linear hyperdoctrine

Linear hyperdoctrines


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


(0,1)(0,1)-Category theory

Linear hyperdoctrines


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 ,1,,,\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 ,1,,,&,,,0,,\otimes,\mathbf{1},\parr,\bot,\&,\top,\oplus,\mathbf{0},\exists,\forall.


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

A linear-nonlinear hyperdoctrine models full predicate classical linear logic, with the exponential modality modeled as the comonad FGF 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)

Last revised on July 7, 2019 at 09:09:10. See the history of this page for a list of all contributions to it.