Kerodon

Variant 9.1.2.3. Let $A$ be a well-ordered set and let $\alpha $ denote its order type. Then there is a unique order-preserving bijection $\mathrm{Ord}_{ < \alpha } \simeq A$, which determines an isomorphism of simplicial sets $u: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \simeq \operatorname{N}_{\bullet }(A)^{\triangleright }$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category containing a morphism $f$ and a collection of morphisms $W$, we will say that a diagram $F: \operatorname{N}_{\bullet }(A)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ exhibits $f$ as a transfinite composition of morphisms of $W$ if the composition $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \simeq \operatorname{N}_{\bullet }(A)^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}}$ exhibits $f$ as a transfinite composition of morphisms of $W$, in the sense of Definition 9.1.2.1.