This article is about structure on a closed interval of real numbers, generally taken to be , that is derivable from a coalgebraic perspective. This topic was introduced by Freyd.
For the moment we work classically, over the category . A bipointed set is a cospan of the form
where and might coincide. There is a monoidal product on given by cospan composition (formed by taking pushouts); this monoidal product is denoted . The monoidal unit is a 1-element set with its unique bipointed structure. The category of such cospans or bipointed sets is denoted .
Inside is the full subcategory of two-pointed sets, where and are distinct. (N.B. When we go beyond the classical case, we need a more refined analysis involving notions of apartness, separation, etc.) Let be the category of two-pointed sets. The monoidal product restricts to a functor
and one can define the square
A -coalgebra is a two-pointed set together with a map . An example is given by , where is identified with the interval and the coalgebra structure is identified with multiplication by , . This map will be denoted .
is terminal in the category of -coalgebras.
We now define a number of operations on . For , define and . These give unary operations on which can also be defined as maps in using the coalgebra structure :
We similarly define unary operations , for any -coalgebra . For any coalgebra and , either or . Moreover, if and are the evident pushout inclusions, we have if , and if . This means that coalgebra structures can be recovered from algebraic structures consisting of two constants and two unary operations , , although we must consider a coherent but non-algebraic axiom
Next, we define meet and join operations on , making it a lattice, by exploiting corecursion. A slick corecursive definition of the order is that
if and , or
if and .
If one prefers to work with operations, one could define the meet operation by putting a suitable coalgebra structure on and using terminality of the coalgebra to define as a coalgebra map. A coalgebra structure
which works is
if or ;
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