Let $C$ be an $(\infty,1)$-category (e.g. quasicategory) having all small colimits. If $f_i:x_i \to y_i$ are equivalences in $C$ indexed by a small set $I$, is $f:=\mathrm{colim}_{i \in I} f_i$ an equivalence?
Intuitively, I think the answer is no because you can glue a bunch of contractible simplices to get a non-contractible simplex...but maybe my intuition is wrong since I might need to take homotopy colimits (which I don't really understand) to fix this.
I ask because I'm trying to find the justification for a claim a proof on p.410 of Lurie's Higher Topos Theory, here is the relevant sentence (the definition of the category $P^K_R(C)$ I don't think is relevant for my question. The set $\mathcal{K}$ is a small collection of small simplicial sets):
My question really is, why is $\mathcal{X}$ is stable under $\mathcal{K}$-indexed colimits?
Yeah, since colimits in infinity-categories are homotopy colimits.