Often to prove that the Kanification of a simplicial set $X_\bullet$ is contractible, we instead prove that $X_\bullet$ is a contractible Kan complex (i.e. satisfies the extension property for inclusions $\partial\Delta^n\hookrightarrow\Delta^n$), which is clearly a stronger property.
I find myself in the situation of proving that a simplicial set $X_\bullet$ is a filtered $\infty$-category by proving an analogous stronger property, whose name I don't know. Does this property have a name? The property (of a simplicial set $X_\bullet$) is:
- A map $\partial\Delta^n\to X_\bullet$ with $n\geq 2$ extends to $\Delta^n$.
- The relation $\leq$ on $X_0$ defined by $p\leq q$ iff there is a $1$-simplex from $p$ to $q$ is transitive, i.e. $p\leq q\leq r\implies p\leq r$.
It is an easy exercise to show that these two properties imply that $X_\bullet$ is an $\infty$-category. Moreover, they imply that $\left|X_\bullet\right|$ is a poset and that the functor $X_\bullet\to\left|X_\bullet\right|$ ($\left|X_\bullet\right|$ denotes the homotopy category of $X_\bullet$, and a category $\mathcal C$ is called a poset iff there is at most one morphism $x\to y$ for any $x,y\in\mathcal C$). We thus conclude that:
- $X_\bullet$ is a filtered $\infty$-category iff $\left|X_\bullet\right|$ is a filtered poset.
Of course, $X_\bullet$ satisfying the above two conditions and $\left|X_\bullet\right|$ being a poset are much stronger conditions than $X_\bullet$ being a filtered $\infty$-category. Do they have a name?