Definition 3.1.2.1 (Anodyne Morphisms). Let $T$ be the smallest collection of morphisms in the category $\operatorname{Set_{\Delta }}$ with the following properties:
For each $n > 0$ and each $0 \leq i \leq n$, the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T$.
The collection $T$ is weakly saturated (Definition 1.5.4.12). That is, $T$ is closed under pushouts, retracts, and transfinite composition.
We say that a morphism of simplicial sets $i: A_{} \rightarrow B_{}$ is anodyne if it belongs to the collection $T$.