297 problems

Let $\cat{C}$ be a $tt$-category. The nerves of steel condition holds when the map … is a bijection. Nerves of Steel Conjecture. Every $tt$-category satisfies the nerves of steel c…

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 $\mathcal{C}$ be a locally presentable linear category over a field. An object is compact projective if it is both compact and projective, and $\mathcal{C}$ is strongly generat…

Let a semi-abelian category and a quasi-abelian category be understood in the sense described above; in particular, a quasi-abelian category is both left and right quasi-abelian. R…

Small-limit conjecture. The category $n\operatorname{Pr}^L$ has all small limits for all $n\geq 0$.

Kumjian–Pask–Sims conjecture. The cubical and categorical cohomology groups should be isomorphic in all dimensions:

Let $X$ be the underlying complex manifold, and let $\mathcal{E}_X$ and $\widehat{\mathcal{E}}_X$ denote the two sheaves of microdifferential operators appearing in the paper. Cons…

Let $I$ be a finite category with series Euler characteristic. Write $A_I$ for its adjacency matrix and $\chi_{\Sigma}(I)$ for its series Euler characteristic. The zeta function of…

Functoriality conjecture. The geometry generated by the connected component should depend functorially on the associative pair, yielding an equivalence of categories between associ…

Superfiniteness conjecture. Any finite von Neumann category is superfinite.

Let $\mathbf{CH}$ be the category of compact Hausdorff spaces and continuous maps. A category is Stone-like when its abstract ultracoproduct construction satisfies the relevant Sto…

Let ${\cal H}$ be the Heegaard category, and let $T(V)$ denote the relevant target associated with a vector space $V$. Residual representability conjecture. If $a$ and $b$ are dist…

Let $\mathcal O_{m|\infty}^{++}$ and $\mathcal O_{m+\infty}^{++}$ be the two categories introduced in the paper, with their Grothendieck groups related by the preceding isomorphism…

Measurable functor conjecture. The class of fiberwise norm-continuous ${\mathbb C}$-linear functors is the same as the structurally defined class of measurable functors.

Higher category modelling conjecture. Category theory and higher dimensional category theory could be necessary for modelling this kind of behaviour.

Colimit analogy conjecture. The notion of a colimit may give useful analogies to the way complex biological systems operate.

Uniqueness conjecture. There are no other braids underlying the composition of a product of enriched categories besides braid (1) and its inverse that fulfill both obligations.

Giraud's theorem for Segal topoi. $A$ is a $\mathbb{U}$-Segal topos if and only if:

Let two derivations have the same premises and conclusions, and let their generality record which occurrences of variables must remain occurrences of the same variable under every…

Similarity–equivalence conjecture. $H_1$ and $H_2$ are similar if and only if they are equivalent.

Let $\2K$ be a pre-modular tensor category. A fully embedded subcategory is one included as a full tensor subcategory of a modular category $\2C$. Write $\2Z_2(\2K)$ for the center…

Harish-Chandra diagram-category conjecture. For any semisimple group, the category of Harish-Chandra modules is equivalent to a certain category of diagrams in the category of fini…

Let a directed $A_{\infty}$-pre-category be an $A_{\infty}$-pre-category whose objects are indexed by ${{\bf Z}}$ with transversal sequences ordered by the indices, and consider…

Let an $A_{\infty}$-pre-category be understood with the equivalence relation defined in the source, and let an $A_{\infty}$-category with strict identity morphisms be a non-unital…

Non-equivalence conjecture. The categories $\widetilde{\bf Theories}$ and $Ho(\widetilde{\bf 2Theories})$ are not equivalent.