebras, and let … be the augmentation ideal functor. Right-transfer conjecture. There i…

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 $G$ be a finite group and let ${\mathscr{O}}_G$ be its orbital category. For a $G$-transfer system ${\mathcal{T}}$, let…

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 $\mathcal{E}$ be a locally cartesian closed locos with coequalisers, and let the effective model structure of Gambino, Henry, Sattler and Szumilo be given on simplicial objects…

Triangulated homotopy category conjecture. The category $\operatorname{Ho}(\mathcal T)$ is always a triangulated category for an arbitrary proper class $\xi$ of triangles.

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 $f:X\to Y$ be a cellular map of finite CW complexes. Let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ denote the finite-space maps implicit in the definition of a n…

Model structure conjecture. A similar model structure should exist for weak $\omega$-groupoids and weak $\omega$-categories.

Let $(P,A)$ be a $\Sigma$-free substitude that is non-quasi-tame, meaning that \Pi_1(\text{{bf T}^{scriptstyle tt T+{1}}}) is not equivalent to a discrete groupoid. Let…

Let $P$ be a polynomial monad, and suppose that a transferred model structure on its category of algebras is under consideration. A polynomial monad is quasi-tame when its canonica…

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 ${\rm{\mathbf{{\mathcal{G}}dTop}}}$…

Let the $\tau_n$ tower be the tower of model structures introduced in the paper, and let its limit be denoted by the limit of the $\tau_n$ tower. Let the coinductive localization b…