13 problems

Let a strict $\infty$-category be a higher category in which composition, associativity, and the exchange law are strict. Consider the weaker structures obtained by replacing stric…

The Generalized Pushout Conjecture. The following statements hold: first, the map $p:X\to X_+$ is a weak equivalence; second, given a connected diagram of controlled theories $J$,…

Let $\mathsf{S}_{\rm ana}$ denote the proposed left derivator of “infinity-groupoids and anafunctors”, and let $\mathsf{Gpd}_{\rm ana}$ denote the corresponding groupoid-level deri…

Let $\mathfrak{D}$ be a coherator for $infty$-categories, and suppose it can be endowed with a system of inverses. A coherator is contractible when it is a coherator for $infty$-gr…

Let Grothendieck $infty$-groupoids be the weak $infty$-groupoids defined using coherators, and let Batanin $infty$-groupoids denote Batanin's notion of $infty$-groupoid. Comparison…

Let Grothendieck $infty$-groupoids be the algebraic structures introduced to model weak $infty$-groupoids, and let homotopy types be represented by spaces or topological spaces. Gr…

Let $\Delta$ be the simplex category, let $\mathbf{Set}$ be a category of sets, and let a simplicial set be a contravariant functor $X:\Delta\to\mathbf{Set}$. Simplicial sets form…

A globular weak $\infty$-groupoid model is any of the unspecified models of globular weak $\infty$-groupoids considered in the conjecture. Let the category of such models be equipp…

Grothendieck's homotopy hypothesis. The functor

Grothendieck's conjecture. The infinity-groupoid $\Pi_\infty(X)$, up to weak equivalence, classifies the homotopy type of $X$.

Grothendieck's conjecture. The functor $\overline{\Pi_\infty}$ is an equivalence of categories.

Let $\mathbf{C}$ be a $\mathsf{Gr}$-coherator with stages $\mathbf{C}_n$, and let \mathchoice{\text{\boldmath Theta}}{\text{\boldmath Theta}}{\text{\boldmath scriptstyle Thet…

Let $\mathbf{C}$ be a $\mathsf{Gr}$-coherator with a tower of definition … Here each $\mathbf{C}_n$ is a stage in the tower and the maps $\mathbf{C}_n\longrightarrow\mathbf{C}$ are…