Linked Questions
12 questions linked to/from Compelling evidence that two basepoints are better than one
157
votes
48
answers
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Generalizing a problem to make it easier
One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you ...
63
votes
22
answers
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What's a groupoid? What's a good example of a groupoid? [closed]
Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?
23
votes
5
answers
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Has any attempt been made to classify finite groupoids?
I recently stumbled upon the Mathieu groupoid and I found them fascinating.
It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
- 2,040
16
votes
6
answers
6k
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Fundamental group of the line with the double origin.
In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole".
This ...
- 3,699
27
votes
3
answers
7k
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Why are we interested in the Fundamental Groupoid of a Space?
The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$.
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- 609
21
votes
6
answers
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Is there a good notion of morphism between orbifolds?
Following Thurston, an orbifold is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable ...
- 10.5k
5
votes
2
answers
1k
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What was Seifert's contribution to the Seifert-van Kampen theorem?
The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van ...
- 1,488
7
votes
3
answers
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What is the geometric realization of the the nerve of a fundamental groupoid of a space?
It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows:
Obj: $X \mapsto \pi_{\leq 1}(X)$, ...
- 1,215
5
votes
2
answers
581
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Can we define fundamental groups functorially for non-pointed path connected topological spaces?
Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{...
- 9,019
11
votes
1
answer
1k
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Descent theorems for fundamental groups and groupoids?
Grothendieck in his 1984 "Esquisse d'un programme" (Section 2) wrote (English translation):
" ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base ...
- 12.3k
6
votes
1
answer
406
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Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
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2
votes
1
answer
384
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Dimension leaking in homology as opposed to homotopy
In homotopy theory we have the Seifert van-Kampen theorem, which is a clean statement about the fundamental groupoid of a pushout in $\mathsf{Top}$. There is also a 2d version of SvK in R Brown's ...
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