Let's view the category of algebraic spaces as a full subcategory of the category of "spaces" over the opposite category of commutative rings, that is, the category of sheaves on $CRing^{op}$ in the étale topology. Is there any sort of interesting model structure on this category, or a suitable enlargement of it (perhaps by looking at simplicial sheaves or by changing the topology), capturing the theory of formally smooth morphisms (as fibrations)? As a bonus, is there any way to describe the interesting spaces of this category, say algebraic spaces, as a category of fibrant or cofibrant objects?

If there is some obvious failure that I'm overlooking, is there any way to rescue the idea? Is there any homotopical content in the definition of formally smooth morphisms by a lifting property?