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I am trying to remove the erroneous shifts in degree by $\pm 1$ that inevitably I have been making at simplicial skeleton and maybe at truncated.
So a Kan complex is the nerve of an $n$-groupoid iff it is $(n+1)$-coskeletal, I hope ;-)
At truncated in the examples-section i want to be claiming that the truncation adjunction in a general (oo,1)-topos is in the case of $\infty$Grpd the $(tr_{n+1} \dashv cosk_{n+1})$-adjunction on Kan complexes. But I should be saying this better.
I typed out now what I think is a complete proof, see Truncation in ooGrpd.
added some propositions and proofs on properties of truncation to Truncation - Properties - General
added more details on the proof of the recursive definition of k-truncated morphisms here.
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