My lecture notes mentoins the canonical isomorphism between $H_0(X, A)$ and $A$ for $X$ a topological space. When I try to work this out it becomes very messy and I don't see anything canonical.
$S(X)_0 = \{f: \{1\} \to X \}$, which is isomorphic to $X$. So an element of $A[S(X)_0]$ is a formal sum $\sum_{x \in X} a_x x$ where almost all $a_x$ are 0.
$S(X)_1 = \{f: [0, 1] \to X\}$ (or an embedding of $[0, 1]$ into $\mathbb{R}^2$ to be precise), which are all paths in $X$. So $d_0(f) = f(1)$ and $d_1(f) = f(0)$. This means that $\partial_1(f) = f(1) - f(0)$.
So if I understand the material correctly $H_0(X, A)$ is $\{\sum_{x \in X}a_xx\}/\{\sum_{\gamma \textrm{ is a path in }X}a_\gamma (\gamma(1) - \gamma(0))\}$. Where is the canonical isomorphism to $A$?
$H_0(X, A)$ is a direct sum of copies of $A$ each corresponding to a path-connected component of $X$.
If we assume that $X$ is path-connected, then for any two points $x_1, x_2 \in X$, we can find $f \in S_1(X)$ such that $f(0) = x_1$ and $f(1) = x_2$. It follows that all points in $X$ get identified to a single point $x_0$ in $H_0(X, A)$. The desired isomorphism is then $A \to H_0(X, A)$ given by $a \mapsto ax$.