I asked this question fourteen days ago on MathStackexchange (see here). I have not received any answers or comments until now. It seems to me that on MathStackexchange not many people are familiar with star-autonomous categories. Following Nick Champion's advice, I therefore have decided to cross-post the question on this site.
1. Question
The n-Lab article on the Chu-construction says:
"Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of Boolean algebras, or perhaps better Boolean rigs), the star-autonomous structure on $\operatorname{Chu}(C,d)$ can pretty much be deduced (or strongly guessed) […]."
How do star-autonomous categories behave as categorified versions of Boolean algebras or Boolean rigs? In what way is the term categorification used here?
2. Wikipedia says
One explanation might be given on wikipedia:
"A degenerate example [of a star-autonomous category] (all homsets of cardinality at most one) is given by any Boolean algebra (as a partially ordered set) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object."
I suppose the internal hom of two objects $a,b$ in this category is $\neg a \lor b $, correct? The dual functor is the complement?
Edit for future readers: The quoted statement on the nLab seems to have been inaccurate. The nLab entry has now been changed to: "Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of linear logic), the star-autonomous structure on $\operatorname{Chu}(C,d)$ can pretty much be deduced (or strongly guessed).“