[For part I, cf. T. Beke, Math. Proc. Camb. Philos. Soc. 129, No. 3, 447-475 (2000; Zbl 0964.55018)].
Suppose one has a category \({\mathcal C}\) and a functor \(F\) from \({\mathcal C}\) to \({\mathcal S}\)ets. Then there is an induced functor from the category \(s{\mathcal C}\) of simplicial objects in \({\mathcal C}\) to simplicial sets and, in his original book on closed model categories [“Homotopical algebra”, Lect. Notes Math. 43, Springer-Verlag, Berlin (1967; Zbl 0168.20903)], D. G. Quillen wrote down some hypotheses under which \(sC\) has a closed model category with weak equivalences and fibrations created by \(F\). This question can easily be generalized to sheaves: does the induced functor from simplicial sheaves in \(G\) to simplicial sheaves create a model category structure in the same way? The difficulty is that Quillen’s hypotheses will never apply to sheaves in some satisfactory way: Quillen required – among other things – that \({\mathcal C}\) have projective generators in a suitable sense. This, of course, does not apply in a topos setting. The purpose of this paper is to find a substitute for Quillen’s hypotheses that do work well for sheaves. The result is a theorem similar in spirit to Quillen’s result and, more, behaves well with respect to topos morphisms. Also, using the language of definable functors it is possible to generalize the base category away from simplicial sets. Nonetheless, it is worth pointing out that the hard part remains the same: at some point one has to verify that push-outs along acyclic cofibrations remain weak equivalences. This is even difficult for simplicial sets: as the author points out there is no known proof of this fact for simplicial sets which is either internal to simplicial sets – that is, does not use geometric realization – or avoids the axiom of choice.