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\begin{document}

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\section*{triangle inequality}

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

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{InterpretationInEnrichedCategoryTheory}{Interpretation in enriched category theory}\dotfill \pageref*{InterpretationInEnrichedCategoryTheory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The key conditions on a [[distance]] function for \emph{[[metric spaces]]} and on \emph{[[norms]]} on [[normed vector spaces]] is that the distance $d(x,z)$ between any two points $x,z$ is no larger than the sum of the distances $d(x,y) + d(y,z)$ via any third point $y$

\begin{displaymath}
d(x,z) \leq d(x,y) + d(y,z)
  \,.
\end{displaymath}
For the usual metric/norm on [[Euclidean space]] and $f$ and $g$ two edges of a [[triangle]]

\begin{displaymath}
\itexarray{
    && y
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    x && \underset{f + g}{\longrightarrow} && z
  }
\end{displaymath}
then this [[inequality]] expresses that the straight path from $x$ to $z$ is always shorter or at worst as long as the path from $x$ to $z$ via $y$.

However, [[normed fields]] and hence [[normed vector spaces]] for which this example gives the correct intuition are rare among all normed fields and vector spaces (they are the \emph{[[archimedean]]} ones). Most norms instead satisfy the stronger [[ultrametric]] triangle inequality which says that

\begin{displaymath}
{\vert f + g \vert} \leq max({\vert f\vert}, {\vert g\vert })
  \,.
\end{displaymath}
A norm with this property is called \emph{[[non-archimedean]].}

\hypertarget{InterpretationInEnrichedCategoryTheory}{}\subsection*{{Interpretation in enriched category theory}}\label{InterpretationInEnrichedCategoryTheory}

One may equivalently regard the triangle equality in [[metric spaces]] as the [[composition]] operation in a certain incarnation of the metric space as an [[enriched category]] (\hyperlink{Lawvere73}{Lawvere 73}). From this perspective some concepts from [[analysis]] usefully generalize to other [[enriched categories]], notably the concept of \emph{[[Cauchy complete categories]]}. For more on this see at \emph{\href{metric+space#LawvereMetricSpace}{Lawvere metric space}}.

Namely regard the [[half-open interval]] $[0,\infty) \subset \mathbb{R}$ as a [[poset]] under the [[relation]] $\geq$, and regard this poset as a [[category]] (see at [[(0,1)-category]]). The operation of addition of real numbers makes this a [[monoidal category]].

This means that a category [[enriched category|enriched over]] this monoidal category $([0,\infty){\geq}, +)$ is

\begin{enumerate}%
\item ([[objects]]) a set $X$;


\item ([[hom objects]]) for every [[pair]] of points $(x,y) \in X \times X$ a real number $d(x,y) \in [0,\infty)$


\item ([[composition]]) for all $x,y,z \in X$ a morphism in $[0,\infty)_{\geq}$ of the form

\begin{displaymath}
\circ_{x,y,z} \;\colon\; d(x,y) + d(x,z) \longrightarrow d(x,z)
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item ([[unitality]]) $d(x,x) = 0$;


\item ([[associativity]]) \ldots{}



\end{enumerate}
Now since the [[category]] $[0,\infty)_{\geq}$ is a [[poset]], there is at most one morphism between any given pair of objects, and hence the choice of [[composition]] morphism $\circ_{x,y,z}$ above is really a condition on $d(-,-)$. Moreover, since a morphism $x \to y$ in $[0,\infty)$ exists precisely if $x \geq y$, then this condition is exactly the triangle identity

\begin{displaymath}
d(x,y) + d(y,z) \geq d(x,z)
  \,.
\end{displaymath}
Moreover, the [[unitality]] condition is part of the non-degeneracy condition on a metric, $d(x, y) = 0$ iff $x = y$, and the [[associativity]] condition is automatically satisfied once [[composition]] is defined.

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

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Triangle_inequality}{Triangle inequality}}


\item [[Bill Lawvere]] (1973). \emph{Metric spaces, generalized logic and closed categories}. Reprinted in [[TAC]], 1986. \href{http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html}{Web}.



\end{itemize}
[[!redirects triangle inequalities]]



\end{document}