Lecture notes for Tuesday, Feb. 28
Table of Contents
This was a shorter lecture where all we did was prove the title result in all its gory details. This cruel and unusual step was taken to give a full example of a “diagram chase”, to lend credence to the claim that such proofs are straightforward and tedious [1].
Left adjoints preserve pushouts
We still haven’t completely finished with our calculation of the fundamental groupoid of \(S^1\). Recall what we did: We took two basepoints on \(S^1\), \(a\) and \(b\), and imagine them as the endpoints of a horizontal diameter of the circle. Let \(U\) be the upper half of the circle and \(V\) the vottom [sic] half. We have \(U \cup V = S^1\) and \(U \cap V = \{a,b\}\). Applying van Kampen, we get that the following is a pushout square of groupoids.
| \(\pi_{\le 1}(U \cap V, \{a,b\})\) | \(\to\) | \(\pi_{\le 1}(U, \{a,b\})\) |
| \(\downarrow\) | \(\downarrow\) | |
| \(\pi_{\le 1}(V, \{a,b\})\) | \(\to\) | \(\pi_{\le 1}(S^1, \{a,b\})\) |
[1] I feel I managed to convey the tedium very convincingly.