While studying homology in algebraic topology, I sometimes see the notation $H_n(S_*(X))$, and sometime the notation $H_n(X)$. I think these are supposed to be the same, but I'm not sure.
The first notation is mostly used when we consider a chain complex other then $\{S_n(X)\}$. For example if $U$ is an open covering of $X$, and we consider the subgroups $S_n^{U}$ of $S_n$ consisting of simplices that are contained completely in some open set of the covering, we write $$i_*:H_n(S_*^U(X))\to H_n(X)$$ Now I find this slightly confusing, since $H_n$ take a chain complex first, and then a topological space later. So to me the only sensible way to interpret this would be if $H_n(X)=H_n(S_*(X))$ is just a shorthand. Is this correct?
It is confusing, because the notation $H_n$ is being used ambiguously for two different functors:
#1. In the notation $H_n(S_*(X))$, $H_n$ is a functor from chain complexes to sequences of abelian groups.
#2. In the notation $H_n(X)$, $H_n$ is a functor from topological spaces to sequences of abelian groups.
The equation $H_n(X) = H_n(S_*(X))$ means that $H_n$ version #2, when applied to the topological space $X$, gives the same result as $H_n$ version #1, when applied to the chain complex $S_*(X)$.