Definition 9.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that a morphism $f$ of $\operatorname{\mathcal{C}}$ is a transfinite composition of morphisms of $W$ if there exists an ordinal $\alpha $ and a functor $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ with the following properties:
- $(a)$
For every nonzero limit ordinal $\lambda \leq \alpha $, the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \lambda }) }$ is a colimit diagram: that is, it exhibits $F( \lambda )$ as a colimit of the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \lambda } )}$.
- $(b)$
For every ordinal $\beta < \alpha $, the morphism $F(\beta ) \rightarrow F(\beta +1)$ belongs to $W$.
- $(c)$
The morphism $F(0) \rightarrow F(\alpha )$ coincides with $f$.
In this case, we will say that $F$ exhibits $f$ as a transfinite composition of morphisms of $W$.
We say that $W$ is closed under transfinite composition if it contains every morphism which is a transfinite composition of morphisms of $W$.