Definition 1.5.4.10. For every ordinal $\alpha $, let $\mathrm{Ord}_{\leq \alpha } = \{ \beta : \beta \leq \alpha \} $ denote the collection of all ordinal numbers which are less than or equal to $\alpha $, regarded as a linearly ordered set.
Let $\operatorname{\mathcal{C}}$ be a category and let $S$ 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 $S$ if there exists an ordinal $\alpha $ and a functor $F: \mathrm{Ord}_{\leq \alpha } \rightarrow \operatorname{\mathcal{C}}$, given by a collection of objects $\{ C_{\beta } \} _{ \beta \leq \alpha }$ and morphisms $\{ f_{\gamma , \beta }: C_{\beta } \rightarrow C_{\gamma } \} _{\beta \leq \gamma }$ with the following properties:
- $(a)$
For every nonzero limit ordinal $\lambda \leq \alpha $, the functor $F$ exhibits $C_{\lambda }$ as a colimit of the diagram $( \{ C_{\beta } \} _{\beta < \lambda }, \{ f_{\gamma , \beta } \} _{\beta \leq \gamma < \lambda } )$.
- $(b)$
For every ordinal $\beta < \alpha $, the morphism $f_{\beta +1, \beta }$ belongs to $S$.
- $(c)$
The morphism $f$ is equal to $f_{\alpha ,0}: C_0 \rightarrow C_{\alpha }$.
We will say that $S$ is closed under transfinite composition if, for every morphism $f$ which is a transfinite composition of morphisms of $S$, we have $f \in S$.