I’ve spoken recently about how arbitrary the unit interval is in the definition of homotopy. I’ve also tried to point out how replacing it with different topologies might give invariants suited to different classification problems. For instance, while $[0, 1]$ is best suited to classifying CW-complexes, the affine line $\mathbb{A}^1_k$ for a field $k$ is suitable for algebraic geometry in $k$.

In order for the above statements to make sense generally however, a few issues need to be ironed out:

If we want to figure out what a general homotopy theory should be, we should isolate exactly what we want it to do. Henceforth, let $P$ (short for path) be the identified topological space to serve as the interval, in which one map will vary to become another.

What form of homotopy should we use for maps $f, g : X \rightarrow Y$? A map $X \times P \rightarrow Y$? How about $X \rightarrow Y^P$? Perhaps we could use $P \rightarrow Y^X$? These are not going to be equivalent in general, if $P$ is not as nicely behaved as we’d like. Let’s use the first one, for consistency with classical homotopy theory.

Which two maps does a morphism $H : X \times P \rightarrow Y$ identify? We need two identified points $a, b : \ast \rightarrow P$. With these chosen, $H$ will be a relation between the maps $f = H(-, a)$ and $g = H(-, b)$. We’ll call $H$ a $P$-homotopy from $f$ to $g$.

Now, we want an equivalence relation out of $P$-homotopy, which we’ll write as $\sim_P$. Reflexivity is immediate, by the bang morphism $P \rightarrow \ast$ that always exists uniquely. How do we get symmetry and transitivity? We need a map $P \rightarrow P$ swapping $a$ and $b$. Indeed note that $a \sim_P b$ by the homotopy $1_P : \ast \times P \rightarrow P$. If $\sim_P$ is symmetric then $b \sim_P A$, so there must be a map $K : \ast \times P \rightarrow P$ sending $a$ and $b$ to each other. Hence, $\sim_P$ is symmetric precisely when such a swap map exists.

For transitivity, consider the pushout $Q := P \bigsqcup_{b, a} P$ of the span $P \overset{b}{\leftarrow} \ast \overset{a}{\rightarrow} P$. There are three induced points $x, y, z : \ast \rightarrow Q$, where $x$ is the first point in the first $P$, $y$ is shared between the two $P$’s and $z$ is the second point in the second $P$. A map $P \rightarrow Q$ sending $a$ to $x$ and $b$ to $z$ will suffice to induce transitivity in $\sim_P$. Conversely, suppose $\sim_P$ is transitive. Then note that $x \sim_P y$ and $y \sim_P z$, implying that $x \sim_P z$. This induces a map $L : \ast \times P \rightarrow Q$ sending $a \mapsto x$ and $b \mapsto z$ as needed.

To summarize, for a complete $P$-homotopy theory, we need two maps $a, b : \ast \rightarrow P$, a swap map $S : P \rightarrow P$ sending $a \mapsto b$ and $b \mapsto a$, and a transitive map $T : P \rightarrow P \bigsqcup_{b, a} P$ sending $a$ to the $a$ in the first $P$ and $b$ to the $b$ in the second $P$. (For the record, if we ever wanted to take this rather abstract categorical structure out of $\textbf{Top}$, we would need the category we use to have a terminal object for the points and for reflexivity, as well as finite pushouts for transitivity.)

It’s pretty obvious what $P$-homotopy equivalences should be. These are maps $f : X \rightarrow Y$ for which there exists some $g : Y \rightarrow X$ such that $f \circ g \sim_P 1_Y$ and $g \circ f \sim_P 1_X$. The induced relation $\cong_P$ on spaces is guaranteed to be an equivalence relation if $\sim_P$ is, though the implication isn’t two-way. What kind of spaces $P$ exist where $\cong_P$ is an equivalence relation but $\sim_P$ isn’t? Such situations seem a bit pathological, so we’ll ignore them here. Go study them yourself if you care.

Perhaps it would be reassuring to show that the $P$-homotopy theory of $\textbf{Top}$ forms a model category structure. Let’s try just that. As a matter of fact, we have two options for a model category structure, akin to using either Hurewicz or Serre fibrations. We’ll start with the former, as the latter needs us to think a bit about some analogy to homotopy groups. (Of course, we absolutely should think about these - it’s half the reason to even consider $P$-homotopy theory!)

The weak equivalences will be $P$-homotopy equivalences and the fibrations will be the $P$-Hurewicz fibrations. These are maps $f : X \rightarrow Y$ where, for any space $Z$, the morphism $Z \rightarrow Z \times P$ sending $Z$ to $Z \times {a}$ has a lifting. The cofibrations are then induced by the left lifting property. This is all rather directly analogous to the standard model structure on $\textbf{Top}$.

We need to make sure this actually does give a model category. For starters, we have that $\textbf{Top}$ has all small limits and colimits, so we’re safe on that front. Next, $P$-homotopy equivalences need to satisfy the 2-out-of-3 property. This takes a bit of algebra.

Consider maps $X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} Z$ with $h := g \circ f$. If we have $P$-homotopies $H : X \times P \rightarrow Y$ from $f$ to $f'$ and and $K : Y \times P \rightarrow Z$ from $g$ to $g'$, then $g \circ H$ is a $P$-homotopy from $g \circ f$ to $g' \circ f$ and $K \circ (f\times 1_P)$ is a $P$-homotopy from $g \circ f$ to $g' \circ f$. We can use these to show $P$-homotopy equivalences are closed under composition. Furthermore, if there are functions $f^* :Y \rightarrow X$ and $h^* : Z \rightarrow X$ such that $f^* \circ f \sim_P 1_X$ and $h^* \circ (g \circ f) \sim_P 1_X$, then $$ (f \circ h^*) \circ g \sim_P f \circ (h^* \circ g \circ f) \circ f^* \sim_P f \circ f^* \sim_P 1_X $$exhibiting $g^* := f \circ h^*$ as a homotopy inverse to $g$, with $g \circ g^* \sim_P 1_Z$ being similar. We can repeat this method when we have $g^*$ and $h^*$ too, so $P$-homotopy equivalences satisfy the 2-out-of-3 condition as required.

When it comes to showing (trivial) cofibrations and (trivial) fibrations form weak factorization systems… I’ve got nothing. It’ll take me some time to figure out how that works in classical homotopy theory, so hopefully I can make some progress and post about it here later. Until then, I remain confident that $P$-homotopy theory always gives a neat model category - it just fits together too well.