\(n\)-groupoids from \(n\)-truncated simplicial objects as a solution to a universal problem (Q1588063)

The author develops a comparison theory relating split \(n\)-truncated simplicial objects in a category \({\mathcal B}\) with the normalized \(n\)-groupoid objects in \({\mathcal B}\). \smallskip Homotopy theorists have to be careful about the meaning of the word ``split'' in this paper: it does not mean split up to a certain level in the usual sense. It involves the existence of an extra degeneracy, but at the wrong end of the list of degeneracies, so that for a graph \(X\) a splitting has the effect of determining a canonical choice of path from a vertex \(x\) to a canonical choice of representative for \(x\) in \(\pi_{0}X\). The term ``normalized'' for groupoids carries essentially the same meaning. \smallskip An \(n\)-groupoid is the standard thing, namely a category enriched in \((n-1)\)-groupoids, but defined internally. The main result of the paper is a recognition principle for nerves of \(n\)-groupoids which is also internal, and avoids reference to pasting schemes: it is shown (theorem 5.4) that the category of \(n\)-groupoids in \({\mathcal B}\) is equivalent to a full subcategory of a suitably defined category of \(\theta\)-complexes in the category of simplicial objects in \({\mathcal B}\). The central proof technique is an iterated comparison of algebras for certain monads.