Quasi-category

Remarks

The vertices of $X$ are its objects and the $1$-simplices its morphisms. For $n = 2$, $k = 1$, the condition says that two morphisms $f \colon x \to y$ and $g \colon y \to z$ have a composite: a $2$-simplex with edges $f$, $g$ and some $h \colon x \to z$. Example: the nerve $N(\mathcal{C})$ of a category $\mathcal{C}$ is a quasi-category, and in it every inner horn has exactly one filler. Example: every Kan complex is a quasi-category, since it has fillers for all horns, inner or not; the singular simplicial set of a topological space is one. Non-example: the horn $\Lambda^2_1$ itself is not a quasi-category. Its identity map $\Lambda^2_1 \to \Lambda^2_1$ does not extend to $\Delta^2$, because every $2$-simplex of $\Lambda^2_1$ is degenerate. A degenerate $2$-simplex cannot have the two distinct nondegenerate edges $0 \to 1$ and $1 \to 2$ as faces. Quasi-categories are also called weak Kan complexes or $\infty$-categories.