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\begin{document}

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\section*{categorical semantics}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

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

[[!include type theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{of_dependent_type_theory}{Of dependent type theory}\dotfill \pageref*{of_dependent_type_theory} \linebreak
\noindent\hyperlink{ContextsAndTypeJudgements}{Contexts and type judgements}\dotfill \pageref*{ContextsAndTypeJudgements} \linebreak
\noindent\hyperlink{terms}{Terms}\dotfill \pageref*{terms} \linebreak
\noindent\hyperlink{variables}{Variables}\dotfill \pageref*{variables} \linebreak
\noindent\hyperlink{substitution}{Substitution}\dotfill \pageref*{substitution} \linebreak
\noindent\hyperlink{of_homotopy_type_theory}{Of homotopy type theory}\dotfill \pageref*{of_homotopy_type_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

One may interpret mathematical \emph{[[logic]]} as being a formal language for talking about the collection of [[monomorphisms]] into a given [[object]] of a given [[category]]: the [[poset of subobjects]] of that object.

More generally, one may interpret [[type theory]] and notably [[dependent type theory]] as being a formal language for talking about [[slice categories]], consisting of all morphisms into a given object.

Conversely, starting with a given theory of logic or a given type theory, we say that it has a \emph{categorical semantics} if there is a category such that the given theory is that of its slice categories, if it is the \emph{[[internal logic]]} of that category.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For the general idea, for the moment see at \emph{[[type theory]]} the section \emph{\href{type+theory#CategoricalSemantics}{An introduction for category theorists}} and see at \emph{[[relation between type theory and category theory]]}.

\hypertarget{of_dependent_type_theory}{}\subsubsection*{{Of dependent type theory}}\label{of_dependent_type_theory}

We discuss how to interpret judgements of [[dependent type theory]] in a given [[category]] $\mathcal{C}$ with [[finite limits]]. For more see \emph{[[categorical model of dependent types]]}.

Write $cod : \mathcal{C}^I \to \mathcal{C}$ for its [[codomain fibration]], and write

\begin{displaymath}
\chi : \mathcal{C}^{op} \to Cat
\end{displaymath}
for the corresponding [[Grothendieck construction|classifying functor]], the self-[[indexed category|indexing]]

\begin{displaymath}
\chi : \Gamma \mapsto \mathcal{C}_{/\Gamma}
\end{displaymath}
that sends an [[object]] of $\mathcal{C}$ to the [[slice category]] over it, and sends a morphism $f : \Gamma \to \Gamma'$ to the [[pullback]]/[[base change]] functor

\begin{displaymath}
f^\ast : \mathcal{C}_{/\Gamma'} 
  \to 
  \mathcal{C}_{/\Gamma}
  \,.
\end{displaymath}
We give now rules for choices ``$[x y z]$'' that associate with every string ``$x y z$'' of symbols in [[type theory]] objects and morphisms in $\mathcal{C}$. A collection of such choices following these rules is \emph{an interpretation} / a choice of \emph{categorical semantics} of the type theory in the category $\mathcal{C}$.

\hypertarget{ContextsAndTypeJudgements}{}\paragraph*{{Contexts and type judgements}}\label{ContextsAndTypeJudgements}

\begin{enumerate}%
\item The empty [[context]] $()$ in [[type theory]] is interpreted as the [[terminal object]] of $\mathcal{C}$

\begin{displaymath}
[ () ] := *
  \,.
\end{displaymath}

\item If $\Gamma$ is a context which has already been given an interpretation $[\Gamma] \in Obj(\mathcal{C})$, then a judgement of the form

\begin{displaymath}
\Gamma \vdash A : Type
\end{displaymath}
is interpreted as an object in the slice over $\Gamma$

\begin{displaymath}
[\Gamma \vdash A : Type] \in Obj(\mathcal{C}_{/\Gamma})
  \,,
\end{displaymath}
hence as a choice of morphism

\begin{displaymath}
\itexarray{
    [(\Gamma, x : A)]
    \\
    \downarrow^{\mathrlap{[\Gamma \vdash A : Type]}}
    \\
    [\Gamma]
  }
\end{displaymath}
in $\mathcal{C}$.


\item If a judgement of the form $\Gamma \vdash A : Type$ has already found an interpretation, as above, then an extended [[context]] of the form $(\Gamma, x : A)$ is interpreted as the domain object $[(\Gamma, x : A)]$ of the above choice of morphism.



\end{enumerate}
\hypertarget{terms}{}\paragraph*{{Terms}}\label{terms}

Assume for a context $\Gamma$ and a judgement $\Gamma \vdash A : Type$ we have already chosen an interpretation $[\Gamma, x : A] \stackrel{[\Gamma \vdash A : Type]}{\to} [\Gamma]$ as above.

