Abstract
We generalize the adjoint logics of Benton and Wadler [1994, 1996] and Reed [2009] to allow multiple different adjunctions between the same categories. This provides insight into the structural proof theory of cohesive homotopy type theory, which integrates the synthetic homotopy theory of homotopy type theory with the synthetic topology of Lawvere’s axiomatic cohesion. Reed’s calculus is parametrized by a preorder of modes, where each mode determines a category, and there is an adjunction between categories that are related by the preorder. Here, we consider a logic parametrized by a 2-category of modes, where each mode represents a category, each mode morphism represents an adjunction, and each mode 2-morphism represents a pair of conjugate natural transformations. Using this, we give mode theories that describe adjoint triples of the sort used in cohesive homotopy type theory. We give a sequent calculus for this logic, show that identity and cut are admissible, show that this syntax is sound and complete for pseudofunctors from the mode 2-category to the 2-category of adjunctions, and investigate some constructions in the example mode theories.
This material is based on research sponsored by The United States Air Force Research Laboratory under agreement number FA9550-15-1-0053. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Air Force Research Laboratory, the U.S. Government, or Carnegie Mellon University.
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Acknowledgments
We thank Jason Reed for helpful discussions about this paper and work, and we thank the anonymous reviewers for helpful feedback on a previous draft.
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Licata, D.R., Shulman, M. (2016). Adjoint Logic with a 2-Category of Modes. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_16
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DOI: https://doi.org/10.1007/978-3-319-27683-0_16
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