interval

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 atinterval object.

In the general context of posets, an *interval* is an under category, over category, or under-over category.

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 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.

Intervals of real numbers are important in analysis and topology. The bounded closed intervals in the real line are the original compact spaces.

The interval in the reals has a universal characterization: it is the terminal coalgebra 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.

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 $1$-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) may also be interpreted as $[y,x]$ with the reverse orientation. This also matches the traditional notation for the integral.

The classifying topos for linear intervals is the category sSet of simplicial sets. See the section *For intervals* at *classifying topos*.

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*.

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, “cellular” models for the intervals play a central role. See interval object.

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 at cohesive homotopy type theory.

Revised on January 3, 2018 13:00:00
by Mike Shulman
(76.167.222.204)