Model categories and simplicial methods

Abstract

Abstract. There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We're going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well.There are now any number of excellent sources for getting into the subject and since this monograph is not intended to be complete, perhaps the reader should have some of these nearby. For example, the paper of Dwyer and Spalinski [16] is a superb and short introduction, and the books of Hovey [22] and Hirschhorn [21] provide much more in-depth analysis. For a focus on simplicial model categories -model categories enriched over simplicial sets in an appropriate way -one can read [20]. Reaching back a bit further, there's no harm in reading the classics, and Quillen's original monograph [30] certainly falls into that category.

Derived Galois deformation rings

“…Since it is not standard in the number-theory literature, let us briefly recall this concept. This is intended informally, and not as a substitute for a rigorous introduction; for that see [12,24]. Also, note that we are only ever interested in simplicial commutative rings, and occasionally will drop the word "commutative.…”

On Koszul–Tate resolutions and Sullivan models

“…Theorem 22. The category DGDA of differential non-negatively graded commutative unital algebras over the ring D = D X (X) of total sections of the sheaf D X of differential operators of a smooth affine variety X, is a finitely ( and thus a cofibrantly ) generated model category ( in the sense of [GS06] and in the sense of [Hov07] ). The weak equivalences are the DGDAmorphisms that induce an isomorphism in homology, the fibrations are the DGDA-morphisms that are surjective in all positive degrees p > 0, and the cofibrations are exactly the retracts of the relative Sullivan D-algebras.…”

Weak quantization of Poisson structures

“…For any simplicial set X • one can define define its realization |X| as a certain colimit in a category of topological spaces (see [10,Definition 1.19] ; simply note that |∆ n | = σ n ). Then one can define the set of path components π 0 (X) := π 0 (|X|) and homotopy groups π i (X, x) := π i (|X|, x) (i > 0) for any x ∈ π 0 (X).…”