The Stacks project

Remark 14.19.11. We do not need all finite limits in order to be able to define the coskeleton functors. Here are some remarks

1. We have seen in Example 14.19.1 that if $\mathcal{C}$ has products of pairs of objects then $\text{cosk}_0$ exists.

2. For $k > 0$ the functor $\text{cosk}_ k$ exists if $\mathcal{C}$ has finite connected limits.

This is clear from the inductive procedure of constructing coskeleta (Remarks 14.19.8 and 14.19.9) but it also follows from the fact that the categories $(\Delta /[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ used in Lemma 14.19.2 are connected. Observe that we do not need the categories for $n \leq k$ by Lemma 14.19.3 or Lemma 14.19.4. (As $k$ gets higher the categories $(\Delta /[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ are more and more connected in a topological sense.)