**natural deduction** metalanguage, practical foundations

**type theory** (dependent, intensional, observational type theory, homotopy type theory)

**computational trinitarianism** =

**propositions as types** +**programs as proofs** +**relation type theory/category theory**

A *linear hyperdoctrine* is a hyperdoctrine that is adapted for modeling first-order linear logic.

Since there are many variants of linear logic, there are correspondingly many variants of linear hyperdoctrines, but all of them are some kind of indexed monoidal category, or more precisely indexed monoidal poset. We mention only a few of the most important.

A **MILL hyperdoctrine** is a closed indexed monoidal poset with indexed products? and indexed coproducts satisfying the Beck-Chevalley condition and the Frobenius reciprocity condition.

A MILL hyperdoctrine models predicate intuitionistic linear logic, with $\otimes,\mathbf{1},\multimap,\exists,\forall$.

A **MALL hyperdoctrine** is a MILL hyperdoctrine whose fibers are *-autonomous lattices and whose reindexing functors preserve all the $\ast$-autonomous and lattice structure.

A MALL hyperdoctrine models predicate classical linear logic without exponentials, with $\otimes,\mathbf{1},\parr,\bot,\&,\top,\oplus,\mathbf{0},\exists,\forall$.

A **linear-nonlinear hyperdoctrine** is a MALL hyperdoctrine $L$ together with a first-order hyperdoctrine $M$ and a fiberwise monoidal adjunction $F : M \rightleftarrows L : G$.

A linear-nonlinear hyperdoctrine models full predicate classical linear logic, with the exponential modality modeled as the comonad $F G$ and $?$ as its de Morgan dual.

- R. A. G. Seely,
*Linear logic, $\ast$-autonomous categories and cofree coalgebras*,*Contemporary Mathematics*92, 1989. (pdf, ps.gz)

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