nLab 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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity 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 set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
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


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



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 algebraization of logic.



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.


Classes of hyperdoctrines

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

Related notions

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


The notion was introduced in

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

and further developed in

and with further emphasis on the role of the Beck-Chevalley condition:

  • Robert A. G. Seely, Hyperdoctrines, Natural Deduction and the Beck Condition, Zeitschr. f. math Logik und Grundlagen d. Math. 29 (1983) 505-542 [pdf]

Surveys and reviews include

  • Anders Kock, Gonzalo Reyes, Doctrines in categorical logic, in J. Barwise (ed.) Handbook of Mathematical Logic (North Holland, Amsterdam, 1977) 283-313

  • 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

A concise introduction to posetal hyperdoctrines is given in

Last revised on April 4, 2024 at 22:14:57. See the history of this page for a list of all contributions to it.