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\begin{document}

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\section*{hyperdoctrine}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classes_of_hyperdoctrines}{Classes of hyperdoctrines}\dotfill \pageref*{classes_of_hyperdoctrines} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of a \emph{hyperdoctrine} is essentially an axiomatization of the collection of [[slice category|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'' (\hyperlink{Lawvere69}{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 [[quantifier|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]]

\begin{displaymath}
P \colon T^{op} \to \mathbf{C}
\end{displaymath}
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 logic|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

\begin{itemize}%
\item $T$ as a category of [[types]] or rather [[contexts]];


\item $P$ as assigning

\begin{itemize}%
\item to each context $X \in T$ the [[lattice]] of [[propositions]] in this context;


\item to each morphism $f \colon X \to Y$ in $T$ the operation of ``substitution of variables'' / ``extension of contexts'' for propositions $P(Y) \to P(X)$;



\end{itemize}

\item the left adjoint to $P(f)$ gives the application of the [[existential quantifier]];


\item the right adjoint to $P(f)$ gives the application of the [[universal quantifier]] (see there for the interpretation of quantifiers in terms of adjoints).


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


\item Frobenius reciprocity expresses the derivation rules.



\end{itemize}
So, in particular, a hyperdoctrine is a kind of [[indexed category]] or [[Grothendieck fibration|fibered category]].

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

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\begin{theorem}
\label{}\hypertarget{}{}
The [[functors]]

\begin{itemize}%
\item $Cont$, that form a [[category of contexts]] of a [[first-order logic|first-order theory]];


\item $Lang$ that forms the [[internal language]] of a [[hyperdoctrine]]



\end{itemize}
constitute an [[equivalence of categories]]

\begin{displaymath}
FirstOrderTheories
    \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}}
  Hyperdoctrines
  \,.
\end{displaymath}
\end{theorem}
This is due to (\hyperlink{SeelyA}{Seely, 1984a}). For more details see \emph{[[relation between type theory and category theory]]}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{classes_of_hyperdoctrines}{}\subsubsection*{{Classes of hyperdoctrines}}\label{classes_of_hyperdoctrines}

\begin{itemize}%
\item [[first-order hyperdoctrine]]


\item [[modal hyperdoctrine]]


\item [[coherent hyperdoctrine]]


\item [[Boolean hyperdoctrine]]


\item [[linear hyperdoctrine]]



\end{itemize}
\hypertarget{special_cases}{}\subsubsection*{{Special cases}}\label{special_cases}

\begin{itemize}%
\item $T$ = the category of contexts, $P(X)$ is the category of formulas. ``Given any theory (several sorted, intuitionistic or [[first-order hyperdoctrine|classical]]) \ldots{}''


\item $T$ = 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'').


\item $T$ = the category of small sets, $P(X) = Set^X$ \ldots{} ``This hyperdoctrine may be viewed as a kind of set-theoretical surrogate of proof theory''


\item ``honest proof theory would presumably yield a hyperdoctrine with nontrivial $P(X)$, but a syntactically presented one''.


\item $T$ = [[Cat]], the category of small categories, $P(B) = 2^B$


\item $T$ = [[Cat]] the category of small categories, $P(B) = Set^B$


\item $T$ = [[Grpd]] the category of small groupoids, $P(B) = Set^B$



\end{itemize}
\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

\begin{itemize}%
\item [[comprehension]]


\item [[tripos]]


\item [[modal hyperdoctrine]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion was introduced in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Adjointness in Foundations}, (\href{http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html}{TAC}), Dialectica 23 (1969), 281-296

\end{itemize}
and further developed in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]}, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14. (\href{https://ncatlab.org/nlab/files/LawvereComprehension.pdf}{pdf})


\item [[R. A. G. Seely]], \emph{Hyperdoctrines, natural deduction, and the Beck condition}, Zeitschrift f\"u{}r math. Logik und Grundlagen der Math., Band 29, 505-542 (1983). (\href{http://www.math.mcgill.ca/~rags/ZML/ZML.PDF}{pdf})



\end{itemize}
Surveys and reviews include

\begin{itemize}%
\item [[Anders Kock]], [[Gonzalo Reyes]], \emph{Doctrines in categorical logic}, in J. Barwise (ed.) \emph{Handbook of Mathematical Logic} (North Holland, Amsterdam, 1977) 283-313


\item [[Peter Dybjer]], \emph{(What I know about) the history of the identity type} (2006) (\href{http://www.cse.chalmers.se/~peterd/papers/historyidentitytype.pdf}{pdf slides})



\end{itemize}
A [[string diagram]] calculus for monoidal hyperdoctrines is discussed in

\begin{itemize}%
\item Geraldine Brady, [[Todd Trimble]], \emph{[[A string diagram calculus for predicate logic]]}

\end{itemize}
[[!redirects hyperdoctrines]]



\end{document}