*-Autonomous Envelopes and Conservativity

Abstract: We prove 2-categorical conservativity for any {0,T}-free fragment of MALL over its corresponding intuitionistic version: that is, that the universal map from a closed symmetric monoidal category to the *-autonomous category that it freely generates is fully faithful, and similarly for other doctrines. This implies that linear logics and graphical calculi for *-autonomous categories can also be interpreted canonically in closed symmetric monoidal categories. In particular, every closed symmetric monoidal category can be fully embedded in a *-autonomous category, preserving both tensor products and internal-homs. 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 Hyland-Schalk double gluing and a technique of Lafont. The same is true for other fragments of *-autonomous structure, such as linear distributivity, and the embedding can be enhanced to preserve any desired family of nonempty limits and colimits.