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The new notion of weak $*$-autonomous category is a very simple generalization of the one of $*$-autonomous category, and a specialization of the one of closed monoidal category, which allows to speak about duality in more situations. For example, the category $Vec_{\mathbb{K}}$ is a weak $*$-autonomous category but is not a $*$-autonomous category.
Any return/collaboration would be very much appreciated.
The morphism $d : A \multimap (A \multimap \bot) \multimap \bot$ is the unit for the continuation monad, and so this is asking that the unit of this monad is a mono. I’m not sure this has much in common with *-autonomous categories, but it is typical in programming language semantics that the unit of the monad is mono, or even stronger that the unit of the monad is the equalizer of $\eta T$ and $T\eta : T \to T^2$ (see Moggi ’88, Computational lambda-calculus and monads)
The idea comes from models of differential linear logic which are $*$-autonomous categories but this is often too restrictive. For instance a lot of categories of topological vector spaces must verify this definition. So this is already linked with programming language semantics. I’m going to take a look at what you say! It seems very to the point in fact.
I’m wondering if the unit of the monad is this equalizer in $Vec_{\mathbb{K}}$.
I don’t know what are the multiplication and the unit of the continuation monad?
Also, it seems to be related to $*$-autonomous categories by the paper “Linear continuation and duality” (Paul-André Melliès, Nicolas Tabareau), 2008 on the nlab page.
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