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\section*{interval}

\begin{quote}%
This entry is about the notion in [[order theory]]. For the related concept in [[topology]] see at \emph{[[topological interval]]}, and for concept in [[homotopy theory]] see at \emph{[[interval object]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

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

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

\noindent\hyperlink{in_posets}{In posets}\dotfill \pageref*{in_posets} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{intervals_in_the_real_line}{Intervals in the real line}\dotfill \pageref*{intervals_in_the_real_line} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classifying_topos}{Classifying topos}\dotfill \pageref*{classifying_topos} \linebreak
\noindent\hyperlink{RelationToSimplices}{Relation to simplices}\dotfill \pageref*{RelationToSimplices} \linebreak
\noindent\hyperlink{intervals_as_generators_of_the_incidence_algebra}{Intervals as generators of the incidence algebra}\dotfill \pageref*{intervals_as_generators_of_the_incidence_algebra} \linebreak
\noindent\hyperlink{in_homotopy_theory}{In homotopy theory}\dotfill \pageref*{in_homotopy_theory} \linebreak
\noindent\hyperlink{in_geometry}{In geometry}\dotfill \pageref*{in_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{in_posets}{}\subsection*{{In posets}}\label{in_posets}

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

In the general context of [[posets]], an \emph{interval} is an [[under category]], [[over category]], or under-over category. They are closed under betweenness: if two points belong to an interval and a third point is between them, then that third point also belongs to the interval.

\hypertarget{definitions}{}\subsubsection*{{Definitions}}\label{definitions}

Given a [[poset]] $P$ and an element $x$ of $P$, the \textbf{upwards unbounded interval} $[x,\infty[$ (also $[x,\infty)$, $[x,\infty[_P$, etc) is the [[subset]]

\begin{displaymath}
{[x, \infty[} = \{ y : P \;|\; x \leq y \} ;
\end{displaymath}
the \textbf{downwards unbounded interval} $]{-\infty}, x]$ (also $(-\infty,x]$, $]{-\infty},x]_P$, etc) is the subset

\begin{displaymath}
]{-\infty}, x] = \{ y : P \;|\; y \leq x \} ;
\end{displaymath}
and given an element $y$ of $P$, the \textbf{bounded interval} $[x,y]$ (also $[x,y]_P$) is the subset

\begin{displaymath}
[x,y] = \{ z : P \;|\; x \leq z \leq y \} .
\end{displaymath}
Thinking of $P$ as a [[category]] and subsets of $P$ as [[full subcategory|subcategories]], $[x,\infty[$ is the [[coslice category]] $(x/P)$, $]{-\infty},x]$ is the [[slice category]] $(P/x)$, and $[x,y]$ is the bislice category $(y/P/x)$.

An interval with distinct [[top]] and [[bottom]] element in a [[total order]] is also called a \textbf{linear interval}. (Sometimes this is called a \textbf{strict linear interval} and just ``linear interval'' then refers to the situation where top and bottom may coincide.)

Besides the \textbf{closed intervals} above, we also have the \textbf{open intervals}

\begin{itemize}%
\item ${]x, \infty[} = {[x,\infty[} \setminus \{x\} = \{ y : P \;|\; x \lt y \} ,$
\item ${]{-\infty}, x[} = {]{-\infty}, x]} \setminus \{x\} = \{ y : P \;|\; y \lt x \} ,$
\item ${]x, y[} = [x, y] \setminus \{x, y\} = \{ z : P \;|\; x \lt z \lt y \} ,$

\end{itemize}
as well as the \textbf{half-open intervals}

\begin{itemize}%
\item ${[x,y[} = [x,y] \setminus \{y\} = \{ z : P \;|\; x \leq z \lt y \} ,$
\item $]x,y] = [x,y] \setminus \{x\} = \{ z : P \;|\; x \lt z \leq y \} .$

\end{itemize}
These are important in analysis, and more generally whenever the [[quasiorder]] $\lt$ is at least as important as the [[partial order]] $\leq$.

The entire poset $P$ is also considered an \textbf{unbounded interval} in itself.

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\hypertarget{intervals_in_the_real_line}{}\paragraph*{{Intervals in the real line}}\label{intervals_in_the_real_line}

Intervals of [[real numbers]] are important in [[analysis]] and [[topology]]. They may be succinctly characterized as the [[connected space|connected]] [[subspaces]] of the real line. The bounded closed intervals in the real line are the original [[compact spaces]].

The interval in the reals has a [[universal property|universal]] characterization: it \href{http://ncatlab.org/nlab/show/terminal+coalgebra+of+an+endofunctor#UnitInterval}{is the terminal coalgebra} of the [[endofunctor]] on the category of all intervales that glues an interval end-to-end to itself.

