Can singular 2-simplices be identified with homotopies?

Question

In what follows, let $\sigma_p$ be the standard p-simplex in $\mathbb{R}^{p+1}$. This question is inspired by the following quote from Vick's Homology Theory, p.3:

Let $X$ be a topological space. A singular p-simplex in $X$ is a continuous function $$\phi: \sigma_p \to X.$$ Note that singular 0-simplices may be identified with the points of $X$, the singular 1-simplices with the paths in $X$, and so forth [emphasis mine].

This immediately made me wonder if singular 2-simplices may be identified with the homotopies of $X$. Now obviously the standard 2-simplex is homeomorphic to $[0,1]\times[0,1]$, so there exists a continuous map $\psi: [0,1]\times[0,1]\to\sigma_2$, and thus by composition, for any singular 2-simplex $\phi:\sigma_2\to X$, we automatically have a continuous map $\phi\circ\psi:[0,1]\times[0,1]\to X$.

However, in order for this to be a homotopy, we need additionally that the maps $\phi\circ\psi(0,-):[0,1]\to X$ and $\phi\circ\psi(1,-):[0,1]\to X$ be constants (at least according to the definition of homotopy given in Hatcher's Algebraic Topology).

Question: does there exist such a continuous map $\psi:[0,1]\times[0,1]\to \sigma_2$ which satisfies these two additional conditions?

EDIT: This page claims that we can replace continuous maps from the standard $k-$simplex with continuous maps from the closed $k-$ball, although it makes the mistake of saying that $\phi$ has to be a homeomorphism (it just has to be continuous), and it doesn't mention $[0,1]^k$ ($k-$cubes) at all, which would be necessary to define a $k-$homotopy.

This page also states the claim that singular 1-simplices can be identified with paths, and it says that

We could generalize the fundamental group by taking homotopy classes of singular $n-$simplices in $X$, and making an appropriate definition of the 'product' of two singular simplicies. This can indeed be done, but the resulting groups are the homotopy groups $\pi_n(X)$.

This would seem to corroborate my suspicion that singular $2-$simplices are either related or equivalent to the set of all path homotopies in $X$, but I am not really sure.

This webpage claims that singular 0-simplices can be identified with points, and that singular 1-simplices can be identified with paths, but it says "Fill this in later" for singular 2-simplices.

This webpage states that singular $n-$simplices generalize paths in $X$, but it doesn't clarify how, specifically whether this generalization is the same as homotopies. Although it does say that "this construction... is one of the classical approaches to determining invariants of the homotopy type of the space". And it says that the singular simplicial complex of $X$ is its nerve.

This document gives a nice intuition of how in other contexts homotopies are a natural generalization of points and paths to higher dimensions.

Answer 1

(Speaking as someone who hasn't really had time to digest all of this stuff)

Let $I$ be the interval and assume that we are working in a cartesian closed subcategory of Top. (this assumption can be removed if needed, by taking these arguments as merely inspiration and only using the function-space-free statements they inspire)

The intuition about a homotopy is to view them not as maps $I \times X \to Y$, but as maps $I \to Y^X$: a homotopy between two functions is just a path in the corresponding function space.

To any space $X$, the homotopies in $X$ are the maps $I \to X$, and we can collect them into a space $X^I$. For any particular points $a,b \in X$, we can also consider paths between them:

$$ \text{Path}(a,b) = \{ f \in X^I | f(0) = a \wedge f(1) = b \} $$

Now, if we are to speak of a homotopy of homotopies, we might thus consider maps $I \to X^I$. These are (I think) called "free" homotopies. Of course, these correspond to maps $I^2 \to X$.

But to any points $a,b \in X$, we may also consider maps $I \to \text{Path(a,b)}$ — this is what is usually meant when one speaks of a "homotopy between paths". These correspond precisely to those maps $I^2 \to X$ whose left and right sides are the constant functions to $a$ and $b$.

Since we insist on the two sides being constant functions, we might as well work in the quotient space that collapses each side to a point. The end result is still homeomorphic to $I^2$, but the result is maybe easier to express by constructing a homeomorphism to $D^2$ (the unit disc) instead.

There are lots of spaces homeomorphic to $D^2$ that we could use. Each gives different flavor.

The unit disc $D^2$ emphasizes the nature of a homotopy as being from one path to another; more generally the $n$-globe (i.e. unit $n$-ball) $D^n$ can be viewed as a generalized path from its upper and lower hemispheres, which are copies of $D^{n-1}$.

The simplices $\Delta^n$ better emphasize composition; e.g. $\Delta^2$ can be viewed as relating a composite of two paths to a third path. $\Delta^3$ gives a sort of ternary composition of a triangle being subdivided by adding a new point in its middle. These are historically important as their combinatorial theory (simplicial sets) was worked out first, and is maybe simpler to calculuate with?

The cubes $I^n$ arise naturally from the sort of reasoning I give above, and allow for the obvious binary compositions along all of their axes. As for regarding $I^2$ as a path between paths, I think the better picture is not to think of it as going from the South edge to the North edge, but as a path between composites $\text{South}\cdot\text{East} \to \text{West}\cdot\text{North}$.

Other shapes could be used, of course; but these shapes are the most popular and studied the most, and the corresponding combinatorial versions have names: "globular sets", simplicial sets", and "cubical sets".