nLab
interval

An interval is an under category, over category, or under-over category in a poset.

Definitions

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

$$[x,\mathrm{\infty }[=\{y:P\mid x\le y\};$${[x, \infty[} = \{ y : P \;|\; x \leq y \} ;

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

$$]-\mathrm{\infty },x]=\{y:P\mid y\le x\};$$]{-\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

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

Thinking of $P$ as a category and subsets of $P$ as subcategories, $[x,\mathrm{\infty }[$ is the coslice category $(x/P)$, $]-\mathrm{\infty },x]$ is the slice category $(P/x)$, and $[x,y]$ is the bislice category $(y/P/x)$.

Besides the closed intervals above, we also have the open intervals

• $]x,\mathrm{\infty }[=[x,\mathrm{\infty }[\setminus \{x\}=\{y:P\mid x<y\},$
• $]-\mathrm{\infty },x[=]-\mathrm{\infty },x]\setminus \{x\}=\{y:P\mid y<x\},$
• $]x,y[=[x,y]\setminus \{x,y\}=\{z:P\mid x<z<y\},$

as well as the half-open intervals

• $[x,y[=[x,y]\setminus \{y\}=\{z:P\mid x\le z<y\},$
• $]x,y]=[x,y]\setminus \{x\}=\{z:P\mid x<z\le y\}.$

These are important in analysis, and more generally whenever the quasiorder $<$ is at least as important as the partial order $\le$.

The entire poset $P$ is also considered an unbounded interval in itself.

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 unit interval $[\mathrm{0,1}]$ is primary in homotopy theory; a homotopy from $f$ to $g$ (themselves continuous maps from $A$ to $B$) is a continuous map $h:A\times [\mathrm{0,1}]\to B$ such that $h(x\mathrm{,0})=f(x)$ and $h(x\mathrm{,1})=g(x)$ 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 $1$-form on the real line requires orienting an interval; the standard orientation is from $x$ to $y$ in $[x,y]$. If $x>y$, 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.