Lemma 3.7.1. Suppose that $T = \mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } T_\alpha$ is a colimit of sets indexed by ordinals less than a given ordinal $\beta$. Suppose that $\varphi : S \to T$ is a map of sets. Then $\varphi$ lifts to a map into $T_\alpha$ for some $\alpha < \beta$ provided that $\beta$ is not a limit of ordinals indexed by $S$, in other words, if $\beta$ is an ordinal with $\text{cf}(\beta ) > |S|$.

Proof. For each element $s \in S$ pick a $\alpha _ s < \beta$ and an element $t_ s \in T_{\alpha _ s}$ which maps to $\varphi (s)$ in $T$. By assumption $\alpha = \sup _{s \in S} \alpha _ s$ is strictly smaller than $\beta$. Hence the map $\varphi _\alpha : S \to T_\alpha$ which assigns to $s$ the image of $t_ s$ in $T_\alpha$ is a solution. $\square$

Comment #143 by on

Replace $T_{\beta}$ by $T$.

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