Tower techniques for cosimplicial resolutions (Q1762699)

By combining Quillen model structures and simplicial structures, the authors define a class of functors called pseudo-simplicial functors, and for a pseudo-simplicial functor \(\Phi\) they define a notion of weak \(\Phi\)-equivalence. In particular, let \({\mathcal M}\) be a simplicial model category; then the forgetful functor \(\text{ pro}\) from towers to pro-objects is a pseudo-simplicial functor. In this case, if \(X\to R\) is a trivial cofacial resolution of fibrant objects in \({\mathcal M}\), then the authors obtain a pro-weak equivalence between the constant tower \(X\) and the tower \(\text{ tot}.(R)\) associated to \(R\). A cofacial resolution of an object \(X\) can be obtained from a simplicial augmented functor \(J\), and the associated tower is denoted by \(J.(X)\) The resolution is trivial, and therefore gives a pro-weak equivalence, provided that \(X\) is \(J\)-injective. In this connection let \(F\) be a functor on a category with finite-dimensional nerve whose values are fibrant \(J\)-injective objects in \({\mathcal M}\); the authors show that the homotopy limit of \(F\) is \(J_s\)-injective for some finite \(s\).