nLab
hyperdoctrine

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 $T$ 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 $T$ for a given notion of logic $L$ is a functor

$$P \colon T^{op} \to \mathbf{C}$$

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

In the most classical case, $L$ is propositional logic, and $\mathbf{C}$ 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 $P$ sends $A \in T$ to the poset of subobjects $Sub_T(A)$ of $A$. The predicate logic we obtain in this way is the usual sort of internal logic of $T$.

We generally require also that for every morphism $f \colon A \to B$ the morphism $P(f)$ 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 $f$. Thus, a hyperdoctrine can also be regarded as a way of “adding quantifiers” to a given kind of logic.

More precisely, one thinks of

• $T$ as a category of types or rather contexts;

• $P$ as assigning

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

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

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

• the right adjoint to $P(f)$ 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

• $Cont$, that form a category of contexts of a first-order theory;

• $Lang$ that forms the internal language of a hyperdoctrine

constitute an equivalence of categories

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

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

Examples

Classes of hyperdoctrines

• first-order hyperdoctrine

• modal hyperdoctrine

• coherent hyperdoctrine

• Boolean hyperdoctrine

• linear hyperdoctrine

Special cases

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

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

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

• $T$ = the category of small sets, $P(X) = Set^X$ … “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 $P(X)$, but a syntactically presented one”.

• $T$ = Cat, the category of small categories, $P(B) = 2^B$

• $T$ = Cat the category of small categories, $P(B) = Set^B$

• $T$ = Grpd the category of small groupoids, $P(B) = Set^B$

Related notions

• comprehension

• tripos

• modal hyperdoctrine

References

The notion was introduced in

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

and further 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. (pdf)

• 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

• 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

• Geraldine Brady, Todd Trimble, A string diagram calculus for predicate logic