Week 6 Notes

• EX 4.4:
1. The diagram is

2. Define $$\Lambda : \mathcal{X} \unicode{x21F8} \mathcal{Y}$$ as a monotone map from the diagram in 1. to the diagram below

The preimage of $$\texttt{true}$$ forms an upper set in $$\mathcal{X}^\text{op} \times \mathcal{Y}$$, meaning if "my aunt can explain an $$x$$ given $$\text{nothing}$$", then "my aunt can explain an $$x$$ given $$\text{book}$$", and if "my aunt can explain an $$x \in \mathcal{X}^\text{op}$$ given $$y \in Y$$ where $$x \leq x'$$ for some $$x' \in \mathcal{X}^\text{op}$$", then "my aunt can explain an $$x' \in \mathcal{X}^\text{op}$$ given $$y \in Y$$". This corresponds to $$\Lambda$$ being a monotone map (see remark after D 4.2).

Created: 2020-01-13 Mon 14:36