7 problems

An $n$-type is a homotopy type whose homotopy groups vanish in degrees greater than $n$. An $n$-groupoid is an algebraic structure intended to encode homotopical information throug…

Consider the categories of special $\operatorname{cat}^{n-1}\text{-}\operatorname{group}$s and fair $n$-groupoids, together with the strongly contractible faces of a special…

A fair $n$-groupoid is Joachim Kock's formalization of a strict $n$-groupoid equipped with a suitable notion of weak identity arrow. Kock-Simpson conjecture. Every homotopy $n$-typ…

A fair $n$-groupoid is Joachim Kock's semistrict structure in which composition is strict while identity arrows are suitably weak. Kock's conjecture. Fair $n$-groupoids should mode…

Let $A{\rightarrow} W$ be a map of $n$-groupoids, and let $j$ be a nonnegative integer. For each $k$, write $\pi_k$ for the corresponding homotopy group of an $n$-groupoid. Strong…

Simpson's conjecture. There exists a notion of strict $n$-groupoid with weak identity arrows and a notion of geometric realisation such that every homotopy $n$-type appears in this…

Let $n$ be a nonnegative integer. Consider the category of weakly unital strict $n$-groupoids, its equivalences, and the category of $n$-truncated spaces with weak equivalences. Le…