Kerodon

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Construction 4.3.5.12. Let $f: K \rightarrow X$ be a morphism of simplicial sets. We define a morphism of simplicial sets $c: X_{/f} \star K \rightarrow X$ as follows:

• The restriction of $c$ to the simplicial subset $X_{/f} \subseteq X_{/f} \star K$ is equal to the projection map $X_{/f} \rightarrow X$ of Remark 4.3.5.2.

• The restriction of $c$ to the simplicial subset $K \subseteq X_{/f} \star K$ is equal to $f$.

• Let $\sigma : \Delta ^{n} \rightarrow X_{/f} \star K$ be an $n$-simplex which does not belong to $X_{/f}$ or $K$, so that $\sigma$ factors (uniquely) as a composition

$\Delta ^{n} \simeq \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X_{/f} \star K$

for $p+1+q=n$ (see Remark 4.3.3.15). Using the definition of the simplicial set $X_{/f}$, we can identify $\sigma _{-}$ with a morphism of simplicial sets $\overline{f}: \Delta ^{p} \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$. We then define $c(\sigma )$ to be the $n$-simplex of $X$ given by the composite map

$\Delta ^{n} \simeq \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \operatorname{id}\star \sigma _{+} } \Delta ^{p} \star K \xrightarrow { \overline{f} } X.$

We will refer to $c$ as the slice contraction morphism. Applying a similar construction to the opposite simplicial sets, we obtain a morphism $c': K \star X_{f/} \rightarrow X$ which we will refer to as the coslice contraction morphism.