Is there a scheme corresponding to the unit interval? Can someone complete the following table?
$\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & \mathbb{C} & \mathbb{A}^1 \\\\  \mathbb{R} P^1 \cong S^1 & \mathbb{C} P^1 \cong S^2 & \mathbb{P}^1 \\\\ {}D^1(\mathbb{R}) \cong [0,1] & D^1(\mathbb{C}) \cong  [0,1] \times [0,1] & ?  \end{array}$
I agree that this question is quite vague. Possible precise interpretations would be:
Is there a scheme $X$ satisfying $X(R) = D^1(R)$ for nice metric rings $R$? Or: Does the endofunctor $(X,x_0,x_1) \mapsto (X \cup_{x_0 \simeq x'_1} X',x_0,x'_1)$ (with a copy $X'$ of $X$) of the category of $k$-schemes (or locally ringed spaces over $k$) equipped with two $k$-rational points have a terminal coalgebra, similar to Freyd's characterization of $[0,1]$?
But I don't want to limit my question to this interpretation. I also would be happy with an algebraic space or stack corresponding to $[0,1]$. Note that $[-1,1]$ "is" the generalized ring and therefore affine scheme $|\mathbb{Z}_{\infty}|$ à la Durov defined as an algebraic submonad of $\mathbb{R}$.
Background: In $\mathbb{A}^1$-homotopy theory the affine line plays the role of the unit interval, right? But when we define homotopies between morphisms of schemes $X \to Y$ in the naive way as morphisms $X \times \mathbb{A}^1 \to Y$, we won't get a transitive relation. The reason is that, although $\mathbb{A}^1$ has two distinguished rational points $0,1$, we don't get $\mathbb{A}^1$ when we glue two copies of $\mathbb{A}^1$, as in the case of the unit interval.
 A: You can define a unit interval $I$ as a co-presheaf: the set of maps from $I$ to a connected scheme $X$ is the set of triples $(x,y,\phi)$, where $x$ and $y$ are geometric points in $X$, and $\phi$ is a natural isomorphism between the corresponding fiber functors (from the category of finite étale covers of $X$ to finite sets).
I'm pretty sure this is not co-represented by a scheme (for example it needs to have maps to the spectra of fields of arbitrarily large transcendence degree), but it does have nice composition properties.
There is a discussion of objects similar to this in section 10 of Deligne's "Le Groupe Fondamental de la Droite Projective Moins Trois Points", but I think there may be later papers that are more explicit.
A: The spectrum of a standard affinoid algebra (in one variable)?
A: This is really more deserving of being a comment(but I don't have enough points to make them), check out this MO post What properties make $[0,1]$ a good candidate for defining fundamental groups?
Perhaps you will find some inspiration in terms of other ways to think about the role of $[0,1]$ and what its analogue should be in scheme-land.
