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\begin{document}

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\section*{free topos}

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

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos 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{foundations_2}{Foundations}\dotfill \pageref*{foundations_2} \linebreak
\noindent\hyperlink{remark_two_properties_of_the_free_topos}{Remark: Two properties of the free topos}\dotfill \pageref*{remark_two_properties_of_the_free_topos} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{links}{Links}\dotfill \pageref*{links} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[higher-order logic|higher-order]] [[intuitionistic type theory]] $\mathfrak{L}$, it is possible to construct a [[topos]] $T(\mathfrak{L})$ out of the [[syntax]] of $\mathfrak{L}$. The \textbf{free topos} $T(\mathfrak{L}_0)$ is the result of this construction when $\mathfrak{L}_0$ is `pure' type theory i.e. the only types are 1 ([[unit type]]), $N$ ([[natural numbers type]]), and $\Omega$ ([[type of propositions]]) lacking relations beyond the bare necessities.

As $\mathfrak{L}_0$ is an [[initial object]] in the appropriate category of type theories, the free topos $T(\mathfrak{L}_0)$ is itself initial in the corresponding category of toposes (with a natural numbers object) and suitable [[logical morphisms]] and is, therefore, also known as the \textbf{initial topos}\footnote{This is not to be confounded with the gadget of the same name in [[SGA4]] (1972, p.313) i.e. the topos $sh(\emptyset)$ of sheaves on the empty topological space aka the one point category, also called the \emph{empty topos} there. In this context the appropriate maps are [[geometric morphisms]].} .

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

The [[internal language]] of the free topos is precisely pure (intuitionistic) higher order type theory. In 1978, [[Jim Lambek]] and [[Phil Scott]] exploited this connection in order to prove properties of intuitionistic type theory by [[proof theory|proof-theoretic]] means. It was observed by [[Peter Freyd]] then that the concept of a [[Freyd cover]] permits to give conceptual proofs of their findings. The following lemma and proposition is a replication of his ideas.

\begin{lemma}
\label{}\hypertarget{}{}
For any [[category]] $C$ with a [[terminal object]] $\mathbf{1}$, the terminal object of the [[Freyd cover]] $\widehat{C}$ is [[small-projective]], i.e., the representable $\Gamma = \widehat{C}(1, -) \colon \widehat{C} \to Set$ preserves any colimits that exist.

\end{lemma}
\begin{proof}
To check that $\Gamma^{op} \colon \widehat{C}^{op} \to Set^{op}$ preserves limits, it suffices to check that the composite

\begin{displaymath}
\widehat{C}^{op} \stackrel{\Gamma^{op}}{\to} Set^{op} \stackrel{2^-}{\to} Set
\end{displaymath}
preserves limits, because the contravariant power set functor $P = 2^-$ is monadic. But it is easily checked that this composite is the contravariant representable given by $(2, \mathbf{1}, 2 \to \Gamma(\mathbf{1}))$.

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
The [[terminal object]] in the free topos $\mathcal{T}$ is [[connected object|connected]] and [[projective object|projective]] in the sense that $\Gamma = \hom(1, -) \colon \mathcal{T} \to Set$ preserves finite [[colimits]].

\end{theorem}
\begin{proof}
We divide the argument into three segments:

\begin{itemize}%
\item The [[hom-functor]] preserves [[finite limits]], so by general properties of [[Artin gluing]], the [[Freyd cover]] $\widehat{\mathcal{T}}$ is also a topos. Observe that $\mathcal{T}$ is equivalent to the [[slice category|slice]] $\widehat{\mathcal{T}}/M$ where $M$ is the object $(\emptyset, \mathbf{1}, \emptyset \to \Gamma(\mathbf{1}))$. Since pulling back to a slice is a logical functor, we have a logical functor

\begin{displaymath}
\pi \colon \widehat{\mathcal{T}} \to \mathcal{T}
\end{displaymath}
Since $\mathcal{T}$ is initial, $\pi$ is a retraction for the unique logical functor $i \colon \mathcal{T} \to \widehat{\mathcal{T}}$.


\item We have maps $\mathcal{T}(1, -) \to \widehat{\mathcal{T}}(i 1, i-) \cong \widehat{\mathcal{T}}(1, i-)$ (the isomorphism comes from $i 1 \cong 1$, which is clear since $i$ is logical), and $\widehat{\mathcal{T}}(1, i-) \to \mathcal{T}(\pi 1, \pi i-) \cong \mathcal{T}(1, -)$ since $\pi$ is logical and retracts $i$. Their composite must be the identity on $\mathcal{T}(1, -)$, because there is only one such endomorphism, using the Yoneda lemma and terminality of $1$.



\end{itemize}
Finally, since $\mathcal{T}(1, -)$ is a retract of a functor $\widehat{\mathcal{T}}(1, i-)$ that preserves finite colimits (by the lemma, and the fact that the logical functor $i$ preserves finite colimits), it must also preserve finite colimits.

\end{proof}
This is important because it implies that the [[internal logic]] of the free topos satisfies the following properties:

\begin{itemize}%
\item The \emph{disjunction property}: if ``P [[or]] Q'' is provable in the empty [[context]], then either P is so provable, or Q is so provable. (Note that this clearly fails in the presence of [[excluded middle]].)


\item The \emph{existence property}: if ``there exists an $x\in A$ such that $P(x)$'' is provable in the empty context, then there exists a [[global element]] $x\colon 1\to A$ such that $P(x)$ is provable in the empty context. (Again, this is clearly a [[constructive mathematics|constructivity]] property.)


