# Week 6 Notes

- EX 4.4:
The diagram is

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).