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\begin{document}

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\section*{Lawvere&#39;s fixed point theorem}

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

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

[[!include category theory - contents]]

\hypertarget{logic}{}\paragraph*{{Logic}}\label{logic}

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

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{precise_statement}{Precise statement}\dotfill \pageref*{precise_statement} \linebreak
\noindent\hyperlink{History}{History}\dotfill \pageref*{History} \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}

Various [[diagonal arguments]], such as those found in the proofs of the [[halting theorem]], [[Cantor's theorem]], and [[Gödel]]`s [[incompleteness theorem]], are all instances of the \emph{Lawvere fixed point theorem} (\hyperlink{Lawvere69}{Lawvere 69}), which says that for any [[cartesian closed category]], if there is a suitable notion of [[epimorphism]] from some [[object]] $A$ to the [[exponential object]]/[[internal hom]] from $A$ into some other object $B$

\begin{displaymath}
A \longrightarrow B^A
\end{displaymath}
then every [[endomorphism]] $f \colon B \to B$ of $B$ has a [[fixed point]].

\hypertarget{precise_statement}{}\subsection*{{Precise statement}}\label{precise_statement}

Let us say that a map $\phi: X \to Y$ is \emph{point-surjective} if for every point $q: 1 \to Y$ there exists a point $p: 1 \to X$ that lifts $q$, i.e., $\phi p = q$.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Lawvere's fixed-point theorem)} In a cartesian closed category, if there is a point-surjective map $\phi: A \to B^A$, then every morphism $f: B \to B$ has a fixed point $s: 1 \to B$ (so that $f s = s$).

\end{theorem}
\begin{proof}
Given $f: B \to B$, let $q: 1 \to B^A$ name the composite map

\begin{displaymath}
A \stackrel{\delta}{\to} A \times A \stackrel{\phi \times 1_A}{\to} B^A \times A \stackrel{eval}{\to} B \stackrel{f}{\to} B
\end{displaymath}
from $A$ to $B$. In [[lambda calculus]] notation, $q = \lambda a: A. f \phi(a)(a)$. Let $p: 1 \to A$ lift $q$. Then calculate

\begin{displaymath}
\phi(p)(p) = q(p) = (\lambda a: A. f \phi(a)(a))(p) = f \phi(p)(p)
\end{displaymath}
where the last equation is a [[beta-reduction]]. Hence $s \coloneqq \phi(p)(p)$ is a fixed point of $f$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
As pointed out by Lawvere, a hypothesis even weaker than point-surjectivity will do. Namely, $g: X \to Y^A$ is called \emph{weakly point-surjective} iff for every $f: A \to Y$ there is $x: 1 \to X$ such that for every $a: 1 \to A$, we have $g(x)(a) = f(a)$. (This is weaker because $g(x) = f$ cannot be inferred from $g(x)(a) = f(a)$ for all \emph{[[global elements]]} $a$.)

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The statement need not hold if ``(weakly) point-surjective'' is replaced by ``epimorphism''. For example, in the cartesian closed category of compactly generated Hausdorff spaces and continuous maps, with $S = S^1$ the circle, the [[Polish space]] $S^\mathbb{N}$ is compactly generated under the product topology; this is the exponential where $\mathbb{N}$ is given the discrete topology. There is a countable dense subspace $i: \mathbb{N} \to S^\mathbb{N}$, but recall that for any full subcategory of the category of Hausdorff spaces, a map $f: X \to Y$ is an epimorphism iff it has a dense [[image]]. On the other hand, there are obvious rotations of $S$ that have no fixed points.

\end{remark}
Thus epimorphisms need not be (weakly) point-surjective. Nor are point-surjective maps necessarily epimorphisms; for example, if $U \hookrightarrow V$ is a proper inclusion between proper subobjects of the terminal object $1$ (as may happen in a [[sheaf topos]]), then this is vacuously point-surjective but not an epimorphism.

Point-surjectivity may seem like an inadequate notion of ``epimorphism'', but it suffices for many purposes. For example,

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(Cantor's theorem in a topos)} For any object $X$, there is an epimorphism $f: X \to \Omega^X$ only if the topos is degenerate.

\end{prop}
\begin{proof}
Suppose there existed such an epi. In a topos, a map $f: X \to Y$ is epi iff the [[image|direct image]] map $\exists_f: \Omega^X \to \Omega^Y$ retracts the [[preimage|inverse image]] map $\Omega^f: \Omega^Y \to \Omega^X$, i.e., $\exists_f \circ \Omega^f = 1_{\Omega^Y}$. Putting $Y = \Omega^X$, the supposition implies that $\exists_f: Y \to \Omega^Y$ is a retraction. But retractions are automatically point-surjective.

