This entry is about the notion in order theory. For a related but different notion in homotopy theory see at interval object.
In the general context of posets, an interval is an under category, over category, or under-over category.
Given a poset and an element of , the upwards unbounded interval (also , , etc) is the subset
the downwards unbounded interval (also , , etc) is the subset
and given an element of , the bounded interval (also ) is the subset
Thinking of as a category and subsets of as subcategories, is the coslice category , is the slice category , and is the bislice category .
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
as well as the half-open intervals
These are important in analysis, and more generally whenever the quasiorder is at least as important as the partial order .
The entire poset 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 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 is primary in homotopy theory; a homotopy from to (themselves continuous maps from to ) is a continuous map such that and 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 -form on the real line requires orienting an interval; the standard orientation is from to in . If , then (which by the definition above would be empty) may also be interpreted as 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.
Relation to simplices
Let be the category of finite linear intervals.
There is an equivalence of categories
from the opposite category of the simplex category to .
and the inverse is
See also at Simplex category – Duality with intervals.
Intervals as generators of the incidence algebra
Recall that the incidence algebra of a poset (relative to some commutative ring ) is an associative unital algebra containing all functions such that implies . For any pair of elements related by the order , we can define an element of the incidence algebra by:
and the collection of such functions form a basis of as an -module. So, the dimension of the incidence algebra is equal to the total number of (non-empty) intervals in .
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.
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.