The Stacks project

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. $\square$

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