The \textbf{unit interval} $[0,1]$ is primary in [[homotopy theory]]; specifically in [[topological homotopy theory]] a \textbf{[[left homotopy]]} from a [[continuous function]] $f$ to a continuous function $g$ is a continuous function $h \colon A \times I \to B$ out of the [[product topological space]] with the [[topological interval]] $I = [0,1]$ such that $h(x,0) = f(x)$ and $h(x,1) = g(x)$. More generally this is the concept of left homotopy for an [[interval object]] in a suitable ([[model category|model]]) [[category]].

The usual [[integral]] in ordinary calculus is done over an interval in the real line, a compact interval for a `proper' integral, or any interval for an `improper' integral. The theory of [[Lebesgue measure]] removes this restriction and allows integrals over any [[measurable subset]] of the real line. Still, the Lebesgue measure on intervals (even compact intervals) generates all of the rest.

To integrate a $1$-[[differential form|form]] on the real line requires orienting an interval; the standard orientation is from $x$ to $y$ in $[x,y]$. If $x \gt y$, then $[x,y]$ (which by the definition above would be [[empty set|empty]]) may also be interpreted as $[y,x]$ with the reverse orientation. This also matches the traditional notation for the integral.

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

\hypertarget{classifying_topos}{}\paragraph*{{Classifying topos}}\label{classifying_topos}

The [[classifying topos]] for linear intervals is the category [[sSet]] of [[simplicial sets]]. See the section \emph{\href{http://ncatlab.org/nlab/show/classifying+topos#ForIntervals}{For intervals}} at \emph{[[classifying topos]]}.

\hypertarget{RelationToSimplices}{}\paragraph*{{Relation to simplices}}\label{RelationToSimplices}

Let $\mathbb{I}$ be the category of \emph{finite} linear intervals.

There is an [[equivalence of categories]]

\begin{displaymath}
\widehat{(-)} : \Delta^{op} \stackrel{\simeq}{\to} \mathbb{I}
\end{displaymath}
from the [[opposite category]] of the [[simplex category]] to $\mathbb{I}$.

Here

\begin{displaymath}
\widehat{[n]} \coloneqq Hom_{\Delta}([n],[1]) \simeq [n+1]
\end{displaymath}
and the inverse is

\begin{displaymath}
[n] \mapsto Hom_{\mathbb{I}}([n],[1])
  \,.
\end{displaymath}
See also at \emph{\href{simplex+category#DualityWithIntervals}{Simplex category -- Duality with intervals}}.

\hypertarget{intervals_as_generators_of_the_incidence_algebra}{}\paragraph*{{Intervals as generators of the incidence algebra}}\label{intervals_as_generators_of_the_incidence_algebra}

Recall that the [[incidence algebra]] $I(P)$ of a poset $P$ (relative to some commutative ring $R$) is an [[associative unital algebra]] containing all functions $f : P \times P \to R$ such that $x \nleq y$ implies $f(x,y) = 0$. For any pair of elements related by the order $x \leq y$, we can define an element $\epsilon_{x,y}$ of the incidence algebra by:

\begin{displaymath}
\epsilon_{x,y}(u,v) = \begin{cases}1 & u = x, w = y \\ 0 & \text{otherwise}\end{cases}
\end{displaymath}
and the collection of such functions $\epsilon_{x,y}$ form a [[basis]] of $I(P)$ as an $R$-[[module]]. So, the [[dimension]] of the incidence algebra $I(P)$ is equal to the total number of (non-empty) intervals in $P$.

Information about the number of intervals in a finite poset is also encoded in its [[zeta polynomial]].

\hypertarget{in_homotopy_theory}{}\subsection*{{In homotopy theory}}\label{in_homotopy_theory}

In [[homotopy theory]], ``cellular'' models for the intervals play a central role. See [[interval object]].

\hypertarget{in_geometry}{}\subsection*{{In geometry}}\label{in_geometry}

The [[geometry]] (for instance [[differential geometry]]) of intervals, for instance in the [[real line]], are often relevant.

See for instance \href{http://ncatlab.org/nlab/show/cohesive+homotopy+type+theory#GeometricSpacesAndTheirHomotopyTypes}{Geometric spaces and their homotopy types} at [[cohesive homotopy type theory]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[interval type]], [[interval object]]


\item [[abstract circle]]


\item [[twisted arrow category]]


\item [[t-norm]]



\end{itemize}
[[!redirects interval]] [[!redirects intervals]]

[[!redirects unit interval]]

[[!redirects under-over category]]

[[!redirects linear interval]] [[!redirects linear intervals]]

[[!redirects open interval]] [[!redirects open intervals]]

[[!redirects closed interval]] [[!redirects closed intervals]]

[[!redirects half-open interval]] [[!redirects half-open intervals]]



\end{document}