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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?

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    $\begingroup$ Isn't ∞-groupoid just a synonym of Kan complex? Moreover $C\to hC$ is an equivalence iff $C$ is (the nerve of) an ordinary category. $\endgroup$ Jan 22, 2017 at 17:18
  • $\begingroup$ Yes, indeed! I will edit . . . $\endgroup$ Jan 22, 2017 at 18:38

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An $\infty$-category $\mathsf C$ satisfies the extension property for the pairs $(\Delta^n,\partial\Delta^n)$ for all $n\geq 2$ iff $\mathrm{Hom}(x,y)\in\{*,\varnothing\}$ for all objects $x,y\in\mathsf C$.

So this is a partial answer to the question: this property is equivalent to $\mathsf C$ being an $\infty$-category which is equivalent to a poset.

This is roughly analogous to the fact that a simplicial set has the extension property for the pairs $(\Delta^n,\partial\Delta^n)$ for all $n\geq 0$ iff it is a contractible (i.e. htpy equiv to $*$) Kan complex.

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