nLab 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 at interval object.

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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 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 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 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.

Properties

Classifying topos

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

Relation to simplices

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.

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 at cohesive homotopy type theory.

Related concepts

• interval arithmetic

• interval type, interval object

• abstract circle

• twisted arrow category

• t-norm