Definition 4.2.4.1 (Left Anodyne Morphisms). Let $T_{L}$ be the smallest collection of morphisms in the category $\operatorname{Set_{\Delta }}$ with the following properties:
For every pair of integers $0 \leq i < n$, the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T_{L}$.
The collection $T_{L}$ is weakly saturated (Definition 1.5.4.12). 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 $T_{L}$.