I'm trying to understand singular complexes via Hatcher page 108. Here is my understanding so far and where I'm not getting it.
Given a space $X$, a singular n-simplex is just a $\sigma \in C^1(\Delta^n, X)$ and the boundary maps are linear maps based on (writing $\Delta^n = [v_0, ..., v_n]$):
$$\partial_n(\sigma) = \sum_{j=0}^{n}{(-1)^j}\sigma|[v_0, ...,\hat{v_j}, ... v_n]$$
So, now, the definition of a singular complex, $S(X)$ is:
...the $\Delta$-complex with one $n$-simplex $\Delta_{\sigma}^{n}$ for each singular $n$-simplex $\sigma:\Delta^n\to X$, with $\Delta_{\sigma}^{n}$ attached in the obvious way to the $(n-1)$-simplicies in $\partial \Delta^{n}$.
According to this post, $S(X)$ is a $\Delta$-complex, but not for the original space $X$. So, I'm trying to figure out what this is. Here is what I've got so far:
- We have $C^1(\Delta^0, X)$ which is basically just one constant function for each $x \in X$.
- We have $C^1(\Delta^1, X)$ which, essentially, is continuous maps from the unit interval to $X$. Also, for each $\sigma \in C^1(\Delta^1, X)$, $\partial(\sigma) = \lambda_1-\lambda_0$ where $\lambda_1, \lambda_0 \in C^1(\Delta^0, X)$ are $\lambda_1(x) = \sigma(1)$ and $\lambda_0(x) = \sigma(0)$
- We have $C^1(\Delta^2, X)$ which are continuous maps from a triangle with unit side lengths to $X$. Also, for each $\sigma \in C^1(\Delta^2, X)$, $\partial(\sigma)$ is the alternating sum of three elements of $C^1(\Delta^1, X)$, each of which corresponds to where $\sigma$ sends one side of the triangle.
Question 1: is this right so far?
If so, then I'm confused by this part:
One can regard $S(X)$ as a $\Delta$-complex model for $X$ ...
Question 2: is this the same as a $\Delta$-complex structure for $X$? It seems not, because this would violate the condition that every $x \in X$ is in the interior of exactly one of the maps of each dimension. For example, if $X = [0, 1]$, we have $\sigma_1(x) = x/2$ and $\sigma_2(x) = 1/4 + x/2$ and, so, $1/3$ is in the interior of both of these.
Question 3: so, if it isn't a $\Delta$-complex structure for $X$, what Hatcher mean by "$\Delta$-complex model for $X$"?