*-autonomous envelopes and 2-conservativity of duals
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We show the doctrine of ∗-autonomous categories is "2-conservative" over the doctrine of closed symmetric monoidal categories, i.e. the universal map from a closed symmetric monoidal category to the ∗-autonomous category that it freely generates is fully faithful. This implies that linear logics and graphical calculi for ∗-autonomous categories can also be interpreted canonically in closed symmetric monoidal categories. In particular, our result implies that every closed symmetric monoidal category can be fully embedded in a ∗-autonomous category, preserving both tensor products and internal-homs. But in fact we prove this directly first with a Yoneda-style embedding (an enhanced "Hyland envelope" that can be regarded as a polycategorical form of Day convolution), and deduce 2-conservativity afterwards from double gluing and a technique of Lafont. Since our method uses polycategories, it also applies to other fragments of ∗-autonomous structure, such as linear distributivity. It can also be enhanced to preserve any desired family of nonempty limits and colimits.
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