The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective on points in the sense that the induced map $\text{Hom}(1, X) \to \text{Hom}(1, Y^X)$ is surjective, then $Y$ has the fixed point property: for every morphism $g : Y \to Y$ there exists a point $y : 1 \to Y$ such that $g \circ y = y$.

The Brouwer fixed point theorem asserts that the closed $n$-disks, all of which I will denote by $D$ for ease of notation, have the fixed point property as objects of $\text{Top}$.

Seeing these two theorems together, it is tempting to try to prove the latter from the former by finding a topological space $X$ such that the exponential $D^X$ exists, together with a surjective continuous map $X \to D^X$. Does there in fact exist such an $X$?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective on points in the sense that the induced map $\text{Hom}(1, X) \to \text{Hom}(1, Y^X)$ is surjective, then $Y$ has the fixed point property: for every morphism $g : Y \to Y$ there exists a point $y : 1 \to Y$ such that $g \circ y = y$.

The Brouwer fixed point theorem asserts that the closed $n$-disks, all of which I will denote by $D$ for ease of notation, have the fixed point property as objects of $\text{Top}$.

Seeing these two theorems together, it is tempting to try to prove the latter from the former by finding a topological space $X$ such that the exponential $D^X$ exists, together with a surjective continuous map $X \to D^X$. Does there in fact exist such an $X$?

Edit, 4/13/17: I'm still interested in this question, and so are some people associated with MIRI (at least when $n = 1$); for some details about why see here.