The next step (in our discussion, started yesterday, of the cotangent complex) will be to define a model structure on the category of algebras over a fixed ring. Model structures allow one to define derived functors in a non-abelian setting. The key idea is that, when you want to derive an additive functor on an object in some abelian category, you replace by a projective resolution and evaluate the functor on this resolution. (And then, take its homology; in the setting of derived categories, though, one usually just takes of the projective resolution and leaves it at that.) Because on projective resolutions is much better behaved than simply on modules, the derived functor is a nice replacement.
The intuition is that a projective resolution is a cofibrant approximation to the initial object, in the language of model categories (which is often seen as a non-abelian version of classical homological algebra). This is actually precisely true if one imposes the usual model structure on bounded-below chain complexes for modules over a ring, for instance.
In constructing the cotangent complex, we are trying to derive the (highly non-abelian) functor of abelianization, which as we saw was closely related to the construction of differentials. This functor was defined on rings under a fixed ring and over a fixed ring , which is not anywhere near an abelian category. So we will need the language of model categories, and today we shall construct a model structure on a certain class of categories.
In deriving an additive functor, one ultimately applies it not on the initial abelian category, but the larger category of chain complexes. Here the analogy extends again: by the Dold-Kan correspondence (which I recently talked about), this is equivalent to the category of simplicial objects in that category. The appropriate approach now seems to be to define a model structure not on -algebras over , but on the category of simplicial -algebras over .
1. Lifting model structures via adjunctions
We are now interested in obtaining a model structure on the category of simplicial -algebras. We shall take as known that there exists a cofibrantly generated model structure on the category of simplicial sets, whose cofibrations are the injections, fibrations are Kan fibrations, and weak equivalences are maps inducing homotopy equivalences on the geometric realizations.
Now we know that there is an adjunction
that consists of the forgetful functor and the free algebra functor. From this adjunction, we want to induce a model structure on . We will do so using the following statement:
Theorem 5 (Quillen) Suppose are complete and cocomplete categories all of whose objects are small, and suppose are a pair of adjoint functors.Suppose is a cofibrantly generated model category with a set of generating cofibrations and a set of generating acyclic cofibrations. Suppose moreover is a fibrant object in for any . Then there exists a cofibrantly generated model structure on , generated by , such that
- The weak equivalences in are those such that is a weak equivalence.
- The fibrations in are those such that is a fibration.
- The cofibrations are thus determined (as those having the left lifting property with respect to the acyclic fibrations).
The pair is then a Quillen adjunction between .
Proof: There are several axioms to check. First, we have assumed is complete and cocomplete, so the axioms of the additional model structure are left to check.
We start by noting that if is an acyclic cofibration in , then has the llp (left lifting property) with respect to all fibrations in . For indeed, if is a fibration in , then by definition is a fibration in , and the lifting problem
is equivalent to the lifting problem
The latter, however, will admit a solution if is an acyclic cofibration in , because already satisfies the model category axioms. It follows that sends acyclic cofibrations in to maps having the llp with respect to all fibrations in . Similarly sends cofibrations in to cofibrations in (because will send a cofibration in to something having the llp for all acyclic fibrations in , which is the definition of a cofibration).
The first claim is that any morphism in admits a functorial factorization into a cofibration followed by an acyclic fibration. To see this, apply the small object argument to the set of maps . We find that any map in can be factored such that is a transfinite composition of pushouts of maps in (hence a cofibration in ) and has the left lifting property with respect to .
But this implies that is a fibration; indeed, by the same argument as before, we see that since has the llp with respect to , has the llp with respect to . Thus is an acyclic fibration, and by definition so is . So we get one of the (functorial) factorizations.
By the same reasoning, applying the small object argument to , we find that any map in can be factored as such that is a transfinite composition of maps in (in particular, a cofibration) and has the llp with respect to ; in particular, has the llp with respect to , and is thus a fibration. We want to say that is an acyclic cofibration, but in general we don’t know that preserves weak equivalences. However, at least we can say that has the llp with respect to all fibrations, because it is a transfinite composition of pushouts of maps of , and every map in has the llp with respect with respect to all fibrations. We will show below that anything with the llp with respect to all fibrations (in ) is in fact a weak equivalence. So is, indeed, an acyclic cofibration.
