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We give a natural-deduction-style type theory for symmetric monoidalcategories whose judgmental structure directly represents morphisms with tensorproducts in their codomain as well as their domain. The syntax is inspired bySweedler notation for coalgebras, with variables associated to types in thedomain and terms associated to types in the codomain, allowing types to betreated informally like "sets with elements" subject to global linearity-likerestrictions. We illustrate the usefulness of this type theory with variousapplications to duality, traces, Frobenius monoids, and (weak) Hopf monoids.

A practical type theory for symmetric monoidal categories