# Joyal's CatLab Quasi-categories

• Recall that a simplicial set $X$ is called a Kan complex if every horn $\Lambda^k[n]\to X$ has a filler $\Delta[n]\to X$.
• We shall say that a horn $\Lambda^k[n]$ is inner if $0 \lt k \lt n$, otherwise we shall say that is outer.
• We shall say that a simplicial set $X$ is a quasi-category if every inner horn $\Lambda^k[n]\to X$ has a filler $\Delta[n]\to X$.