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Definition Let $X_{\bullet }$ be a simplicial set. We will say that a $2$-simplex $\sigma $ of $X_{\bullet }$ is thin if it satisfies the following condition:

$(\ast )$

Let $n \geq 3$, let $0 < i < n$, and let $\tau $ denote the $2$-simplex of $\Lambda ^{n}_{i}$ given by the map

\[ [2] \simeq \{ i-1, i, i+1 \} \subseteq [n]. \]

Then any map of simplicial sets $f_0: \Lambda ^{n}_{i} \rightarrow X_{\bullet }$ satisfying $f_0( \tau ) = \sigma $ can be extended to an $n$-simplex of $X_{\bullet }$.