E2 model structures for presheaf categories

The purpose of this paper is to develop analogs of the E2 model structures of Dwyer, Kan and Stover for categories related to pointed bisimplicial presheaves and simplicial presheaves of spectra. The development is by analogy with and builds on the work of Dwyer, Kan and Stover [1], [2], along with later work of Goerss and Hopkins [3]. The technical challenge met in the present paper is that not all objects in the categories under consideration are fibrant. This is overcome with the introduction of a bounded cofibration approximation technique which builds on an approach to constructing localizations that appears in [4]. The results proven here are completely combinatorial and apply, in particular, to pointed bisimplicial presheaves, as well as simplicial spectra, simplicial symmetric spectra and their motivic analogs. The first section of this paper gives a list of general results which hold for simplicial objects in a proper closed simplicial model category M. The basic notions of A-fibration and A-equivalence are given there, where A is a small diagram consisting of homotopy cogroup objects Ai of M. Generally a map g : X → Y of simplicial objects of M is defined to be an A-equivalence if and only if the induced map of simplicial groups