In his lecture, [*The Categorical Origins of Lebesgue Measure*](https://www.maths.ed.ac.uk/~tl/cambridge_ct14/cambridge_ct14_talk.pdf), Professor Tom Leinster mentions the following theorem:



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> **Theorem 1:** (Freyd; Leinster) The topological space $[0, 1]$ comes equipped with two distinct basepoints $0$ and $1$, and a map $[0, 1] \rightarrow [0, 1] \amalg [0, 1] / \text{first }1 \sim \text{second } 0$. $[0, 1]$ is terminal as such.
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Leinster's theorem is about a universal property of the banach space $L^1 [0, 1]$ and the integration function $\int : L^1 [0, 1] \rightarrow \mathbb{R}$. It goes like this: let $\mathcal{A}$ be the category of Banach spaces, whose objects are banach spaces and whose maps are maps $\phi : X \rightarrow Y$ of banach spaces such that $||\phi(x)|| \leq ||x||$. Let $\mathcal{A}/\mathbb{R}$ be the under-category (objects are maps $\mathbb{R} \rightarrow X$ in $\mathcal{A}$; this is also called the coslice category). There is a functor $T : \mathcal{A} \rightarrow \mathcal{A}$ defined where $T(X) = X \prod X$, where $X$ has measure $||(x, y)|| = \frac{1}{2} (||x|| + ||y||)$ with corresponding map $\mathbb{R} \rightarrow X \prod X$ induced canonically by the map $\mathbb{R} \rightarrow X$. The initial $T$-algebra is 
$$L^1 [0, 1] = \frac{\text{Lebesgue-integrable functions } [0, 1] \rightarrow \mathbb{R} }{\text{equality almost everywhere}}$$
Note: we can still form $T$-algebras when $T$ is not a monad.


The unique map $\int : L^1 [0, 1] \rightarrow \mathbb{R}$ comes from the universal property now!


My question is this:



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> Can we rearrange theorem 1 above in terms of continuous functions $C([0, 1]) = [[0, 1], \mathbb{R}]\_{\text{Top}}$ and an endofunctor $T$ where $T(A) = A \prod A$, just like in the measure theory case?
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From a categorical perspective, this might be nice since we would no longer have to require two *distinct* points in the topological space.


So the rearrangement of Professor Leinster's theorem would go like this:


Let $\mathcal{B}$ be the category of topological vector spaces (some tweaking might be necessary, maybe topological algebras would be better? Maybe the Lawvere theory of $C^0$-algebras?). Let $\mathcal{B}/\mathbb{R}$ be the under-category (objects are maps $\mathbb{R} \rightarrow X$ in $\mathcal{B}$). There is a functor $T : \mathcal{B} \rightarrow \mathcal{B}$, where $T(X) = X \prod X$, just as before. Perhaps the initial $T$-algebra is $C([0, 1])$!




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Now that I think about it, if measure theory and topology have their own versions of this theorem, maybe there is one for differential geometry and $C^{\infty}$-algebras. Just a thought.



On the universal property for interval objects
The obvious thing is to consider the category of unital $C^\*$-algebras $A$ equipped with two distinct characters $\chi\_0,\chi\_1 : A \to \mathbb{C}$ (let me stick to $\mathbb{C}$ since strange things might happen over $\mathbb{R}$, but I am not sure). There is a non-unital monoidal structure $\otimes$ given by pullback "along the middle character". Consider the category of $B=(A,\chi\_0,\chi\_1)$ equipped with a morphism $B \otimes B \to B$. For example, we have $C[0,1]$ with its two evaluation characters $e\_0,e\_1$ and the evident "pasting" (iso)morphism $C[0,1] \times\_{e\_1,e\_0} C[0,1] \cong C[0,1] $. I think it is an initial object because of the adjunction between $C(-)$ and the character space $\Phi(-)$ of a unital $C^\*$-algebra and the mentioned universal characterization of $[0,1]$.


