Lemma 3.7.1 (05N2)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 3: Set Theory
Section 3.7: Cofinality
Lemma 3.7.1 (cite)

next proposition

numberstags

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. $\square$

Comments (2)

Comment #143 by Fred Rohrer on February 21, 2013 at 06:36

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

Comment #148 by Johan on February 21, 2013 at 18:10

Fixed, see here. Thanks!

There are also:

12 comment(s) on Section 3.7: Cofinality

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (2)

Post a comment