Clarification regarding the functor $\Pi|\mathcal{O}:\mathcal{O}\to \mathcal{GP}$ in May's "Algebraic Topology"

Question

This question has to do with the following functor $$\Pi|\mathcal{O}:\mathcal{O}\to \mathcal{GP}$$ on pg 17 of Peter May's book on Algebraic Topology. $\mathcal{O}$ is the category of subsets of a topological space $X$, where morphisms are inclusions. $GP$ is the category of groupoids. What are the objects of $\mathcal{O}$ being mapped to? The objects of $\mathcal{O}$ are subsets and the objects of $\mathcal{GP}$ are points!

Answer 1

If $\mathcal{GP}$ is the category of groupoids, then the objects of $\mathcal{GP}$ are groupoids, not points. I think you may be confusing $\mathcal{GP}$ with the fundamental groupoid $\Pi(X)$ of a space, whose objects are points. In any case, the functor $$\Pi|\mathcal{O}:\mathcal{O}\to\mathcal{GP}$$ sends an open subset $U\in\mathcal{O}$ to its fundamental groupoid $\Pi(U)$, and an inclusion $i:V\hookrightarrow{U}$ to the induced map $\Pi(i):\Pi(V)\to\Pi(U)$. It is just as Peter May says; $\Pi|\mathcal{O}$ is the restriction of the fundamental groupoid functor to the category $\mathcal{O}$.