nLab
coalgebra of the real interval

Introduction

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.

Coalgebraic description of $I$

For the moment we work classically, over the category $\mathrm{Set}$. A bipointed set is a cospan of the form

$$\begin{array}{ccccc}1& & & & 1\\ & {}^{x_0}\searrow & & {\swarrow }^{x_1}& \\ & & X& & \end{array}$$\array{ 1 & & & & 1 \\ & ^\mathllap{x_0} \searrow & & \swarrow^\mathrlap{x_1} & \\ & & X & & }

where $x_0$ and $x_1$ might coincide. There is a monoidal product on $\mathrm{Cospan}(1,1)$ 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 $\mathrm{Cos}$.

Inside $\mathrm{Cos}$ 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 $\mathrm{Twop}$ be the category of two-pointed sets. The monoidal product $\vee$ restricts to a functor

$$\vee :\mathrm{Twop}\times \mathrm{Twop}\to \mathrm{Twop}$$\vee \colon Twop \times Twop \to Twop

and one can define the square

$$\mathrm{sq}=(\mathrm{Twop}\stackrel{\Delta }{\to }\mathrm{Twop}\times \mathrm{Twop}\stackrel{\vee }{\to }\mathrm{Twop})$$sq = (Twop \stackrel{\Delta}{\to} Twop \times Twop \stackrel{\vee}{\to} Twop)

A $\mathrm{sq}$-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 $\alpha$.

Theorem (Freyd)

$(I,\alpha )$ is terminal in the category of $\mathrm{sq}$-coalgebras.

Corecursively defined operations on $I$

We now define a number of operations on $I$. For $0\le x\le 1$, define $x\uparrow ≔\min (2x,1)$ and $x\downarrow ≔\max (2x-1,0)$. These give unary operations on $I$ which can also be defined as maps in $\mathrm{Cos}$ using the coalgebra structure $\alpha$:

$$I\stackrel{(-)\uparrow }{\to }I=(I\stackrel{\alpha }{\to }I\vee I\stackrel{I\vee !}{\to }I\vee 1\cong I)$$I \stackrel{{(-) \uparrow}}{\to} I = (I \stackrel{\alpha}{\to} I \vee I \stackrel{I \vee !}{\to} I \vee 1 \cong I)

$$$$\,

$$I\stackrel{(-)\downarrow }{\to }I=(I\stackrel{\alpha }{\to }I\vee I\stackrel{!\vee I}{\to }1\vee I\cong I)$$I \stackrel{{(-) \downarrow}}{\to} I = (I \stackrel{\alpha}{\to} I \vee I \stackrel{! \vee I}{\to} 1 \vee I \cong I)

We similarly define unary operations $(-)\uparrow$, $(-)\downarrow$ for any $\mathrm{sq}$-coalgebra $(X,\xi )$. For any coalgebra $X$ and $x\in X$, either $x\downarrow =x_0$ or $x\uparrow =x_1$. Moreover, if $i_0:X\to X\vee X$ and $i_1:X\to X\vee X$ 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

$$⊢x\uparrow =x_1\mathrm{or}x\downarrow =x_0$$\vdash {x \uparrow} = x_1 or {x \downarrow} = x_0

Order-theoretic operations

Next, we define meet and join operations on $I$, making it a lattice, by exploiting corecursion. A slick corecursive definition of the order $\le$ is that $x\le y$

• if $x\downarrow =0$ and $x\uparrow \le y\uparrow$, or

• if $y\uparrow =1$ and $x\downarrow \le y\downarrow$.

If one prefers to work with operations, one could define the meet operation $\wedge :I\times I\to I$ 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

$$\xi :I\times I\to (I\times I)\vee (I\times I)$$\xi \colon I \times I \to (I \times I) \vee (I \times I)

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 $x\uparrow =1=y\uparrow$.

Midpoint operations

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

$$l=(I\stackrel{i_0}{\to }I\vee I\stackrel{{\alpha }^{-1}}{\to }I)$$l = (I \stackrel{i_0}{\to} I \vee I \stackrel{\alpha^{-1}}{\to} I)

and the right midpoint operation is defined by

$$r=(I\stackrel{i_1}{\to }I\vee I\stackrel{{\alpha }^{-1}}{\to }I).$$r = (I \stackrel{i_1}{\to} I \vee I \stackrel{\alpha^{-1}}{\to} I).

References

• Peter Freyd, Reality Check, post to the categories list, July 31, 2000 (web)