18 problems

Let $k\geq 2$, and consider the complex … Assume that $\beta$ is surjective and that $\alpha$ degenerates in codimension $2$; equivalently, the cokernel in degree $0$ is…

Let $E$ be a rank $2$ torsion-free sheaf on $\mathbb{P}^k$ that is the cohomology of a special monad. Special-monad semistability conjecture. Then $E$ is semistable. This extends t…

Fibrational internal-category conjecture. A fibrational generalization of the notion of internal category should be suitable to provide a generalization of the equivalence between…

Dibinary extension-monad conjecture. The extension monad of $cmathbb{A}$ has the same underlying set as the extension monad of its $cbeta$ reduct.

Maybe-monad procedure conjecture. A similar procedure may exist in Kleisli categories of Maybe monads on other Markov categories.

A graded distributive Markov category is a Markov category equipped with the grading and distributive structure described above, and a commutative affine graded relative monad is a…

Equivalence conjecture. The assignment sending $(X,(P,f))$ to the corresponding lax algebra for $F^\kappa$,

Let $t$ be a monad equipped with an exceptional structure, and let the purported trace on the category denoted by $\mathbf{DK}_{t}$ be the trace constructed above from the correspo…

Positivity correspondence conjecture. The monad $T$ is positive if and only if its Kleisli category $\mathrm{Kl}_T$ is positive.

A continuous equation scheme is understood in the sense of MPP16, and each such scheme induces a monad on the relevant category of relational structures. Finitarity conjecture. Mon…

Let the Abelian group monad and the list monad be the monads on the category $\mathbf{Set}$ of sets and functions obtained from the corresponding free constructions. Beck's conject…

Let $\mathbb{C}$ be a category with the structures and notation used above, let $C$ be an object of $\mathbb{C}$, and let $\bar{A}=(A,f_A)$ and $\bar{B}=(B,f_B)$ be algebras in…

Let $\mathsf{T}$ be a monad and let $\mathsf{Fix}(\mathsf{T})$ denote its category of fixed points. The codensity monad of a functor is its right Kan extension along itself, when t…

Let $\mathsf{K}$ be a complete category and let $\mathsf{T}$ be a generically idempotent monad on $\mathsf{K}$. Its fixed points are the objects on which the monad is idempotent, f…

A compact max-min convex set is a compact subset $A\subset\mathbb R^\tau$ closed under max-min convex combinations, and a mathbb J-algebra is an algebra $(A,\xi)$ for the max-min m…

Let $R(X)=X\amalg\{*\}$ be the monad on sets obtained by adjoining a distinguished point, and let $CC(R,R)$ denote the associated C-system. A subsystem is a C-system contained in…

Fusion-operator conjecture. The morphism $h_{M,N}$ is a fusion operator.

End formula conjecture. $T(X)$ is the end