\item The \emph{negation property}: False is not provable in the empty context.


\item All numerals in the free topos are ``standard'', i.e., the global sections functor $\Gamma = \hom(1, -): \mathcal{T} \to Set$ preserves the [[natural numbers object]] $N$ (because $N$ can be characterized in terms of finite colimits and $1$, by a theorem of Freyd).



\end{itemize}
\hypertarget{foundations_2}{}\subsubsection*{{Foundations}}\label{foundations_2}

In the [[foundations of mathematics]], [[Jim Lambek]] proposed to use the free topos as ambient world to do mathematics in; see (\hyperlink{LambekWorld}{Lambek 2004}). Being syntactically constructed, but universally determined, with higher-order [[intuitionistic type theory]] as [[internal language]] he saw it as a reconciliation of the three classical schools of [[philosophy of mathematics]], namely [[formalism]], [[platonism]], and [[intuitionism]]. His latest views on this variant of categorical foundations can be found in (\hyperlink{LS11}{Lambek-Scott 2011}).

\hypertarget{remark_two_properties_of_the_free_topos}{}\subsubsection*{{Remark: Two properties of the free topos}}\label{remark_two_properties_of_the_free_topos}

Lambek and Scott mention in their 1986 monograph (\hyperlink{LambekScott86}{pp.viii, 233, 250}) two further remarkable properties of the free topos $\mathcal{T}$:

\begin{itemize}%
\item All maps $R\to R$ in the free topos represent continuous functions. Note that this is not the same as to say that Brouwer's theorem (``all functions $R\to R$ are continuous'') is true in the free topos, which is not the case.


\item The natural numbers object $N$ is an (external) [[projective object]].



\end{itemize}
The first result is attributed to [[André Joyal]], presumably unpublished, and for the second claim they refer to an unpublished manuscript by [[Michael Makkai]] and a manuscript by Friedman and Scedrov, presumably (\hyperlink{FriedmanScedrov83}{1983}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Freyd cover]]


\item [[Artin gluing]]


\item [[syntactic category]]


\item [[adjoint modality]]



\end{itemize}
\hypertarget{links}{}\subsection*{{Links}}\label{links}

\begin{itemize}%
\item \href{http://mathoverflow.net/questions/264652/brouwers-theorem-in-the-free-topos}{MO Brouwer's theorem in the free topos?}


\item \href{https://nforum.ncatlab.org/discussion/6665/free-topos/#Item_0}{nForum discussion}



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

The free topos was first constructed by H. Volger. A second construction appears in

\begin{itemize}%
\item [[Jim Lambek]], \emph{From Types to Sets} , JPAA \textbf{36} (1980) pp.113-164.

\end{itemize}
Freyd's proof of the above properties appears first in

\begin{itemize}%
\item [[Jim Lambek]], [[Phil Scott]], \emph{Intuitionist Type Theory and the Free Topos} , JPAA \textbf{19} (1980) pp. 215-257.

\end{itemize}
On the relation between the proof and the topos-theoretic techniques in the proof see

\begin{itemize}%
\item A. Scedrov, [[Phil Scott]], \emph{A note on the Friedman Slash and Freyd Covers} , pp. 443-452 in \emph{The LEJ Brouwer Centenary Symposium} , North Holland Amsterdam 1982.


\item [[Jim Lambek]], [[Phil Scott]], \emph{New proofs of some intuitionistic principles} , Zeit. f\"u{}r Math. Logik \textbf{29} (1983) pp. 493-504.



\end{itemize}
For textbook accounts of the free topos see

\begin{itemize}%
\item [[Jim Lambek]], [[Phil Scott]], \emph{[[Introduction to Higher-Order Categorical Logic]]} , Cambridge UP 1986. (\href{https://synrc.com/publications/cat/Category%20Theory/Categorical%20Logic/Lambek%20J.,%20Scott%20P.J.%20Introduction%20to%20Higher%20Order%20Categorical%20Logic.pdf}{pdf})


\item [[Peter Freyd|P. Freyd]], A. Scedrov, \emph{[[Categories, Allegories]]} , North-Holland Amsterdam 1990. (1.(10)31, p.192)


\item [[Elephant]] F3, to appear.



\end{itemize}
For dependent choice in intuitionistic type theory:

\begin{itemize}%
\item H. M. Friedman, A. Scedrov, \emph{Set existence property for intuitionistic theories with dependent choice} , APAL \textbf{25} no.2 (1983) pp.129-140.

\end{itemize}
For J. Lambek's views on the role of the free topos in [[foundations of mathematics]] see:

\begin{itemize}%
\item J. Couture, [[Jim Lambek]], \emph{Philosophical Reflections on the Foundations of Mathematics} , Erkenntnis \textbf{34} (1991) pp.187-209.


\item [[Jim Lambek]], \emph{Are the traditional philosophies of mathematics really incompatible?} , Math. Intelligencer \textbf{16} (1994) pp.56--62.


\item [[Jim Lambek]], \emph{What is the world of mathematics?} , APAL \textbf{126} (2004) pp.149-158. (\href{http://www.math.mcgill.ca/barr/lambek/pdffiles/WorldMath.pdf}{draft})


\item [[Jim Lambek]], [[Phil Scott]], \emph{Reflections on Categorical Foundations of Mathematics} , pp.171-185 in Sommaruga (ed.), \emph{Foundational Theories of Classical and Constructive Mathematics}, Springer New York 2011. (\href{https://www.site.uottawa.ca/~phil/papers/LS11.final.pdf}{draft})



\end{itemize}
[[!redirects initial topos]]



\end{document}