I am trying to understand why the usual definition of chain groups goes
- Define $n$-chains as maps from $n$-simplices to $\mathbb{Z}$ that vanish cofinitely
- Prove $C_n$ is free Abelian
Why not start with definition of $C_n$ as free Abelian? I saw this in a bunch of different lecture notes on the internet, but textbooks seem to prefer the definition in terms of maps. I would appreciate if someone could point me to where this simplification fails or what is the "profit" gained in intuition or otherwise? Thanks!
You could do so.
An Abelian group $F$ is called free if it has a basis. This is a subset $B \subset F$ such that each $x \in F$ has a unique representation as finite sum $x = \sum_{i=1}^n n_i b_i$ with $n_i \in \mathbb Z$ and $b_i \in B$. Frequently one wants to start with a set $X$ and find a free Abelian group $F$ having $X$ as a basis. This leads to the following definition:
Given a set $X$, an Abelian group $F(X)$ together with a function $\iota : X \to F(X)$ is called freely generated by $X$ if for every Abelian group $G$ and every function $f : X \to G$ there exists a unique group homomorphism $\bar f : F(X)\to G$ such that $f = \bar f \circ \iota$.
This definition via a universal property determines the pair $(F(X),\iota)$ uniquely up to canonical isomorphism. In fact, if $\iota' : X \to F'(X)$ has the same property, then we get unique group homomorphims $\bar \iota' : F(X) \to F'(X)$ such that $\iota' = \bar \iota' \circ \iota$ and $\bar \iota' : F(X) \to F'(X)$ such that $\iota = \bar \iota \circ \iota'$. Hence $\bar \iota \circ \bar \iota' \circ \iota = \bar \iota \circ \iota' = \iota = id_{F(X)} \circ \iota$ which implies $\bar \iota \circ \bar \iota' = id_{F(X)}$ via uniqueness. Similarly $\bar \iota' \circ \bar \iota = id_{F'(X)}$.
It remains to prove the existence of such a pair $(F(X),\iota)$. The standard construction is $F(X) = \{ \phi : X \to \mathbb Z \mid \phi(x) \ne 0 \text{ only for finitely many } x \in X \}$ and $\iota(x')(x) = 1$ for $x' = x$, $\iota(x')(x) = 0$ for $x' \ne x$. We do not go into details here, you can consult a textbook. Note that this $F(X)$ is easily seen to be a free Abelian group with basis $\iota(X)$ which justifies the notation freely generated by $X$.
Now let us come to your question. Defining $C_n$ as the free Abelian group generated by the $n$-simplices is not really precise because there are many models for such an object. For example, if you have only one $0$-simplex, you could simply take $C_0 = \mathbb Z$. But you want to obtain a functor $C_n$ and therefore it is not expedient to say "pick any free Abelian group generated by the $n$-simplices" - doing so would involve a variant of the axiom of choice for proper classes. Concerning classes have a look at Can we say," The set of all compact metric spaces"?
This problem can easily be avoided by taking a concrete construction of free Abelian groups, for example that as maps as above. We do not need any choice here.