I asked this question fourteen days ago on MathStackexchange (see [here](https://math.stackexchange.com/questions/4460652/star-autonomous-categories-are-categorifications-of-boolean-algebras)). 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](https://meta.mathoverflow.net/questions/5012/can-i-ask-a-question-on-mathoverflow-and-also-on-another-site/5013#5013), I therefore have decided to cross-post the question on this site.


**1. Question**  

The [n-Lab article](https://ncatlab.org/nlab/show/Chu+construction) on the Chu-construction says:



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> "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) […]."
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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](https://en.wikipedia.org/wiki/*-autonomous_category#Examples):



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> "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."
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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).“



Star-autonomous categories are categorifications of Boolean algebras?
The starting point for decategorification is the observation that a category in which any parallel arrows are equal must necessarily be a preorder.
Restricting to skeletal categories makes it a poset.


Thus, we might as well ask: what is a \*-autonomous category whose underlying category is a poset?


For the specific case when the poset is a Boolean algebra,
we set $A→B=A⊸B$, and $A⊢B$ means there is a morphism $A→B$.
We have $A⊗X≅(A⊸X^\*)^\*$, which means $A⊗X=A∧B$.
Most axioms of propositional logic are now straightforward to verify.
In particular, we can identify the global dualizing object $⊥$:
if for all objects $A$ the canonical map $A→(A⊸⊥)⊸⊥$
is an isomorphism, then $⊥$ must be the initial object,
since $B⊢(A→B)$ in our case.


More generally, \*-autonomous posets that are not Boolean algebras provide models for [linear logic](https://ncatlab.org/nlab/show/linear%20logic). So in the strict sense \*-autonomous categories do not categorify Boolean algebras (or rigs), but rather the corresponding algebraic structure for linear logic.


