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\begin{document}

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\section*{Freyd cover}

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

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

[[!include category theory - contents]]

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - 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{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_the_initial_topos}{Relation to the initial topos}\dotfill \pageref*{relation_to_the_initial_topos} \linebreak
\noindent\hyperlink{AsALocalTopos}{As a local topos}\dotfill \pageref*{AsALocalTopos} \linebreak
\noindent\hyperlink{as_a_fibration}{As a fibration}\dotfill \pageref*{as_a_fibration} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{Freyd cover} $\overline{\mathcal{T}}$ of a [[topos]] $\mathcal{T}$, or more generally of a [[category]] with a [[terminal object]], is a way to turn $\mathcal{T}$, viewed as an `abstract' category, into a `concrete' category of structured sets by sewing it together with [[Set]] along the [[global section functor]].

As $\overline{\mathcal{T}}$ comes equipped with two well-behaved projection functors the resulting category has many good logical properties making the construction an important tool in [[type theory]] and theoretical computer science.

The Freyd cover is sometimes known as the \textbf{Sierpinski cone} or \textbf{scone}, because in [[topos theory]] it behaves similarly to the [[cone]] on a space, but with the [[interval]] $[0,1]$ replaced by the [[Sierpinski space]].

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

Let $\mathcal{T}$ be a category with a terminal object and $\Gamma = \mathcal{T} (1, -)$ the global section functor . The \textbf{Freyd cover} of $\mathcal{T}$ is the category $\overline{\mathcal{T}}$ whose objects are triples $(X, \xi, U)$ where:

\begin{itemize}%
\item $X$ is a set
\item $U$ is an object of $\mathcal{T}$
\item $\xi$ is a function $X \to \Gamma (U)$

\end{itemize}
A morphism from $(X,\xi, U)$ to $(Y,\eta , V)$ is a pair of morphisms $(\varphi :X\to Y, t:U\to V)$ with $\varphi\in Set$ and $t\in \mathcal{T}$ such that $\eta\varphi=\Gamma(t)\xi$.

\hypertarget{remark}{}\subsubsection*{{Remark}}\label{remark}

The construction is a special case of [[Artin gluing]]: i.e. $\overline{\mathcal{T}}$ is the [[comma category]] $Set \downarrow \Gamma$ with $\Gamma = \mathcal{T} (1, -)$.

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

\hypertarget{relation_to_the_initial_topos}{}\subsubsection*{{Relation to the initial topos}}\label{relation_to_the_initial_topos}

One of the first applications of the Freyd cover was to deduce facts about the initial topos (initial with respect to [[logical morphism|logical morphisms]] --- also called the [[free topos]]). They were originally proved by syntactic means; the conceptual proofs of the lemma and theorem below are due to Freyd.

\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

$\backslash$begin\{center\} $\backslash$begin\{tikzcd\} $\backslash$widehat\{C\}{\tt \symbol{94}}\{op\} $\backslash$arrowr, ``$\backslash$Gamma{\tt \symbol{94}}\{op\}'' \& $\backslash$mathrm\{Set\}{\tt \symbol{94}}\{op\} $\backslash$arrowr, ``2{\tt \symbol{94}}-'' \& $\backslash$mathrm\{Set\} $\backslash$end\{tikzcd\} $\backslash$end\{center\}

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 initial 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 $\widehat{\mathcal{T}}/M$ 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 (which is exactly ``intuitionistic higher-order logic'') 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{AsALocalTopos}{}\subsubsection*{{As a local topos}}\label{AsALocalTopos}

The Freyd cover of a [[topos]] is a [[local topos]], and in fact freely so. Every local topos is a [[retract]] of a Freyd cover.

See (\hyperlink{Johnstone}{Johnstone, lemma C3.6.4}).

\hypertarget{as_a_fibration}{}\subsubsection*{{As a fibration}}\label{as_a_fibration}

The Freyd cover $\overline{\mathcal{T}}$ is fibered over $\mathcal{T}$ as it arises equivalently by change of base of the [[codomain fibration]] of [[Set]] along the [[global section functor]] $\Gamma$.

See (\hyperlink{Jacobs}{Jacobs, p.57}).

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

\begin{itemize}%
\item [[Artin gluing]]


\item [[comma category]]


\item [[Sierpinski topos]]



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

Some of the above material is taken from

\begin{itemize}%
\item [[Tom Leinster]], \href{http://mathoverflow.net/questions/12136/freyd-cover-of-a-category}{reply at MathOverflow}

\end{itemize}
The following two n-caf\'e{} blog posts provide an introductory discussion of the scone construction from a geometric and a logical perspective:

\begin{itemize}%
\item [[Mike Shulman]], \emph{Discreteness, Concreteness, Fibrations, and Scones} , November 2011. (\href{https://golem.ph.utexas.edu/category/2011/11/discreteness_concreteness_fibr.html}{link})


\item [[Mike Shulman]], \emph{Scones, Logical Relations, and Parametricity} , April 2013. (\href{https://golem.ph.utexas.edu/category/2013/04/scones_logical_relations_and_p.html}{link})



\end{itemize}
You can find more on [[Artin gluing]] in this important (and nice) paper:

\begin{itemize}%
\item [[Aurelio Carboni]], [[Peter Johnstone]], \emph{Connected limits, familial representability and Artin glueing} , Mathematical Structures in Computer Science \textbf{5} (1995) pp.441-459.

\end{itemize}
plus

\begin{itemize}%
\item [[Aurelio Carboni]], [[Peter Johnstone]], \emph{Corrigenda to `Connected limits\ldots{}'} , Mathematical Structures in Computer Science \textbf{14} (2004) pp.185-187.

\end{itemize}
See also section C3.6 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] vol. 2} . Oxford UP 2002.


\item [[Bart Jacobs]], \emph{Categorical Logic and Type Theory} , Elsevier Amsterdam 1999.



\end{itemize}
The following paper studies logical properties of the Freyd cover:

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{On the Freyd Cover of a Topos} , Notre Dame J. Formal Logic \textbf{24} no.4 (1983) pp.517-526. (\href{http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093870454}{pdf})

\end{itemize}
The argument given above for the properties of the free topos is an amplification of

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

\end{itemize}
For more on the free topos and the first appearance in print of Freyd's observation see

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

\end{itemize}
[[!redirects scone]] [[!redirects Sierpinski cone]]



\end{document}