> This entry is about the notion in [[order theory]]. For the related concept in [[topology]] see at _[[topological interval]]_, and for concept in [[homotopy theory]] see at _[[interval object]]_. *** +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### $(0,1)$-Category theory +--{: .hide} [[!include (0,1)-category theory - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## In posets ### Idea In the general context of [[posets]], an _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. ### Definitions Given a [[poset]] $P$ and an element $x$ of $P$, the __upwards unbounded interval__ $[x,\infty[$ (also $[x,\infty)$, $[x,\infty[_P$, etc) is the [[subset]] $$ {[x, \infty[} = \{ y : P \;|\; x \leq y \} ;$$ the __downwards unbounded interval__ $]{-\infty}, x]$ (also $(-\infty,x]$, $]{-\infty},x]_P$, etc) is the subset $$ ]{-\infty}, x] = \{ y : P \;|\; y \leq x \} ;$$ and given an element $y$ of $P$, the __bounded interval__ $[x,y]$ (also $[x,y]_P$) is the subset $$ [x,y] = \{ z : P \;|\; x \leq z \leq y \} .$$ 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 **linear interval**. (Sometimes this is called a **strict linear interval** and just "linear interval" then refers to the situation where top and bottom may coincide.) Besides the __closed intervals__ above, we also have the __open intervals__ * $ {]x, \infty[} = {[x,\infty[} \setminus \{x\} = \{ y : P \;|\; x \lt y \} ,$ * $ {]{-\infty}, x[} = {]{-\infty}, x]} \setminus \{x\} = \{ y : P \;|\; y \lt x \} ,$ * $ {]x, y[} = [x, y] \setminus \{x, y\} = \{ z : P \;|\; x \lt z \lt y \} ,$ as well as the __half-open intervals__ * $ {[x,y[} = [x,y] \setminus \{y\} = \{ z : P \;|\; x \leq z \lt y \} ,$ * $ ]x,y] = [x,y] \setminus \{x\} = \{ z : P \;|\; x \lt z \leq y \} .$ 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 __unbounded interval__ in itself. ### Examples #### 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 [is the terminal coalgebra](http://ncatlab.org/nlab/show/terminal+coalgebra+of+an+endofunctor#UnitInterval) of the [[endofunctor]] on the category of all intervales that glues an interval end-to-end to itself. The __unit interval__ $[0,1]$ is primary in [[homotopy theory]]; specifically in [[topological homotopy theory]] a __[[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. ### Properties #### Classifying topos The [[classifying topos]] for linear intervals is the category [[sSet]] of [[simplicial sets]]. See the section _[For intervals](http://ncatlab.org/nlab/show/classifying+topos#ForIntervals)_ at _[[classifying topos]]_. #### Relation to simplices {#RelationToSimplices} Let $\mathbb{I}$ be the category of _finite_ linear intervals. There is an [[equivalence of categories]] $$ \widehat{(-)} : \Delta^{op} \stackrel{\simeq}{\to} \mathbb{I} $$ from the [[opposite category]] of the [[simplex category]] to $\mathbb{I}$. Here $$ \widehat{[n]} \coloneqq Hom_{\Delta}([n],[1]) \simeq [n+1] $$ and the inverse is $$ [n] \mapsto Hom_{\mathbb{I}}([n],[1]) \,. $$ See also at _[Simplex category -- Duality with intervals](simplex+category#DualityWithIntervals)_. #### 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: $$ \epsilon_{x,y}(u,v) = \begin{cases}1 & u = x, w = y \\ 0 & \text{otherwise}\end{cases} $$ 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]]. ## In homotopy theory In [[homotopy theory]], "cellular" models for the intervals play a central role. See [[interval object]]. ## In geometry The [[geometry]] (for instance [[differential geometry]]) of intervals, for instance in the [[real line]], are often relevant. See for instance [Geometric spaces and their homotopy types](http://ncatlab.org/nlab/show/cohesive+homotopy+type+theory#GeometricSpacesAndTheirHomotopyTypes) at [[cohesive homotopy type theory]]. ## Related concepts * [[interval arithmetic]] * [[interval type]], [[interval object]] * [[abstract circle]] * [[twisted arrow category]] * [[t-norm]] [[!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]]