Homotopy theory of \(G\)-diagrams and equivariant excision (Q258846)

Given a finite group \(G\) acting on a category \(I\) by functors \(a(g):I\rightarrow I\), a \textit{\(G\)-diagram} in a category \(\mathcal C\) is a functor \(X : I \rightarrow \mathcal C\) together with natural transformations \(g_X : X \rightarrow X \circ a(g)\) for every \(g\) in \(G\) which are compatible with the group structure, that is, (1) \(e_X = 1_X\), and (2) \((g_X)_{a(h)} \circ h_X = (gh)_X\) for all \(g,h \in G\), where \((g_X)_{a(h)}\) is the natural transformation obtained by restricting \(g_X\) along the functor \(a(h) : I \rightarrow I\). A \textit{map of \(G\)-diagrams} \(f : X \rightarrow Y\) is a natural transformation between the underlying diagrams such that the following diagram NEWLINE\[CARRIAGE_RETURNNEWLINE \begin{tikzcd} X \ar[r, "f"] \ar[d, "g_X" '] & Y \ar[d, "g_Y"] \\ X\circ g \ar[r, "f_g" '] & Y\circ g\end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE is commutative for each \(g \in G\) in the usual category \(\mathcal{C}^I\)of functors from \(I\) to \(\mathcal C\). For a fixed action \(a\) of the group \(G\) on \(I\), \(\mathcal C^I_a\) denotes the category of \(G\)-diagrams and maps of \(G\)-diagrams. A \textit{\(G\)-model category} is a cofibrantly generated simplicial model category \(\mathcal C\) together with the data of a cofibrantly generated model structure on \(\mathcal C^H\) satisfying two conditions, where \(H\) is any subgroup of \(G\). For a finite \(G\)-set \(J\), a \(J\)-cube \(X\) in \(\mathcal C\) is a \(G\)-diagram in \(\mathcal C\) shaped over the poset category \(\mathcal P(J)\) of subsets of \(J\) ordered by inclusion. The \(J\)-cube \(X\) is said to be \textit{homotopy Cartesian} if the canonical map NEWLINE\[CARRIAGE_RETURNNEWLINEX_{\emptyset}\rightarrow\mathrm{holim}_{\mathcal P(J)\setminus \emptyset} ~X CARRIAGE_RETURNNEWLINE\]NEWLINE is a weak equivalence in the model category of \(G\)-objects \(\mathcal C^G\). Dually, it is said to be \textit{homotopy co-Cartesian} if the canonical map NEWLINE\[CARRIAGE_RETURNNEWLINE\mathrm{hocolim}_{\mathcal P(J)\setminus J} ~X \rightarrow X_J CARRIAGE_RETURNNEWLINE\]NEWLINE is an equivalence in \(\mathcal C^G\). A suitably homotopy invariant functor \(\Phi : \mathcal C^G \rightarrow \mathcal D^G\) is called \textit{\(G\)-excisive} if it sends homotopy co-Cartesian \(G_+\)-cubes to homotopy Cartesian \(G_+\)-cubes, where \(G_+\) is the set \(G\) with an added disjoint basepoint on which \(G\) acts by left multiplication.NEWLINENEWLINEThe authors prove the fundamental properties of the equivariant homotopy limits and colimits such as a homotopy cofinality theorem for homotopy limits and colimits of \(G\)-diagrams as a generalization of the results of \textit{J. Thévenaz} and \textit{P. J. Webb} [J. Comb. Theory, Ser. A 56, No. 2, 173--181 (1991; Zbl 0752.05059)], and an Elmendorf theorem showing that for suitable ambient categories one can define the homotopy theory of \(G\)-diagrams by replacing \(G\) with the opposite of its orbit category as a digrammatic analogue of the classical result of \textit{A. D. Elmendorf} [Trans. Am. Math. Soc. 277, 275--284 (1983; Zbl 0521.57027)]. As an application of the model categorical theory of \(G\)-diagrams, the authors study a series of fundamental properties of equivariant excision which appropriately reflect the fundamental properties of excision to a equivariant context. They show that the identity functor on \(G\)-spectra is \(G\)-excisive, and that any \(G\)-excisive reduced homotopy functor \(\Phi : \mathcal C^G \rightarrow \mathcal D^G\) satisfies the Wirthmüller isomorphism theorem, that is, the canonical map \(\Phi(G \otimes_H c) \rightarrow \hom_H (G, \Phi(c))\) is an equivalence in \(\mathcal D^G\) for every subgroup \(H\) of \(G\) and \(H\)-object \(c\) of \(\mathcal C^H\). In this paper, there are too many beautiful results to be quoted here about \(G\)-model categories and equivariant excisions.