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 algebraization 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

• regular hyperdoctrine

• coherent hyperdoctrine

• Boolean hyperdoctrine

• geometric 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

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus

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)

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

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

A concise introduction to posetal hyperdoctrines is given in

• Thomas Streicher, Categorical models of constructive logic.