*-autonomous envelopes

04/17/2020

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by   Michael Shulman, et al.

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We show that every closed symmetric monoidal category can be fully embedded in a *-autonomous category, preserving both tensor products and internal-homs, as well as any desired nonempty limits and colimits. Thus, linear logics and graphical calculi for *-autonomous categories can also be applied to closed symmetric monoidal categories. The embedding uses an enhanced "Hyland envelope" that can be regarded as a polycategorical form of Day convolution. It also applies to other fragments of *-autonomous structure, such as linear distributivity, and can be combined with a form of double gluing to prove full completeness results.