regular hyperdoctrine



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



A regular hyperdoctrine, also called an elementary existential doctrine, is a version of a first-order hyperdoctrine that is appropriate for regular logic.


Let CC be a category with finite limits. A regular hyperdoctrine, or elementary existential doctrine, over CC is a functor

P:C opInfSemiLattice P \;\colon\; C^{op} \to InfSemiLattice

from the opposite category of CC to the category of inf-semilattices, such that for every morphism f:ABf : A \to B in CC, the functor P(A)P(B)P(A) \to P(B) has a left adjoint f\exists_f satisfying

  1. Frobenius reciprocity;

  2. Beck-Chevalley condition.


The original definition of Lawvere did not include the last two conditions and allowed the values of the functor to be arbitrary categories with finite products. Other references sometimes require the adjoints f\exists_f to be defined only along projections (corresponding to ordinary existential quantification in logic) and diagonals (corresponding to equality); Lawvere showed that in the presence of Frobenius and Beck-Chevalley more general quantifiers can be constructed in terms of these.


  • William Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor

  • Davide Trotta, The existential completion, 2021 (arxiv:2108.03416)

  • Davide Trotta, An algebraic approach to the completions of elementary doctrines, 2021 (arxiv:2108.03415)

Last revised on August 19, 2021 at 05:30:06. See the history of this page for a list of all contributions to it.