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\begin{document}

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\section*{steady function}

\hypertarget{steady_functions}{}\section*{{Steady functions}}\label{steady_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relationship_to_constancy}{Relationship to constancy}\dotfill \pageref*{relationship_to_constancy} \linebreak
\noindent\hyperlink{steady_endomaps}{Steady endomaps}\dotfill \pageref*{steady_endomaps} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A steady function is one equipped with a certain sort of ``witness of [[constant function|constancy]]''. However, in [[higher category theory]] and [[homotopy theory]], it is debatable whether or not this witness really exhibits ``constancy'', hence the use of a different word. (The word ``steady'' was suggested by [[Andrej Bauer]].)

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

In [[homotopy type theory]], a function $f:A\to B$ is \textbf{steady} if we have a term of type

\begin{displaymath}
\prod_{(x,y:A)} (f x = f y).
\end{displaymath}
By regarding homotopy type theory as the [[internal logic]] of an [[(∞,1)-topos]], we obtain a definition that makes sense in any higher category with binary products: a morphism $f:A\to B$ is \textbf{steady} if the two composites $A\times A \rightrightarrows A \xrightarrow{f} B$ are equivalent.

\hypertarget{relationship_to_constancy}{}\subsection*{{Relationship to constancy}}\label{relationship_to_constancy}

If $f$ is [[constant function|constant]] in the sense that it factors through the [[terminal object]] (i.e. we have $f = \lambda x. b$ for some $b:B$), then $f$ is obviously steady. The converse holds if we know that the domain $A$ is [[inhabited type|inhabited]], for if $a_0:A$, then $f a = f a_0$ for all $a:A$. However, the [[identity function]] of the [[empty type]] is steady, yet not equal to $\lambda x.b$ for any $b:\emptyset$ (since no such $b$ exists).

More generally, if $f$ factors through the [[propositional truncation]] ${\|A\|}$, then it is steady, since any two elements of ${\|A\|}$ are equal (i.e. it is an [[h-proposition]]). In fact, this is true if $f$ factors through \emph{any} h-proposition (in which case it in fact also factors through ${\|A\|}$, by the universal property of the latter).

The converse to this last implication does hold for some specific $f:A\to B$, such as:

\begin{itemize}%
\item If $B$ is an [[h-set]]. For then $f$ factors through the 0-truncation ${\|A\|_0}$, and the set-[[coequalizer]] of the two projections ${\|A\|_0} \times {\|A\|_0} \to {\|A\|_0}$ is the propositional truncation.


\item If $A$ has [[split support]]. For then we have a composite ${\|A\|} \to A \xrightarrow{f} B$, whose restriction to $A$ is equal to $f$ by steadiness.


\item If $A=P+Q$, with $P$ and $Q$ h-propositions. For then ${\|A\|}$ is the [[join]] $P*Q$ of $P$ and $Q$, i.e. the [[pushout]] of the two projections $P \leftarrow P\times Q \to Q$. The universal property of this pushout says exactly that any steady map $P+Q\to B$ factors through $P*Q = \Vert P+Q\Vert$.


\item If $A=B$ (see below).



\end{itemize}
However, it can fail in general, even when $A$ is [[mere proposition|merely]] inhabited (i.e. ${\|A\|}= 1$). For instance, let $A=P+Q+R$ for h-propositions $P$, $Q$, and $R$, and let $B$ be the triple pushout of $P$, $Q$, and $R$ under $P\times Q$, $P\times R$, and $Q\times R$. Then there is a steady map $f:A\to B$, but there exist models in which ${\|A\|} = 1$ but $B$ has no [[global element]. The most straightforward such model is [[presheaves]] on the [[poset]] of proper [[subsets]] of $\{a,b,c\}$, with $P=\{a,b\}$, $Q=\{b,c\}$, and $R=\{a,c\}$. In this model, we have $B(S) = 1$ for all nonempty proper subsets $S$, while $B(\emptyset) = S^1$, and $B$ has no global sections.