We then conclude from Lawvere's fixed point theorem that every endomorphism on $\Omega$, in particular the negation $\neg: \Omega \to \Omega$, has a fixed point $p: 1 \to \Omega$. Then $0 = p \wedge \neg p = p \wedge p = p$, whence $\neg 0 = 0$, or ``true = false'': the topos is degenerate.

\end{proof}
Once we have a proposition $p$ with $p = \neg p$, another way to conclude the proof is to apply Lawvere's fixed-point theorem again to the surjection $p \to 0^p$, where $p$ is regarded as a [[subsingleton]] and $0$ is the initial object, so that $0^p = \neg p$. This gives a fixed point $1\to 0$ of the identity map $0\to 0$, which again makes the topos degenerate. (The shorter version above is a [[beta-reduction]] of this.) For a formalization of this argument and a generalization thereof to [[universe types]], see \hyperlink{Escardo18}{Escardo18}.

\begin{remark}
\label{}\hypertarget{}{}
Another version of Lawvere's fixed-point theorem requires only finite products for its statement. Namely, in a category with finite products, suppose $\Phi: A \times A \to B$ is a morphism with the property that for each $g: A \to B$ there exists $a: 1 \to A$ such that $g \lambda = \Phi \circ (a \times 1_A)$, where $\lambda: 1 \times A \stackrel{\sim}{\to} A$ is the projection. Then every map $f: B \to B$ has a fixed point. This version of the theorem is emphasized by \hyperlink{Yanofsky03}{Yanofsky}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Many applications of Lawvere's fixed point theorem are in the form of negated propositions, e.g., there is no epimorphism from a set to its power set, or [[Peano arithmetic]] cannot prove its own [[consistency]]. However, there are positive applications as well, e.g., it implies the existence of [[fixed-point combinators]] in [[untyped lambda calculus]].

\end{remark}
\hypertarget{History}{}\subsection*{{History}}\label{History}

In an interview (\href{William+Lawvere#Interview07}{Lawvere 07}) not long after G\"o{}del's 100th birthday, [[William Lawvere]] answered the question

\begin{quote}%
We have recently celebrated Kurt G\"o{}del's 100th birthday. What do you think about the extra-mathematical publicity around his incompleteness theorem?


\end{quote}
by saying (reproduced in \hyperlink{Lawvere69}{Lawvere 69 reprint, p. 2}):

\begin{quote}%
``In `Diagonal arguments and Cartesian closed categories' (\hyperlink{Lawvere69}{Lawvere 69}) we demystified the incompleteness theorem of G\"o{}del and the truth-definition theory of Tarski by showing that both are consequences of some very simple algebra in the Cartesian-closed setting. It was always hard for many to comprehend how Cantor's mathematical theorem could be re-christened as a ''paradox`` by Russell and how G\"o{}del's theorem could be so often declared to be the most significant result of the 20th century. There was always the suspicion among scientists that such extra-mathematical publicity movements concealed an agenda for re-establishing belief as a substitute for science. Now, one hundred years after G\"o{}del's birth, the organized attempts to harness his great mathematical work to such an agenda have become explicit.


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

\begin{itemize}%
\item [[Cantor's theorem]]


\item [[fixed-point combinator]]


\item [[incompleteness theorem]]


\item [[Löb's theorem]]



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

The original article is

\begin{itemize}%
\item [[William Lawvere]], \emph{Diagonal Arguments and Cartesian Closed Categories}, Lecture Notes in Mathematics, 92 (1969), 134-145 (\href{http://tac.mta.ca/tac/reprints/articles/15/tr15abs.html}{TAC})

\end{itemize}
A review and further development is in

\begin{itemize}%
\item [[Noson Yanofsky]], \emph{A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points}, (\href{http://arxiv.org/abs/math/0305282}{arXiv:math/0305282})

\end{itemize}
Expositions include

\begin{itemize}%
\item [[Andrej Bauer]], \emph{On a proof of Cantor's theorem} (\href{http://math.andrej.com/2007/04/08/on-a-proof-of-cantors-theorem/}{web})

\end{itemize}
Other references

\begin{itemize}%
\item [[Martin Escardo]], \emph{On Lawvere's Fixed Point Theorem}, \href{https://www.cs.bham.ac.uk/~mhe/agda-new/LawvereFPT.html}{agda development}, 2018

\end{itemize}
[[!redirects Lawvere fixed point theorem]]



\end{document}