I'm reading section 2.7 (Fundamental Group of the Circle) of the book Algebraic Topology by Tom Dieck. The section mentions the term "topological groupoid" but I cannot find the definition in previous sections.

I tried to think of it as a similar term to "topological group" but I'm still having trouble reading the proof of $\pi_1(S^1,*)\cong \mathbb Z$ without an exact definition and a motivation.

Could anyone please give some clue on how one motivate the definition and why the book uses the term without a definition? (I personally do not think it is an elementary definition that can be omitted.)

Thank you.

  • $\begingroup$ For the definition and a motivation (in terms of the category $\mathcal{Top}$ ) see here. $\endgroup$ – Dietrich Burde Mar 19 at 16:21
  • $\begingroup$ Does the book define groupoid previously in the text? There are two essentially equivalent definitions of groupoid (one category-theoretic, one algebraic) and which one affects how you would normally define the topological groupoid. $\endgroup$ – Thomas Andrews Mar 19 at 16:25
  • $\begingroup$ Thank you for the comment. I looked up for the definition and have read the nLab definition previously, but I'm still having trouble understanding what the source and target map is(why define it from the "morphism space" to "object space") and why it is required to be continuous. Yes it defines groupoid in the previous section, but it only contains the definition of "fundamental groupoid" and the categorical definition of "groupoid". $\endgroup$ – William Sun Mar 19 at 16:26
  • $\begingroup$ It should be helpful to consider the discussion at mathoverflow.net/questions/40945/…. I do not see why topological groupoids have to come in to determine a fundamental group of the circle, or of many other unions of disconnected spaces. $\endgroup$ – Ronnie Brown Mar 19 at 21:29
  • $\begingroup$ I should add that groupoids with various kind of structure, e.g. topological.smooth, measure theoretic, algebraic, ..., are important in other aspects than fundamental groups. $\endgroup$ – Ronnie Brown Mar 19 at 21:40

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