# Functor from fundamental groupoid to first homology

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.

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 $$X$$.)