This question is bit related to my previous question about stacks. After understanding the definition of a stack (to be precise $(2,1)$-sheaf), now I am wondering about $\infty$-stacks. According to my understanding, next generalization of this hierarchy should by sheaves valued in $2$-groupoids (a groupoid enriched in groupoids) and etc, then the ultimate generalization should be sheaves valued in $\infty$-groupoids. According to Wikipedia, there are few models (?) for $\infty$-groupoids such as Kan complexes, Globular sets.
Instead of this this direction, most of the literature is about simplicial sheaves/stacks and their model structures. I would like see why this is so and, where the motivation for this approach comes? Specially, how/why model categories appear here?
Fibrant objects in the local projective model structure on simplicial presheaves are precisely presheaves valued in Kan complexes that satisfy the descent (gluing) condition, so I am not sure what “instead” refers to, since Kan complexes are one of the (geometric) models for ∞-groupoids.
Model structures are already present when you work with stacks in groupoids. The reason why they are not mentioned explicitly that often is because all objects are cofibrant and fibrant in the local model structure on Grothendieck fibrations in groupoids. Thus, one has no need to derive the internal hom functor, for example. However, if one were to compute the limit or colimit of a diagram of stacks in groupoids, the correct notion is not the strict (co)limit, but rather the (co)limit with equalities replaced by isomorphisms at appropriate places. This is precisely the homotopy (co)limit that the machinery of model categories computes.
Model categories appear in the world of simplicial presheaves for exactly the same reason they appear in the world of stacks in groupoids modeled via Grothendieck fibrations: there are maps (weak equivalences, or Morita equivalences in the specific case of stacks in groupoids) that one wants to be invertible up to a homotopy, but which are not invertible in any conventional sense. Hence the need to use the machinery of model categories (more generally, relative categories), which is designed to treat precisely this issue.