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Definition (Left Anodyne Morphisms). Let $T_{L}$ be the smallest collection of morphisms in the category $\operatorname{Set_{\Delta }}$ with the following properties:

  • For each $n > 0$ and each $0 \leq i < n$, the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T_{L}$.

  • The collection $T_{L}$ is weakly saturated (Definition That is, $T_{L}$ is closed under pushouts, retracts, and transfinite composition.

We say that a morphism of simplicial sets $f: A \rightarrow B$ is left anodyne if it belongs to $T_{L}$.