Higher category

$(𝑛, 𝑟)$ -category

Let $𝑛 \geq - 2$ and $𝑟 \geq 0$ .¹ An $(𝑛, 𝑟)$ -category is an $\infty$ -category in which #m/def/ncat

for $𝑘 > 𝑛$ , any two parallel² $𝑘$ -cells are equivalent;
for $𝑘 > 𝑟$ , all $𝑘$ -cells are equivalences.

This classification is a generalization of h-level.

#state/develop | #lang/en | #SemBr

Typically we also take $𝑟 \leq m a x {0, 𝑛 + 1}$ , since increasing $𝑟$ further does not change the resulting structure. ↩
i.e. $𝑘$ -cells with domains the same and co domains the same. ↩