## § Subobject classifier measures how much we need to pay to access fact

- Truths are free. We don't pay any of the monoid (given $A \rightarrow B$, subobj assigns full monoid to image of $A$ in $B$).
- We go bankrupt trying to prove really false things (subobj assigns emptyset)
- To things that are truth adjacent, we spend some of our monoid (by dividing the filter) to get to the element. So we "spend money" to "fix the lie" of the element in $B$ not being truthful, but close enough to the truth.