The \(n\)-cell (or walking \(n\)-morphism) is the \((\infty,n)\)-category \(c_n \coloneqq \Sigma^n[{\sf pt}] \in \mathrm{Cat}_{(\infty, {n})}\).24 Its boundary (or the walking pair of parallel \((n-1)\)-morphisms25) is the \((\infty,n)\)-category \(\partial c_n \coloneqq \partial \Sigma^n[{\sf pt}] \coloneqq \Sigma^n[\emptyset]\) (which is in fact an \((\infty,n-1)\)-category). We use both notations interchangeably, depending on our desired emphasis. We also introduce the notation \[j_n \colon \partial c_n \coloneqq \Sigma^n[\emptyset] \xrightarrow{\Sigma[\emptyset \longrightarrow{\sf pt}]} \Sigma^n[{\sf pt}] \eqqcolon c_n\] for the inclusion, which corepresents the functor taking an \(n\)-morphism to its source and target (which are parallel \((n-1)\)-morphisms).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2