236 problems

Homological 3d mirror symmetry conjecture. There are 3d mirror pairs $X$ and $X^{!}$ whose A-model 2-category and B-model 2-category are interchanged under mirror symmetry.

Let $\mathcal{C}$ be a symmetric monoidal $(\infty,n)$-category with duals. A framed extended $n$-dimensional TFT valued in $\mathcal{C}$ is a symmetric monoidal functor from the f…

Grothendieck 2-topos conjecture. The 2-category of $\operatorname{Cat}$-valued functors on any small 2-category $\mathcal{C}$ is a 2-topos; more generally, every Grothendieck 2-top…

Let $k$ be a perfect field, and let $\mathcal{A}$ be a finite braided tensor category over $k$. The symmetric center of $\mathcal{A}$ is finite semisimple precisely when…

Thin-element boundary-cycle conjecture. The sum $\sum_i\lambda_i x_i$ is a boundary if and only if it is a cycle.

Let $\text{St}_1: (\infty,1)\text{Cat}\to(\omega,1)\text{Cat}$ be the strictification functor constructed by locally strictifying hom $\infty$-groupoids and then strictifying the e…

Let a strict $\infty$-category be a higher category in which composition, associativity, and the exchange law are strict. Consider the weaker structures obtained by replacing stric…

Fix a subgroup $G\subset O_k$, and let ${n}\mathsf{Tang}^G_k$ denote the Whitney category of codimension $k$ tangles whose normal bundles have structure group reducing to $G$. Let…

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 a braided monoidal 2-category be a 2-categorical structure equipped with a braided monoidal product. Baez–Dolan conjecture. Braided monoidal 2-categories are 4-categories with…

Let $H$ be an involutory Hopf category, and let its 2-category of representations be the corresponding representation 2-category. Spherical representation 2-category conjecture. Th…

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 $G$ and $H$ be groups and $R$ a coefficient object, and let ${\mathbb N}(G,H,R)$ denote the associated monoidal 2-category. A semi-weak monoidal 2-category is a structure on…

Let $H$ be an involutory Hopf category whose representations form a spherical 2-category $C_H$. Crane–Frenkel equivalence conjecture. The Crane–Frenkel invariants using $H$ are equ…

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 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…

Baez–Dolan's conjecture. The $n-k$-category of endomorphisms of the $k-1$-fold identity of $s$ in $\sigma^k$ is the Poincaré $n-k$-category $\Pi_{n-k}(X_k)$.

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…

Converse property. If $f$ and $g$ are two parallel $2$-arrows in $\langle \Sigma \rangle$ such that $\overline{\pi}(f)=\overline{\pi}(g)$, then

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…