A judgement of the form $\Gamma \vdash a : A$ (a [[term]] of [[type]] $A$) is to be interpreted as a [[section]] of this morphism, equivalently as a morphism in $\mathcal{C}_{/\Gamma}$

\begin{displaymath}
[\Gamma \vdash : a : A] : * \to [\Gamma, x : A]
\end{displaymath}
from the [[terminal object]] to $[\Gamma \vdash A : Type]$, which in $\mathcal{C}$ is a [[commuting diagram|commuting triangle]]

\begin{displaymath}
\itexarray{
    [\Gamma] &&\stackrel{[(\Gamma \vdash a : A)]}{\to}&&
    [\Gamma, x : A]
    \\
    & {}_\mathllap{[\Gamma]}\searrow
    &&
    \swarrow_{\mathrlap{[\Gamma \vdash A : Type]}}
    \\
    && [\Gamma]
  }
  \,.
\end{displaymath}
\hypertarget{variables}{}\paragraph*{{Variables}}\label{variables}

For a term $\Gamma \vdash  a : A$ the context $\Gamma$ is the collection of [[free variables]] in $a$.

(\ldots{})

\hypertarget{substitution}{}\paragraph*{{Substitution}}\label{substitution}

Assume that interpretations for judgements

\begin{displaymath}
\Gamma , x : A \vdash B(x) : Type
\end{displaymath}
and

\begin{displaymath}
\Gamma \vdash  a : A
\end{displaymath}
have been given as above. Then the substitution judgement

\begin{displaymath}
\Gamma  \vdash B[a/x] : Type
\end{displaymath}
is to be interpreted as follows. The interpretation of the first two terms corresponds to a diagram in $\mathcal{C}$ of the form

\begin{displaymath}
\itexarray{
    &&&& [(\Gamma, x : A, y : B(x))]
    \\
    &&&&
    \downarrow^{\mathrlap{[\Gamma, x : A \vdash B(x) : Type]}}
    \\
    [\Gamma] &&\stackrel{[\Gamma \vdash a : A]}{\to}&&
     [(\Gamma, x : A)]
    \\
    & {}_{id}\searrow && \swarrow_{\mathrlap{[\Gamma \vdash A : Type]}}
    \\
    && [\Gamma]
  }
\end{displaymath}
The interpretation of the substitution statement is then [[generalized the|the]] [[pullback]]

\begin{displaymath}
[\Gamma \vdash B[a/x] : Type]
  :=
  [\Gamma \vdash a : A]^* [\Gamma, x : A \vdash B(x) : Type]
  \,,
\end{displaymath}
hence the morphism in $\mathcal{C}$ that universally completes the above diagram as

\begin{displaymath}
\itexarray{
    [(\Gamma, y : B[x/a])]
    &&\to&& [(\Gamma, x : A, y : B(x))]
    \\
    {}^{\mathllap{[\Gamma \vdash B[a/x] : Type]
}}\downarrow
    &&&&
    \downarrow^{\mathrlap{[\Gamma, x : A \vdash B(x) : Type]}}
    \\
    [\Gamma] &&\stackrel{[\Gamma \vdash a : A]}{\to}&&
     [(\Gamma, x : A)]
    \\
    & {}_{id}\searrow && \swarrow_{\mathrlap{[\Gamma \vdash A : Type]}}
    \\
    && [\Gamma]
  }
\end{displaymath}
\hypertarget{of_homotopy_type_theory}{}\subsubsection*{{Of homotopy type theory}}\label{of_homotopy_type_theory}

See \emph{[[categorical semantics of homotopy type theory]]}.

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

\begin{itemize}%
\item [[models in presheaf toposes]]

\end{itemize}
\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[relation between type theory and category theory]]


\item [[categorical model of dependent types]]


\item [[Awodey's conjecture]]



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

A standard textbook reference for categorical semantics of logic is section D1.2 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
The categorical semantics of [[dependent type theory]] in [[locally cartesian closed categories]] is essentially due to

\begin{itemize}%
\item [[R. A. G. Seely]], \emph{Locally cartesian closed categories and type theory}, Math. Proc. Camb. Phil. Soc. (1984) 95 (\href{http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf}{pdf})

\end{itemize}
For more references on this see at \emph{[[relation between category theory and type theory]]}.

Lecture notes on this include for instance.

\begin{itemize}%
\item [[Martin Hofmann]], \emph{Syntax and semantics of dependent types}, Semantics and Logics of Computation (P. Dybjer and A. M. Pitts, eds.), Publications of the Newton Institute, Cambridge University Press, Cambridge, (1997) pp. 79-130. (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.8985}{web}, )


\item Roy Crole, \emph{Categories for types}



\end{itemize}
See also section B.3 of

\begin{itemize}%
\item [[Michael Warren]], \emph{Homotopy Theoretic Aspects of Constructive Type Theory} (\href{http://www.andrew.cmu.edu/user/awodey/students/warren.pdf}{pdf})

\end{itemize}
A comprehensive definition of semantics of [[homotopy type theory]] in [[type-theoretic model categories]] is in section 2 of

\begin{itemize}%
\item [[Michael Shulman]], \emph{Univalence for inverse diagrams and homotopy canonicity}, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( \emph{From type theory and homotopy theory to Univalent Foundations of Mathematics} ) June 2015 (\href{https://arxiv.org/abs/1203.3253}{arXiv:1203.3253}, \href{https://doi.org/10.1017/S0960129514000565}{doi:/10.1017/S0960129514000565})

\end{itemize}


\end{document}