Functor from fundamental groupoid to first homology

Question

Let $X$ be a topological space, and $x_0 \in X$. There is a natural group homomorphism $\pi_1(X, x_0) \to H_1(X)$ from the fundamental group of $X$ with basepoint $x_0$ to the homology group $H_1(X)$, given by mapping a loop based at $x_0$ to its homology class.

I would like to know if it is possible to extend this to a functor $\Pi_1(X) \to H_1(X)$ from the fundamental groupoid to $H_1(X)$, viewed as a category on one object.

Is there maybe a nice reference where I could read about this?

Answer 1

Yes. We can pretty much just imitate the construction of singular homology, which explicitly comes down to the following: given a groupoid $X$, write $C_0$ for the free abelian group on the objects and $C_1$ for the free abelian group on the morphisms. There is a boundary map $d : C_1 \to C_0$ sending a morphism $f$ to $t(f) - s(f)$ where $t$ is the target and $s$ is the source. Then $H_1(X)$ is the quotient of $\ker(d)$ by the equivalence relation $fg \sim f + g$. (This can be defined in terms of a boundary map from a suitable $C_2$ to $C_1$ also. Really we are working with the free abelian group on the nerve of $X$.)