14 problems

Let $K$ be a $1$-reduced simplicial set and let $EK$ denote its suspension. The cobar construction $\tom\widetilde C(EK)$ admits a natural, strict Alexander–Whitney coalgebra struc…

Let $X$ be a simplicial $R$-module. Write $\mathbf{D}_*X$ for its degeneracy sub-complex and $\mathbf{D}^{\mathrm{sym}}_*X$ for the symmetry sub-complex generated by chains…

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…

Equivalence conjecture. After restricting to sufficiently nice spaces and twists, the construction $f:Y\to |\mathscr{X}|\mapsto\omega_f$ should induce an equivalence between the ca…

Hurewicz fibration conjecture. The maps

Christensen–Wu conjecture. For every diffeological space $X$ and basepoint $x$, one has

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.…

Let $\operatorname{IH}$ be the set of inner horn inclusions, and let $\operatorname{SpOH}$ be the set of special outer horn inclusions. A map of simplicial sets is bijective on $0$…

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

Let $X$ be a cofibrant non-singular simplicial set that is the nerve of a small category. Cofibrant categorical nerve conjecture. $X$ is the nerve of a poset. The conjecture concer…

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…

Nonlinear simplicial Dold–Kan conjecture. Including the higher arity parts of an $E_\infty$-coalgebra structure on the chains of simplicial sets results in a similar nonlinear simp…

Let $\mathcal C$ be a small category, let $N\mathcal C$ denote its nerve, and let $\operatorname{Ex}$ be the Kan subdivision functor. A space has vanishing homotopy groups above di…

Let $n$ be a nonnegative integer. A finitely presentable category is a category presented by finitely many generators and relations; a monoid is regarded as a one-object category.…