97 problems

Generalized pushout conjecture. There is a notion of pushout of ${\cal C}$-categories such that the loop space functor from spaces to ${\cal C}$-groupoids takes pushouts to pushout…

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…

May's generalized Seifert–van Kampen conjecture. The loop space functor from spaces to grouplike $\widehat{{\cal C}}$-spaces takes pushouts of connected spaces to pushouts of…

Effective categorification conjecture. For any $A_{\infty}$-operad ${\cal C}$, there is an operation $Cat _{{\cal C}}$ from $\widehat{{\cal C}}$-precats to $\widehat{{\cal C}}$-cat…

Let $G$ be a group, $H$ an abelian group, and $R$ a commutative ring with unit group $R^*$. Let $B_G$ be the classifying space of $G$, and for each cohomology class…

Let $I^n$ be the $n$-cube, let $2_n$ be the free $\omega$-category generated by an $n$-morphism, and let $G_n$ for $n>0$ be the oriented $n$-globe, namely the free $\omega$-categor…

Let $\mathcal{C}$ be an $\omega$-category and let $I^1$ be the interval in the biclosed monoidal structure on $\omega\operatorname{Cat}$. For either choice of sign, consider the…

Let $\mathcal{C}$ and $\mathcal{D}$ be free $\omega$-categories, let $f,g:\mathcal{C}\to\mathcal{D}$ be homotopic non $1$-contracting $\omega$-functors, and let…

Let $\mathcal{C}$ be a free globular $\omega$-category. For a natural number $n$, let $x_i\in\omega\operatorname{Cat}(I^n,\mathcal{C})^\pm$ and let $\lambda_i$ be natural numbers i…

Let $\mathcal{C}$ be an $\omega$-category, let $I^n$ denote the $n$-dimensional cube, let $A$ and $B$ be composable $n$-morphisms, and let $\widehat{\mathcal{C}}$ denote the modifi…

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 $Y$ be a category and let $A$ be a presheaf of Segal $n$-categories over $Y^o$. Let $horiz$ be the set of horizontal arrows in $\int_Y A$, namely the arrows from $y$ to…

For a commutative ring $A$, let $A\textrm{-}\mathcal{CAT}^{Mor}$ denote the Morita localization of linear categories and let $A\textrm{-}\mathcal{AB}^{proj}$ be the subcategory of…

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…

Let $k$ be a field and let $T$ be a $k$-Tannakian Segal category. Let $\mathcal{H}(T)$ denote a $k$-Tannakian category and let $Ind(\mathcal{H}(T))$ be its ind-category with its in…

Let $A$ be a commutative ring spectrum, let $T$ be an $A$-tensor Segal category, and let $FIB_A(T)$ be its stack of fiber functors. For an $A$-Tannakian Segal category, the source…

Let $Sp^{\Sigma}$ be the category of symmetric spectra and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Let $LSp^{\Sigma}$ be the tensor Segal categor…

Let $Sp^{\Sigma}$ be the category of symmetric spectra with its positive model structure, and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Write…

Gray-operad conjecture. There is a unique contractible $n$-operad $G_n$ such that, for every $k$ with $0\leq k\leq n-1$, \text{cal P}_k(G_n) is the free $k$-fold monoid operad.…

Binary-composition conjecture. Every weak $n$-category is weakly equivalent to an $A$-algebra.

Let $F:A\rightarrow B$ be a morphism of $n+1$-categories, and let $\varphi:A\rightarrow nCAT'$ be a family of $n$-categories over $A$. A direct image is a universal family…

Let $\psi:A\rightarrow nCAT'$ be a functor and let $\mathcal P$ be a property of morphisms from $\psi$ to objects of $nCAT'$. A universal morphism with property $\mathcal P$ is a m…

An $n$-stack is a functor on a site satisfying descent with respect to sieves, and $nSTACK/{\cal X}$ denotes the resulting higher category of $n$-stacks. Let $U'$ be an $n$-stack w…

Suppose $A$ is an $n+1$-category admitting arbitrary direct and inverse limits. A functor $h:A^o\rightarrow nCAT'$ is called representable when it is induced by an object of $A$, a…

Yoneda conjecture. The functors $\alpha$ and $\beta$ are fully faithful. This is the higher-categorical analogue of the corresponding fact for $n=0$; the paper gives no proof or re…