# Homologous quotient of fundamental groupoid

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,

1. 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 $$\Pi_1'$$?
2. 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?

• Thank you! Let's say $X$ is connected. Could you explain the notation $B(H_1(X))$ or mention a reference? I am not an expert about this. – Hang Aug 25 at 22:53
• For a group $G$, here $BG$ means the category with one object and with the hom given by $G$. – Todd Trimble Aug 25 at 22:54