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I was looking for a reference to back up something that Pilar Carrasco and Antonio Cegarra say in their old paper on hypercrossed complexes. (I could not find it on the nLab.) They say that whilst the counit of the adjunction between n-truncated simplicial sets and simplicial sets (that gives the n-skeleton functor on sSets) is a monomorphism, this was not the case if one replaces simplicial sets by simplicial groups. There is probably an obvious example of this but I have not thought one out and wonder if there is a neat one somewhere (in the Lab if possible). The image of that counit is the simplicial subgroup generated by the n-simplices. We do not seem to have a discussion of truncation and skeleta / coskeleta that explicitly mentions cases other than sSets as far as I was able to find. Is there one that I have missed? Perhaps in something by Duskin?
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