ct.category theory - A canonical and categorical construction for geometric realization

As to "why is the unit interval the canonical interval?", there is an interesting universal property of the unit interval given in some observations of Freyd posted at the categories list, characterizing $[0, 1]$ as a terminal coalgebra of a suitable endofunctor on the category of posets with distinct top and bottom elements.

There are various ways of putting it, but for the purposes of this thread, I'll put it this way. Recall that the category of simplicial sets is the classifying topos for the (geometric) theory of intervals, where an interval is a totally ordered set (toset) with distinct top and bottom. (This really comes down to the observation that any interval in this sense is a filtered colimit of finite intervals -- the finitely presentable intervals -- which make up the category $Delta^{op}$.) Now there is a join $X vee Y$ on intervals $X$, $Y$ which identifies the top of $X$ with the bottom of $Y$, where the bottom of $X vee Y$ is identified with the bottom of $X$ and the top of $X vee Y$ with the top of $Y$. This gives a monoidal product $vee$ on the category of intervals, hence we have an endofunctor $F(X) = X vee X$. A coalgebra for the endofunctor $F$ is, by definition, an interval $X$ equipped with an interval map $X to F(X)$. There is an evident category of coalgebras.

In particular, the unit interval $[0, 1]$ becomes a coalgebra if we identify $[0, 1] vee [0, 1]$ with $[0, 2]$ and consider the multiplication-by-2 map $[0, 1] to [0, 2]$ as giving the coalgebra structure.

Theorem: The interval $[0, 1]$ is terminal in the category of coalgebras.

Let's think about this. Given any coalgebra structure $f: X to X vee X$, any value $f(x)$ lands either in the "lower" half (the first $X$ in $X vee X$), the "upper" half (the second $X$ in $X vee X$), or at the precise spot between them. Thus, you could think of a coalgebra as an automaton where on input $x_0$ there is output of the form $(x_1, h_1)$, where $h_1$ is either upper or lower or between. By iteration, this generates a behavior stream $(x_n, h_n)$. Interpreting upper as 1 and lower as 0, the $h_n$ form a binary expansion to give a number between 0 and 1, and therefore we have an interval map $X to [0, 1]$ which sends $x_0$ to that number. Of course, should we ever hit $(x_n, between)$, we have a choice to resolve it as either $(bottom_X, upper)$ or $(top_X, lower)$ and continue the stream, but these streams are identified, and this corresponds to the identification of binary expansions

$$.h_1... h_{n-1} 100000... = .h_1... h_{n-1}011111...$$

as real numbers. In this way, we get a unique well-defined interval map $X to [0, 1]$, so that $[0, 1]$ is the terminal coalgebra.

(Side remark that the coalgebra structure is an isomorphism, as always with terminal coalgebras, and the isomorphism $[0, 1] vee [0, 1] to [0, 1]$ is connected with the interpretation of the Thompson group as a group of PL automorphisms $phi$ of $[0, 1]$ that are monotonic increasing and with discontinuities at dyadic rationals.)