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I did a bit of reorganization of this page, which had a bunch of different subjects all stuck together under "Remarks". It was also inconsistent about its use of and , so I tried to remedy that. I think the stuff about topologizing the fundamental groupoid is kind of confusing right now, but I don't know exactly what it's trying to say so I can't fix it.
I have added to the References-section at fundamental groupoid pointers to
The fundamental groupoid as a terminal costack (arXiv:1406.4419)
The Étale Fundamental Groupoid as a Terminal Costack (arXiv:1412.5473)
This observes that for good topological spaces and for Noetherian schemes the fundamental groupoid assignment $\Pi_1(-)$ is characterized as being 2-terminal among all co-stacks.
Cross-linked with simplicial fundamental groupoid.
moving this ancient query box out of the entry:
+– {: .query} Mike Shulman: Could you say something about what topology you have in mind here? Is the space of objects just $X$ with its original topology?
David Roberts: The short answer is that it is propositions 4.17 and 4.18 in my thesis, but I will put it here soon.
Regarding topology on the fundamental groupoid for a general space; it inherits a topology from the path space $X^I$, but there is also a topology (unless I’ve missed some subtlety) as given in 4.17 mentioned above, but the extant literature on the topological fundamental group uses the first one.
Ronnie Brown: See the account in “Topology and Groupoids” referred to below. But there is also an account using path spaces in Proposition 6.2 of Mackenzie’s 1987 book “Lie groupoids and Lie algebroids in differential geometry”.
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