# Model category with formally smooth morphisms as fibrations?

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?

## 1 Answer

This addresses just the last question "Is there any homotopical content...". It would belong into a comment but doesn't fit.

Mathieu Anel shows in a very recommendable article how two classes of maps in the opposite of a locally presentable category, one having the (left/right) lifting property w.r.t. the other, yield factorization systems - every map of the category factors as a map of the left class followed by a map of the right class. To a factorization system he associates a Grothendieck topology (you can just read the two pages addressing this, the article is very readable). Now a category of presheaves over a site is a model category with weak equivalences those morphisms which become isomorphisms after applying sheafification, cofibrations the monomorphisms and fibrations those with the lifting property. This already gives a homotopical content to a lifting system, but yet not as suggested in your question.

Now if the topology on a site arose via a lifting system, I would guess that the fibrations of the model structure on presheaves can be described in terms of maps from the right class (I am thinking of something like: a map of sheaves is a fibration if every pullback of it into the realm of affines gives a map from the right class). Or maybe one can find a new model structure on presheaves with the same weak equivalences involving both the left and the right class from the site. I guess this is known to someone, but not to me...

• The "map of sheaves is a fibration if every pullback of it into the realm of affines gives a map from the right class" sounds very much like the way one defines a formally smooth map of sheaves, incidentally! – Harry Gindi Sep 19 '10 at 18:58