Group actions on posets (Q1772426)

For basic definitions see, for instance \textit{D. Quillen} [Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85--147 (1973; Zbl 0292.18004)]. Let \({\mathcal C}at\) be the category of all small categories, and \(G\) a group considered as a one-object category. The authors say, that \(G\) acts on \({\mathcal K}\in Ob({\mathcal C}at)\), if there exists a functor \(A_{\mathcal K}:G\longrightarrow{\mathcal C}at\) which takes the unique object of \(G\) to a category \({\mathcal K}\). The colimit of the functor \(A_{\mathcal K}\) is called the quotient of \({\mathcal K}\) by the action of \(G\) (denoted by \({\mathcal K}/G\)). The problem considered in the paper is like that: can we establish any good relations between \(\Delta ({\mathcal K}/G)\) and \(\Delta ({\mathcal K})/G\), where \(\Delta : {\mathcal C}at\longrightarrow{\mathcal S}{\mathcal S}\) is the nerve functor, and \({\mathcal SS}\) is the category of simplicial sets. The main result of the paper shows equivalence of bijectivity of the canonical surjection \(\lambda: \Delta({\mathcal K}) /G \longrightarrow \Delta ({\mathcal K}/G)\) on the \(t\)-skeleton under a certain condition \(C_t\) stated by means of equalities of certain terms in morphisms of the category \({\mathcal K}\). The conditions under which a passage to the quotient commutes with the nerve functor are also established. Some illustrative examples and applications of the results are given.