Does the forgetful functor from an over-$(\infty,1)$-category create weakly contractible limits?

Question

Let's say that a limit diagram $\bar p:K^\triangleleft\to\def\sC{\mathscr{C}}\sC$ is a weakly contractible limit if the simplicial set $K$ is weakly contractible (in that $K\to*$ is a weak homotopy equivalence).

I want to say for an $\infty$-category $\sC$ and an object $x\in\sC_0$ that the forgetful functor $\sC_{/x}\to\sC$ creates (i.e., preserves and reflects) weakly contractible limits. Is there a reference that already has this result?

The nLab shows here that the limit of a diagram $p:K\to\sC_{/x}$ coincides with the limit of the corresponding diagram $p/x:K^\triangleright\to\sC$, so this reduces the problem to showing that the inclusion $K\hookrightarrow K^\triangleright$ is an initial map (using the nLab's convention, and thus is dual to the notion of a "cofinal map" in the sense of Lurie's Higher Topos Theory). By Proposition 4.1.1.3(4) of Higher Topos Theory, this is equivalent to showing that $K\hookrightarrow K^\triangleright$ is left anodyne.

So perhaps an equivalent question is: if $K$ is a weakly contractible simplicial set, then is $K\hookrightarrow K^\triangleright$ left anodyne? Is there an easy way to see this?

Answer 1

You can also use Quillen's theorem A : to prove that $C \to D$ is initial, it suffices to show that for every $d\in D$, $C \times_D D_{/d}$ is weakly contractible.

In the case where $C\to D$ is fully faithful, this is always the case for $d$ in the image of $C$ as this pullback has a terminal object. So here we are left with proving that this is the case for the cone point, which I'm going to call $\infty$. For that one, as it is terminal in $K^\triangleright$, we see that $K \times_{K^\triangleright} (K^\triangleright)_{/\infty} \simeq K$ is weakly contractible, by assumption.