See \href{https://groups.google.com/d/msg/homotopytypetheory/FeBAScTgwzg/Dx6E3-ezdxIJ}{this discussion}.

In general, being steady may be regarded as an ``incoherent approximation'' to constancy in the sense of factoring through an h-proposition. Indeed for a set $A$, its propositional truncation is the set-coequalizer of $A\times A\rightrightarrows A$. However, in general such a construction requires the realization of a whole simplicial diagram (the simplicial kernel of the map $A\to 1$).

\hypertarget{steady_endomaps}{}\subsection*{{Steady endomaps}}\label{steady_endomaps}

While an arbitrary steady function is not very coherent, a steady [[endofunction]] $f:A\to A$ has some extra degree of ``coherence'', as witnessed by the following results of \hyperlink{KECA}{(KECA)}.

\begin{ulemma}
If $f:A\to A$ is steady, then the type $Fix(f) \coloneqq \sum_{x:A} (f x = x)$ is an h-proposition, and equivalent to ${\|A\|}$.

\end{ulemma}
\begin{proof}
Suppose $H: \prod_{(x,y:A)} (f x = f y)$, and let $(a,p),(b,q):Fix(f)$; we want to show $(a,p)=(b,q)$. Let $r:a=b$ be the concatenated path

\begin{displaymath}
a \xrightarrow{p^{-1}} f a \xrightarrow{H_{a,a}^{-1}} f a \xrightarrow{H_{a,b}} f b \xrightarrow{q} b.
\end{displaymath}
It will suffice to show that $p \bullet r = ap_f(r) \bullet q$, where $ap_f$ denotes the action of $f$ on paths. However, the dependent action of $H$ on paths implies that $H_{x,y} \bullet ap_f(s) = H_{x,y'}$ for any $x:A$ and any $s:y=y'$, and in particular $ap_f(r) = H_{a,a}^{-1} \bullet H_{a,b}$. From this $p \bullet r = ap_f(r) \bullet q$ is immediate. Thus, $Fix(f)$ is an h-proposition.

Now we have a map $g:A\to Fix(f)$ defined by $g(x) \coloneqq (f x, H_{f x,x})$, so by the universal property of ${\|A\|}$, we have ${\|A\|} \to Fix(f)$. On the other hand, we have the first projection $pr_1:Fix(f) \to A$, and hence $Fix(f) \to {\|A\|}$. Thus, these two h-propositions are equal.

\end{proof}
\begin{utheorem}
A type $A$ has [[split support]] if and only if it admits a steady endomap $f:A\to A$.

\end{utheorem}
\begin{proof}
Given ${\|A\|} \to A$, the composite $A\to {\|A\|} \to \A$ is steady. Conversely, if $f$ is steady, $Fix(f) = {\|A\|}$ by the lemma, so $pr_1:Fix(f) \to A$ splits the support of $A$.

\end{proof}
Note that $pr_1 \circ g = f$, so that if we start with a steady endomap of $A$, deduce a splitting of the support of $A$, and then reconstruct a steady endomap, we obtain the same map. However, the proof of steadiness is generally different from the one we began with, so this ``logical equivalence'' is not an equivalence of types.

\begin{ucor}
A type $A$ is an [[h-set]] if and only if every identity type $x=_A y$ admits a steady endomap.

\end{ucor}
\begin{proof}
We know that $A$ is an h-set just when all $x=_A y$ have split support.

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

\begin{itemize}%
\item [[Nicolai Kraus]] and [[Martin Escardo]] and [[Thierry Coquand]] and [[Thorsten Altenkirch]], ``Generalizations of Hedberg's theorem'', M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. \href{http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf}{PDF}. In this paper, steady functions are called ``constant''.

\end{itemize}
[[!redirects steady function]] [[!redirects steady functions]]



\end{document}