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).“