And we shouldn’t stand for it.

But I’m getting ahead of myself. At present, all I’ve provided you is a vague-enough-to-be-deep title that really says nothing new at all. Of course homotopy is all about the unit interval. Read the definition.

Let’s do just that, since it will lead us right to the reason why I wrote the first sentence of this post. Consider two topological spaces $X, Y \in \textbf{Top}$ and continuous maps $f, g : X \rightarrow Y$. Suppose they were homotopic, ie. there is a continuous map $H : X \times [0, 1] \rightarrow Y$ such that $H(-, 0) = f$ and $H(-, 1) = g$. If $X$ and $Y$ are nicely behaved, eg. compactly generated and weakly Hausdorff, we’ll get to assume they’re both within a Cartesian closed subcategory of $\textbf{Top}$ consisting of other ‘nicely behaved’ topological spaces. In this scenario, we can replace $H$ with a right homotopy $K : X \rightarrow Y^{[0, 1]}$ where $K(-)(-) = H(-, -)$.

We can do better. Let’s call our Cartesian closed subcategory of nicely behaved spaces $\mathscr{N} \subset \textbf{Top}$ (short for Neverland). The very same Cartesian closed adjunction of $\mathscr{N}$ that let us consider right homotopies will let us instead phrase homotopies as paths $$ L : [0, 1] \rightarrow Y^X $$ in the ‘moduli space of maps’ $Y^X$. We can phrase homotopy equivalences similarly - consider the map $$ X^Y \times Y^X \rightarrow X^X \times Y^Y $$ sending $(f, g) \mapsto (f \circ g, g \circ f)$. Its image $Q_{X, Y} \subseteq X^X \times Y^Y$ becomes a space by the subspace topology (though with no guarantee that it is in $\mathscr{N}$). A homotopy equivalence is now a path $$ M : [0, 1] \rightarrow X^X \times Y^Y $$ starting at $(1_X, 1_Y)$ and ending in $Q_{X, Y}$. We thus see that $E_{X, Y} \subset Q_{X, Y}$, the space of homotopy equivalences, is just the path component of $(1_X, 1_Y)$ in $X^X \times Y^Y$ intersected with $Q_{X, Y}$.

As topologists, we should raise our eyebrows in unison. What’s so special about maps from the unit interval into $Y^X$ or $X^X \times Y^Y$? It panders to our intuition of a ‘path’, but we should be willing to study more exotic objects. Our utopian Neverland of topologies with neat Cartesian closed adjunctions still includes spaces completely alien to the familiar $[0, 1]$, for which $E_{X, Y}$ will be nearly trivial. In these scenarios, we should be willing to find an analogue for the unit interval that will give a more appropriate theory of equivalences.

Some examples are in order. I’ll start with the most boring one. Consider the ‘$S$-sized point’ $S_t$, which most people would call the trivial topology on a set $S$. I think my name is more evocative. A continuous maps $X \rightarrow S_t$ is just any map of sets, so for any two such maps $f, g : X \rightarrow S_t$, there is an automorphism $h : S_t \rightarrow S_t$ such that $f = h \circ g$. One might conclude that any two maps into $S_t$ can only differ in a trivial manner - there is no way to distinguish elements of the $S$-sized point topologically. To its colleagues in $\textbf{Top}$, $S_t$ is an uncanny twin of the singleton $\ast$.

We see almost immediately for any space $X$ and set $S$ that homotopy means nothing. A homotopy $H : X \times [0, 1] \rightarrow S_t$ can have any image, so any two maps $X \rightarrow S_t$ are homotopic. With all my preamble about how no two points in $S_t$ can really be distinguished topologically, is this truly a failure of homotopy? Surely we should rejoice at how homotopy identifies such trivially different functions. I remain unimpressed. In the end, the space $[0, 1]$, in all its cumbersome irrelevance, clouds the deeper similarities between any two maps $X \rightarrow S_t$. Could we not have used a more relevant space to better accentuate this information, like $\{0, 1\}_t$ perhaps?

Let’s explore this suggestion a bit. Let $\textbf{2} := \{0, 1\}$. A $\textbf{2}_t$-homotopy between maps $f, g : X \rightarrow Y$ will denote a map $K: \textbf{2}_t \rightarrow Y^X$ where $K(0) = f$ and $K(1) = g$. If a $\textbf{2}_t$-homotopy from $f$ to $g$ should exist, then for every $x \in X$ the points $f(x)$ and $g(x)$ must have identical neighborhoods in $Y$. In fact, this is precisely when maps $f$ and $g$ will be $\textbf{2}_t$-homotopic. The relation of $\textbf{2}_t$-homotopy now identifies when maps are equivalent ‘up to trivial topologies’. The corresponding notion of $\textbf{2}_t$-homotopy equivalence - a continuous map $f : \textbf{2}_t \rightarrow X^X \times Y^Y$ where $f(0) = (1_X, 1_Y)$ and $f(1) \in Q_{X, Y}$ - identifies spaces that differ up to $S$-sized points, for various $S$. We might have called this pointless topology, if that hilarious name wasn’t already taken.

The relation of $\textbf{2}_t$-homotopy is reflexive by the bang map $\textbf{2}_t \rightarrow \ast$ and symmetric by the swap map $\textbf{2}_t \rightarrow \textbf{2}_t$. For transitivity, we have a map $\textbf{2}_t \rightarrow \{0, 1, 2\}_t$ sending $0 \mapsto 0$ and $1 \mapsto 2$, with which we can concatenate homotopies. We now have an equivalence relation $\sim_{\textbf{2}_t}$ between maps, much like we had the relation $\sim_{[0, 1]}$ for classical homotopy. It is clear that $\sim_{\textbf{2}_t}$ is far stricter than $\sim_{[0, 1]}$; any map $[0, 1] \rightarrow \textbf{2}_t$ (of which there are plenty) can be used to convert a $\textbf{2}_t$-homotopy into a $[0, 1]$-homotopy via precomposition.

We also get a relation $\cong_{\textbf{2}_t}$ between spaces, denoting $\textbf{2}_t$-homotopy equivalence. Again, this is far stricter than the relation $\cong_{[0, 1]}$ corresponding to classical homotopy equivalence. It is this relation that better captures the name ‘$S$-sized point’ so well; for any set $S$, we see immediately that $S_t \cong_{\textbf{2}_t} \ast$. This picture is precisely analogous to how $D^n \cong_{[0, 1]} \ast$; the $S$-sized point has taken the role of a ‘giant point’ that $D^n$ fills in classical homotopy theory. We have successfully reflavored homotopy to better fit the nuance of trivial topologies, the further study of which may reveal results that $[0, 1]$ was completely blind to.

It’s time to return to the title of this post. Classical homotopy theory has concerned itself only with the unit interval as of late. Within the last few decades, Voevodsky made the daring jump into analogizing homotopy theory for algebraic geometry; in the world of schemes and varieties, $\mathbb{A}_1$ is the correct analogy for $[0, 1]$. In this post, I’ve shared a glimpse of the world where $\textbf{2}_t$ is the correct replacement. Indeed, almost any topological space with at least $2$ points could be imagined to have its own homotopy theory, each one bringing insight into a different species of topological space.

We shouldn’t take this as a suggestion that the unit interval is useless. The latter has opened up stunning results around manifolds, CW-complexes and many other spaces whose definitions rely heavily on the topology of the real line. All I’ll say is perhaps it shouldn’t be pushed any further. Neverland is a bizarre and alien place. Those who wish to study it will need bizarre and alien tools.