nLab coherent 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

Definition

Let CC be a category with finite limits. A coherent hyperdoctrine over CC is a functor

P:C opDistLattice P : C^{op} \to DistLattice

from the opposite category of CC to the category of distributive lattices, 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.

Properties

The assignment (here) of a coherent hyperdoctrine S(C)S(C) to a coherent category CC extends to a 2-adjunction

(AS):CoherentCatSACoherentHyperdoctrine (A \dashv S) : CoherentCat \stackrel{\overset{A}{\leftarrow}}{\underset{S}{\hookrightarrow}} CoherentHyperdoctrine

with the right adjoint being a full and faithful 2-functor, hence exhibiting CoherentCatCoherentCat as a reflective sub-2-category of CoherentHyperdoctrineCoherentHyperdoctrine.

(Here CoherentCatCoherentCat has as 2-morphisms those natural transformations that preserve finite products.)

This appears as (Coumans, prop. 8).

Coherent hyperdoctrines are closed under canonical extension () δ:DistLatticeDistLattice(-)^\delta : DistLattice \to DistLattice, in that for P:C opDistLatticP : C^{op} \to DistLattic a coherent hyperdoctrine, so is () δP(-)^\delta \circ P.

This appears as (Coumans, prop. 9).

Examples

Powersets

The powerset functor

P:={0,1} ():Set opDistLattice P := \{0,1\}^{(-)} : Set^{op} \to DistLattice

(sending a set to its power set and a function to the preimage-assignment) is a coherent hyperdoctrine.

Over a coherent category

Let CC be a coherent category. For every object ACA \in C the poset of subobjects Sub C(A)Sub_C(A) is a distributive lattice.

The corresponding functor

C opDistLattice C^{op} \to DistLattice

from the opposite category of CC to the category of distributive lattices is called the coherent hyperdoctrine of CC.

For a coherent theory

Accordingly, there is a coherent hyperdoctrine associated with any coherent theory, where the objects of CC are lists of free variables in the theory, and the lattice assigned to them is that of propositions of the theory in this context.

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

  • Dion Coumans, Generalizing canonical extensions to the categorical setting (Arxiv)

Last revised on November 16, 2022 at 06:02:34. See the history of this page for a list of all contributions to it.