nLab regular hyperdoctrine

Contents

Context

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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

semantics

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

Contents

Idea

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

Definition

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.

Remark

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.

categoryfunctorinternal logictheoryhyperdoctrinesubobject posetcoverageclassifying topos
finitely complete categorycartesian functorcartesian logicessentially algebraic theory
lextensive categorydisjunctive logic
regular categoryregular functorregular logicregular theoryregular hyperdoctrineinfimum-semilatticeregular coverageregular topos
coherent categorycoherent functorcoherent logiccoherent theorycoherent hyperdoctrinedistributive latticecoherent coveragecoherent topos
geometric categorygeometric functorgeometric logicgeometric theorygeometric hyperdoctrineframegeometric coverageGrothendieck topos
Heyting categoryHeyting functorintuitionistic first-order logicintuitionistic first-order theoryfirst-order hyperdoctrineHeyting algebra
De Morgan Heyting categoryintuitionistic first-order logic with weak excluded middleDe Morgan Heyting algebra
Boolean categoryclassical first-order logicclassical first-order theoryBoolean hyperdoctrineBoolean algebra
star-autonomous categorymultiplicative classical linear logic
symmetric monoidal closed categorymultiplicative intuitionistic linear logic
cartesian monoidal categoryfragment {&,}\{\&, \top\} of linear logic
cocartesian monoidal categoryfragment {,0}\{\oplus, 0\} of linear logic
cartesian closed categorysimply typed lambda calculus

References

  • 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 November 16, 2022 at 06:02:17. See the history of this page for a list of all contributions to it.