5 problems

Non-generation conjecture. The set $R_{1}(\widetilde{J})$ is not a generating set of acyclic cofibrations for $\mathcal{SS}\mathrm{ets}_{1}$. The question is motivated by the fact…

Hurewicz fibration conjecture. The maps

Let $\operatorname{SpHorn}$ be the set of special horn inclusions. A map of simplicial sets is bijective on $0$-simplices when it induces a bijection on its sets of $0$-simplices.…

Topological–simplicial correspondence conjecture. The left adjoint functors are either on top or on the left, and the vertical solid arrows are Quillen equivalences.

For simplicial sets $A$ and $B$, let $\mathbf{Cyl}(A,B)$ be the category of cylinders from $A$ to $B$. Thus an object is a simplicial set $X$ equipped with a map $X\to\Delta[1]$ wh…