The Stacks project

Remark 14.21.6. In some texts the composite functor

\[ \text{Simp}(\mathcal{C}) \xrightarrow {\text{sk}_ m} \text{Simp}_ m(\mathcal{C}) \xrightarrow {i_{m!}} \text{Simp}(\mathcal{C}) \]

is denoted $\text{sk}_ m$. This makes sense for simplicial sets, because then Lemma 14.21.5 says that $i_{m!} \text{sk}_ m V$ is just the sub simplicial set of $V$ consisting of all $i$-simplices of $V$, $i \leq m$ and their degeneracies. In those texts it is also customary to denote the composition

\[ \text{Simp}(\mathcal{C}) \xrightarrow {\text{sk}_ m} \text{Simp}_ m(\mathcal{C}) \xrightarrow {\text{cosk}_ m} \text{Simp}(\mathcal{C}) \]

by $\text{cosk}_ m$.

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