nLab
coherent hyperdoctrine

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

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
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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive 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, (idemponent) 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.

References

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

Revised on May 24, 2013 00:46:08 by David Corfield (87.113.28.17)