This article is about structure on a closed interval of real numbers, generally taken to be $I = [0, 1]$, that is derivable from a coalgebraic perspective. This topic was introduced by Freyd.
For the moment we work classically, over the category $Set$. A bipointed set is a cospan of the form
where $x_0$ and $x_1$ might coincide. There is a monoidal product on $Cospan(1, 1)$ given by cospan composition (formed by taking pushouts); this monoidal product is denoted $\vee$. The monoidal unit is a 1-element set with its unique bipointed structure. The category of such cospans or bipointed sets is denoted $Cos$.
Inside $Cos$ is the full subcategory of two-pointed sets, where $x_0$ and $x_1$ are distinct. (N.B. When we go beyond the classical case, we need a more refined analysis involving notions of apartness, separation, etc.) Let $Twop$ be the category of two-pointed sets. The monoidal product $\vee$ restricts to a functor
and one can define the square
A $sq$-coalgebra is a two-pointed set $X$ together with a map $\xi: X \to X \vee X$. An example is given by $I = [0, 1]$, where $I \vee I$ is identified with the interval $[0, 2]$ and the coalgebra structure $I \to I \vee I$ is identified with multiplication by $2$, $[0, 1] \to [0, 2]$. This map will be denoted $\alpha$.
$(I, \alpha)$ is terminal in the category of $sq$-coalgebras.
We now define a number of operations on $I$. For $0 \leq x \leq 1$, define ${x \uparrow} \coloneqq \min(2 x, 1)$ and ${x \downarrow} \coloneqq \max(2 x - 1, 0)$. These give unary operations on $I$ which can also be defined as maps in $Cos$ using the coalgebra structure $\alpha$:
We similarly define unary operations ${(-)\uparrow}$, ${(-) \downarrow}$ for any $sq$-coalgebra $(X, \xi)$. For any coalgebra $X$ and $x \in X$, either ${x \downarrow} = x_0$ or ${x \uparrow} = x_1$. Moreover, if $i_0 \colon X \to X \vee X$ and $i_1 \colon X \to X \vee X$ are the evident pushout inclusions, we have $\xi(x) = i_0({x \uparrow})$ if ${x \downarrow} = x_0$, and $\xi(x) = i_1({x \downarrow})$ if ${x \uparrow} = x_1$. This means that coalgebra structures can be recovered from algebraic structures consisting of two constants $x_0, x_1$ and two unary operations $\uparrow$, $\downarrow$, although we must consider a coherent but non-algebraic axiom
Next, we define meet and join operations on $I$, making it a lattice, by exploiting corecursion. A slick corecursive definition of the order $\leq$ is that $x \leq y$
if ${x \downarrow} = 0$ and ${x \uparrow} \leq {y \uparrow}$, or
if ${y \uparrow} = 1$ and ${x \downarrow} \leq {y \downarrow}$.
If one prefers to work with operations, one could define the meet operation $\wedge \colon I \times I \to I$ by putting a suitable coalgebra structure on $I \times I$ and using terminality of the coalgebra $I$ to define $\wedge$ as a coalgebra map. A coalgebra structure
which works is
$\xi(x, y) = i_0({x \uparrow}, {y \uparrow})$ if ${x \downarrow} = 0$ or ${y \downarrow} = 0$;
$\xi(x, y) = i_1({x \downarrow}, {y \downarrow})$ if ${x \uparrow} = 1 = {y \uparrow}$.
The general midpoint operation is not as easy to construct as one might think, but to start with we do have operations which take the midpoint between a given point and an endpoint. Namely, the left midpoint operation is the unary operation defined by
and the right midpoint operation is defined by