Let $X$ be a connected space and $\Pi_1(X)$ be its fundamental groupoid. We consider the homologous relation $\mathcal R$ on every morphism space: $f,g\in \Pi_1(X)(p,q)$ are related if the singular one-chain $g-f= \partial S$ for a two-chain $S$.

It seems this is a congruence relation: if we further have $f'-g'=\partial S'$ for $f',g'\in \Pi_1(X)(q,r)$ then $f'\circ f-g'\circ g= \partial(S+S')$. So we can consider the quotient category $$ \Pi'_1(X):=\Pi_1(X)/\mathcal R $$ Is this stuff well-studied and is there a good reference? For example,

- It is known that the fundamental groupoid can be viewed as a functor $\Pi_1: \mathcal{Top}\to \mathcal{Grpd}$ from the category of topological spaces to the category of groupoids. Does this property also holds for our quotient $\Pi_1'$?
- The fundamental group $\pi_1(X,p)$ is the object group of $\Pi_1(X)$ at $p\in X$; in analogy, it seems $H_1(X)$ is simply the object group at any point $p\in X$ of our new $\Pi'_1(X)$. Is this correct?