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added to groupoid a section on the description in terms of 2-coskeletal Kan complexes.
Chenchang Zhu kindly writes in to say that she is giving a course on (higher) groupoids and is planning to use relevant nLab pages as course material, and hence panning to edit them further, as need be.
Right now she has added to groupoid the explicit definition.
I have added statement and proof of the relation between equivalence of groupoids and weak homotopy equivalence, in a new section Properties – Equivalences of groupoids.
I have added to groupoid a section “Categories of groupoids” (here) which spells out horizontal composition of homotopies/natural transformations.
Then I used this to spell out the proof that, assuming AC, groupoid representations are euqivalently tuples of group representations (here).
This I also copied over to the entry groupoid representation.
I’ve removed the prime off the lower $F_2$ there. Is that what you meant?
@David_Corfield yes, and also the same diagram is now going $\mathcal{G}_1 \rightarrow \mathcal{G}_2$ but it think it should be $\mathcal{G}_1 \rightarrow \mathcal{G}_4$. I suppose this is just a typo but, since i wasn’t completely sure, i did not correct it myself
giorgio s
@David_Corfield regarding the same lemma (here), i don’t understand the line $(F_2 \cdot \eta \cdot F_1)(x) \;\coloneqq\; F_2(\eta(F_1(x))) \,.$
$\eta$ is said to be a homotopy. Homotopies between groupoids are defined some paragraphs before this lemma, and they can act only on the codomain of the functors they transform, therefore $\eta$ should act on $\mathcal{G}_3$. But writing $\eta(F_1(x))$ means it is acting on $\mathcal{G}_2$
@Marc, i am sorry if i am missing something but your correction (that was also necessary) still doesn’t change the fact that $\eta$ is acting on $F_1(x)\in\mathcal{G}_2$ while i think that $\eta$ can only act on elements of $\mathcal{G}_3$
giorgio s
@giorgio: o.k. let’s dissect the statement of Lemma 2.7 step by step:
(1) $F_2$ and $F^{'}_2$ are functors from $\mathcal{G}_2$ to $\mathcal{G}_3$.
(2) So the natural map $\eta : F^{'}_2 \to F_2$ should assign to every object $y \in \mathcal{G}_2$ a morphism $\eta(y) \in \mathcal{G}_3(F^{'}_2(y),F_2(y))$.
(3) Precomposing with $F_1$ gives for every object $x \in \mathcal{G}_1$ first the object $F(x) \in \mathcal{G}_2$ and then (with $y = F_1(x)$ in (2)) a morphism $\eta(F_1(x)) \in \mathcal{G}_3(F^{'}_2(F_1(x)),F_2(F_1(x)))$.
(4) Applying $F_3$ to $\eta(F_1(x))$ from (3) then gives a morphism $F_3(\eta(F_1(x))) \in \mathcal{G}_4(F_3(F^{'}_2(F_1(x))),F_3(F_2(F_1(x))))$.
So I think indices are the correct ones.
@Marc you are right, thank you for the clarification (i wasn’t thinking of $\eta(y)$ as a morphism)
giorgio s
changed all “delooping groupoid” in the page to “delooping groupoid” and will give this its own little page now.
added pointer to:
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