natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/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 $T$ 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
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
$P$ 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.
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
This is due to (Seely, 1984a). For more details see relation between type theory and category theory.
See also
$T$ = 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, $P(B) = Set^B$
The notion was introduced in
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