That the two-out-of-three axiom holds for weak equivalences in is immediate. Finally, we need to see that the lifting axioms hold: given a diagram in
with a cofibration and a fibration, then if one of is a weak equivalence, we must show that a lift exists.
One direction is straightforward. If is a weak equivalence, then the definition of a cofibration in shows that a lift exists. Now suppose is the acyclic one. We can factor as a composite where is a fibration and has the llp with respect to all fibrations (by using the above small object argument on as above so that is a transfinite composite of pushouts by ). The lemma that we have not proved yet will imply that is a weak equivalence.
Now will be a retract of . Indeed, this is a standard argument. Let and . We have a diagram
By two-out-of-three, is an acyclic fibration, so a lift exists. This means precisely that is a retract of . But as has the llp with respect to all fibrations (in view of its construction), so does . This completes the verification of the lifting axiom.
Hence we need:
Lemma 6 Hypotheses as above, any morphism in with the llp with respect to all fibrations is an acyclic cofibration.
Proof: By assumption it is a cofibration, though we need to see that it is a weak equivalence.
Now by assumption, is fibrant, so the diagram
and the assumption about shows that there is a retraction . We thus get two morphisms given by and . Ideally, we want to claim that the two maps are homotopic. In other words, we want a factorization
Here is a path object for .
In order to do this, we consider the diagram
Here is the composite , which makes sense there is always a natural map (which by definition is required to be a weak equivalence). Since , the diagram commutes.
Since has the llp with respect to all fibrations, we find that a lift exists, which implies that the two maps are indeed (right) homotopic. Thus is a homotopy equivalence, so an isomorphism in the homotopy category; it is in particular a weak equivalence by the next lemma.
2. A converse to the Whitehead theorem
We now need to show that a homotopy equivalence in a model category is a weak equivalence. While this is diagram-chasing, I don’t think it was displayed as prominently as the usual Whitehead theorem (that a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence).
Lemma 7 If is a map in a model category which is left-homotopic to the identity, then is a weak equivalence.
Proof: Indeed, let be a cylinder object of , so there is a morphism
which is a homotopy between the map and the identity. There are inclusions (which are cofibrations if is cofibrant) such that . Now both are weak equivalences; two-out-of-three applied to the second identity shows that is a weak equivalence. Now two-out-of-three again implies that is a weak equivalence. We need the following brief lemma.
Lemma 8 In any model category, a map that induces an isomorphism in the homotopy category is a weak equivalence.
Proof: Without loss of generality, we can assume that are cofibrant and fibrant, by replacing them with better objects. Then we know that the maps in the homotopy category between are just the homotopy classes of maps from .
So we have a homotopy equivalence
between cofibrant-fibrant objects. We need to show that it is a weak equivalence. We can factor as
where is an acyclic cofibration and is a fibration. Clearly is cofibrant-fibrant as well. Now is a homotopy equivalence by the Whitehead theorem, so must be a homotopy equivalence too. We are reduced to showing that is a weak equivalence. In particular, we need only prove the result for fibrations.
Now let be a cylinder object for . There is a map which is a homotopy inverse to . We want a refinement of this map that will also be a section of . So there is a homotopy between and the identity. There are two inclusions ; let be the one corresponding to the identity in the above homotopy.
We get a diagram
There is a lift as is a fibration. Let ; then is homotopic to , and is another homotopy inverse for . But has a redeeming feature that need not have; we have that .
So we are in the following situation. We have a fibration between cofibrant-fibrant objects with a homotopy inverse . We also have that . We are to show that is a weak equivalence.
Now is a weak equivalence by the previous lemma, as it is homotopic to the identity. Moreover, there is a diagram
Here the horizontal maps on the top row are the identities . This commutes because . This is a retract diagram, and we find that is a retract of the weak equivalence ; it is thus a weak equivalence.
3. Simplicial algebras
We shall now use the fact that there is a cofibrantly generated model structure on the category of simplicial sets, as stated above. Let be any ring. Note that there is an adjunction between simplicial -algebras (given by the forgetful functor and the free functor) and simplicial sets. It is a basic fact that every simplicial group (in particular, simplicial ring) is fibrant. As a simple corollary, we find:
Theorem 9 There is a cofibrantly generated model structure on the category of simplicial -algebras. Generating cofibrations are given by where is an inclusion of simplicial sets; the small subset where suffices. A map is a fibration or weak equivalence in iff it is so in .