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At nerve there is the sentence:
It is in this sense that a simplicial set that is a Kan complex but which does not necessarily have the above pullback property that makes it a nerve of an ordinary groupoid models an ∞-groupoid.
This at present is confusing. A bit of punctuation might help, but as ’pullback property’ does not occur earlier on that page, all is not clear. (In fact a ’pullback property’ is referred to later!) I have changed it to:
It suggests the sense that a Kan complex models an ∞-groupoid. The possible lack of uniqueness of fillers in general gives the ’weakness’ needed, whilst the lack of a coskeletal property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.
I am not 100% happy with that wording however.
Hi,
I am studying this entry, and have a problem at this point:
The collection of all functors between linear orders
$\{ 0 \to 1 \to \cdots \to n \} \to \{ 0 \to 1 \to \cdots \to m \}$is generated from those that map almost all generating morphisms $k \to k+1$ to another generating morphism, except at one position, where they
map a single generating morphism to the composite of two generating morphisms
$\delta^n_i : [n-1] \to [n]$ $\delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))$…
It looks to me that here generating is being applied at the same time to the monotone maps between the linear orderings (functors), and also to the arrows of the linear orderings themselves qua posetal categories (perhaps wrongly).
I think that you are complaining about the poor wording at some places. I find the wording heavy and awkward, but would suggest that you try yourself to improve it. Perhaps breaking that part of the definition up into shorter pieces and moving things around a bit might help. Feel free to do this. If it ends up better …. great, if not, you can always rollback to the previous form.
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