nLab
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

Idea

The notion of a hyperdoctrine is essentially an axiomatization of the collection of slices of a locally cartesian closed category (or something similar): a category TT together with a functorial assignment of “slice-like”-categories to each of its objects, satisfying some conditions.

In its use in mathematical logic (“categorical logic” (Lawvere 69)) a hyperdoctrine is thought of (under categorical semantics of logic/type theory) as a collection of contexts together with the operations of context extension/substitution and quantification on the categories of propositions or types? in each context. Therefore specifying the structure of a hyperdoctrine over a given category is a way of equipping that with a given kind of logic.

Specifically, a hyperdoctrine on a category TT for a given notion of logic LL is a functor

P:T opC P \colon T^{op} \to \mathbf{C}

to some 2-category (or even higher category) C\mathbf{C}, whose objects are categories whose internal logic corresponds to LL. Thus, each object AA of TT is assigned an “LL-logic” (the internal logic of P(A)P(A)).

In the most classical case, LL is propositional logic, and C\mathbf{C} is a 2-category of posets (e.g. lattices, Heyting algebras, or frames). A hyperdoctrine is then an incarnation of first-order predicate logic. A canonical class of examples of this case is where PP sends ATA \in T to the poset of subobjects Sub T(A)Sub_T(A) of AA. The predicate logic we obtain in this way is the usual sort of internal logic of TT.

We generally require also that for every morphism f:ABf \colon A \to B the morphism P(f)P(f) has both a left adjoint as well as a right adjoint, typically required to satisfy Frobenius reciprocity and the Beck-Chevalley condition. These adjoints are regarded as the action of quantifiers along ff. Thus, a hyperdoctrine can also be regarded as a way of “adding quantifiers” to a given kind of logic.

More precisely, one thinks of

  • TT as a category of types? or rather contexts;

  • PP as assigning

    • to each context XTX \in T the lattice of propositions in this context;

    • to each morphism f:XYf \colon X \to Y in TT the operation of “substitution of variables” / “extension of contexts” for propositions P(Y)P(X)P(Y) \to P(X);

  • the left adjoint to P(f)P(f) gives the application of the existential quantifier;

  • the right adjoint to P(f)P(f) gives the application of the universal quantifier (see there for the interpretation of quantifiers in terms of adjoints).

  • The Beck-Chevalley condition ensures that quantification interacts with substitution of variables as expected

  • Frobenius reciprocity expresses the derivation rules.

So, in particular, a hyperdoctrine is a kind of indexed category or fibered category.

The general concept of hyperdoctrines does for predicate logic precisely what Lindenbaum-Tarski algebras do for propositional logic, positioning the categorical formulation of logic as a natural extension of the algebraicization of logic.

Properties

Theorem

The functors

constitute an equivalence of categories

FirstOrderTheoriesContLangHyperdoctrines. FirstOrderTheories \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}} Hyperdoctrines \,.

This is due to (Seely, 1984a). For more details see relation between type theory and category theory.

Examples

Classes of hyperdoctrines

See also

Special cases

  • TT = the category of contexts, P(X)P(X) is the category of formulas. “Given any theory (several sorted, intuitionistic or classical) …”

  • TT = the category Set of small sets, P(X)=2 X=P(X) = 2^X = the power set functor, assigning the poset of all propositional functions

    (“or one may take suitable ‘homotopy classes’ of deductions”).

  • TT = the category of small sets, P(X)=Set XP(X) = Set^X … “This hyperdoctrine may be viewed as a kind of set-theoretical surrogate of proof theory”

  • “honest proof theory would presumably yield a hyperdoctrine with nontrivial P(X)P(X), but a syntactically presented one”.

  • TT = Cat, the category of small categories, P(B)=2 BP(B) = 2^B

  • TT = Cat the category of small categories, P(B)=Set BP(B) = Set^B

  • TT = Grpd the category of small groupoids, P(B)=Set BP(B) = Set^B

References

The notion was introduced in

  • Bill Lawvere, Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296

and developed in

  • Bill Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

  • R. A. G. Seely, Hyperdoctrines, natural deduction, and the Beck condition, Zeitschrift für math. Logik und Grundlagen der Math., Band 29, 505-542 (1983). (pdf)

Surveys and reviews include

  • Peter Dybjer, (What I know about) the history of the identity type (2006) (pdf slides)

A string diagram calculus for monoidal hyperdoctrines is discussed in

Revised on February 8, 2014 16:49:59 by Urs Schreiber (89.204.139.189)