nLab
interval

This entry is about the notion in order theory. For a related but different notion in homotopy theory see at interval object.


Contents

In posets

Idea

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

Definitions

Given a poset PP and an element xx of PP, the upwards unbounded interval [x,[[x,\infty[ (also [x,)[x,\infty), [x,[ P[x,\infty[_P, etc) is the subset

[x,[={y:Pxy}; {[x, \infty[} = \{ y : P \;|\; x \leq y \} ;

the downwards unbounded interval ],x]]{-\infty}, x] (also (,x](-\infty,x], ],x] P]{-\infty},x]_P, etc) is the subset

],x]={y:Pyx}; ]{-\infty}, x] = \{ y : P \;|\; y \leq x \} ;

and given an element yy of PP, the bounded interval [x,y][x,y] (also [x,y] P[x,y]_P) is the subset

[x,y]={z:Pxzy}. [x,y] = \{ z : P \;|\; x \leq z \leq y \} .

Thinking of PP as a category and subsets of PP as subcategories, [x,[[x,\infty[ is the coslice category (x/P)(x/P), ],x]]{-\infty},x] is the slice category (P/x)(P/x), and [x,y][x,y] is the bislice category (y/P/x)(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,[=[x,[{x}={y:Px<y}, {]x, \infty[} = {[x,\infty[} \setminus \{x\} = \{ y : P \;|\; x \lt y \} ,
  • ],x[=],x]{x}={y:Py<x}, {]{-\infty}, x[} = {]{-\infty}, x]} \setminus \{x\} = \{ y : P \;|\; y \lt x \} ,
  • ]x,y[=[x,y]{x,y}={z:Px<z<y}, {]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]{y}={z:Pxz<y}, {[x,y[} = [x,y] \setminus \{y\} = \{ z : P \;|\; x \leq z \lt y \} ,
  • ]x,y]=[x,y]{x}={z:Px<zy}. ]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 PP is also considered an unbounded interval in itself.

Examples

Intervals in the real line

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][0,1] is primary in homotopy theory; a homotopy from ff to gg (themselves continuous maps from AA to BB) is a continuous map h:A×[0,1]Bh: A \times [0,1] \to B such that h(x,0)=f(x)h(x,0) = f(x) and h(x,1)=g(x)h(x,1) = g(x) always. This generalises to the notion of interval object in an arbitrary 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 11-form on the real line requires orienting an interval; the standard orientation is from xx to yy in [x,y][x,y]. If x>yx \gt y, then [x,y][x,y] (which by the definition above would be empty) may also be interpreted as [y,x][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 𝕀\mathbb{I} be the category of finite linear intervals.

There is an equivalence of categories

()^:Δ op𝕀 \widehat{(-)} : \Delta^{op} \stackrel{\simeq}{\to} \mathbb{I}

from the opposite category of the simplex category to 𝕀\mathbb{I}.

Here

[n]^Hom Δ([n],[1])[n+1] \widehat{[n]} \coloneqq Hom_{\Delta}([n],[1]) \simeq [n+1]

and the inverse is

[n]Hom 𝕀([n],[1]). [n] \mapsto Hom_{\mathbb{I}}([n],[1]) \,.

See also at Simplex category – Duality with intervals.

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.

Revised on March 31, 2014 00:15:31 by Urs Schreiber (82.113.99.67)