Categorical aspects of equivariant homotopy (Q1923798)

Let \({\mathcal A}\) and \({\mathcal C}\) be simplicially enriched categories and \(T:{\mathcal A}^{op} \times{\mathcal A}\to{\mathcal C}\) a simplicially enriched functor. The authors define a homotopy coherent end \(\oint_A T(A,A)\) and dually a homotopy coherent coend \(\oint^A T(A,A)\). In particular, if \({\mathcal A}\) is the orbit category \({\mathcal O} (G)\) of a discrete group \(G\) and \(T\) an \({\mathcal O} (G)\)-diagram of simplicial sets then Elmendorf's coalescence \(c\) [\textit{A. D. Elmendorf}, Trans. Am. Math. Soc. 277, 275-284 (1983; Zbl 0521.57027)] is obtained as the left homotopy Kan extension \[ \oint^{G/H} T(G/H) \overline \otimes G/H. \] At the end, an excellent enriched version of Elmendorf's results is presented.