In his lecture, *The Categorical Origins of Lebesgue Measure*, Professor Tom Leinster mentions the following theorem:

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.

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:

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?

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 $C([0, 1])$!

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.