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I have a question about a definition used in nLab article on $n$-groupoids: https://ncatlab.org/nlab/show/n-groupoid

What does it mean that "every parallel pair of $j$-morphisms is equivalent for $j>n$? Surely, that's not a research question, but I nowhere found an answer. Does that mean that the corresponending equilizers on $j$-level are equivalences? (just a guess of mine)

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It means that between every two $j$-cells with the same domain and codomain ($j > n$), there is a $(j + 1)$-cell between them (which is necessarily an equivalence by the first condition of the definition on that page) exhibiting them as equivalent.

(Meta note: math.stackexchange is best suited for non-research-level mathematics questions.)

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  • $\begingroup$ Addition to the meta note: also the category theory zulip community might be a good place to ask. $\endgroup$
    – Aurelio
    Commented Jul 16, 2020 at 12:18
  • $\begingroup$ Further to @varkor's answer, I would recommend playing around with a homotopy-oriented proof assistant, for example Globular: it is an excellent way to acquire a hands-on intuition of "being equivalent up to higher cells". You can find a tutorial video for Globular by Jamie Vicary. $\endgroup$
    – Aurelio
    Commented Jul 16, 2020 at 19:59

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