3 problems

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…

Special-cases conjecture. In $\mathfrak{C}\!\operatorname{at}_{\infty}$, the marked $(\infty,2)$-colimit of $F:X^\flat\to\mathfrak{C}\!\operatorname{at}_{\infty}$ coincides with th…

Existence conjecture. There is a theory of marked $(\infty,2)$-colimits in every $(\infty,2)$-category, with input and output as above.