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Definition (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 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 $T$.