As planned, I’m going to try to say a few words on the cotangent complex in the following posts. There is a bit of background I’d like to go through first, starting with the theory of simplicial sets. The reason is that the cotangent complex is most naturally thought of using the model structure on simplicial rings, on which it is a non-abelian version of a derived functor.
1.1. The simplex category
Definition 1 Let be the category of finite (nonempty) ordered sets and order-preserving morphisms. The object will denote the set with the usual ordering. Thus is equivalent to the subcategory consisting of the . This is called the simplex category.
There is a functor from to the category of topological spaces. Given , we send it to the standard topological -simplex that consists of points such that each and . Given a morphism of ordered sets, we define by sending
Here the empty sum is to be regarded as zero.
1.2. Simplicial sets
Definition 2 A simplicial set is a contravariant functor from to the category of sets. In other words, it is a presheaf on the simplex category. A morphism of simplicial sets is a natural transformation of functors. The class of simplicial sets thus becomes a category . Asimplicial object in a category is a contravariant functor .
We have just seen that the category is equivalent to the subcategory consisting of the . As a result, a simplicial set is given by specifying sets for each , together with maps
for each map in . The set is called the set of -simplices of .
Example 1 Let be a topological space. Then we define its singular simplicial set as follows. We let . Using the functoriality of discussed above, it is clear that there are maps for each .
Example 2 Given , we define the standard -simplex via
Given a category , we know that there is a way of generating presheaves on . For each , we consider the presheaf defined as ; the presheaves obtained are the representable presheaves. The standard simplices are a special case of that.
Proposition 3 Let be a simplicial set. Then there is a natural bijection
In other words, mapping from a standard -simplex into is equivalent to giving an -simplex of . Proof: Immediate from Yoneda’s lemma.
1.3. Generalities on presheaves
We are interested in describing functors on the category of simplicial sets. It will be convenient to describe them first on the standard -simplices . In general, this will be sufficient to characterize the functor. We are going to see that the values on the standard -simplices will be enough, in many cases, to determine a functor. We shall discuss this in a general context of presheaves on any small category, though. Let be any small category. We shall, most often, take to be . Let be the category of presheaves on .
Proposition 4 Any presheaf on is canonically the colimit of representable presheaves.
Proof: Let be a presheaf on . For each , we let be the representable presheaf defined, as above, by . Now form the category whose objects are morphisms of presheaves , such that the morphisms between and are given by commutative diagrams
Note that these commutative diagrams depend on nothing fancy: a morphism is just a map , in view of Yoneda’s lemma. There is a functor sending to . The image of this functor consists of representable presheaves (clear) and, by definition of the category , there is a map
that commutes with the diagrams. So there is induced a map
This is a map from a colimit of representable presheaves to . The claim is that it is an isomorphism.
But by the Yoneda lemma, if and , then there is a map in such that the identity in is sent to in . It follows that we can hit any element in any part of the presheaf by a representable presheaf. Thus the map (1) is surjective.
Now let be a fixed object. We want to show that the map
is injective. Note that we can calculate direct limits in “piecewise.”
Suppose two elements and are mapped to the same element of . Then correspond to maps given by elements , and correspond to maps . The fact that they map to the same thing in means that , where the star denotes pulling back. Call .
We are now going to show that are identified in the colimit. To see this, we construct diagrams
The first comes from the map and the map given by ; the map , given by , is just the object . The second diagram is similar. The first shows that the object of the colimit is identified with the identity of by in the diagram (where is an element of by the map given by ). Similarly is identified with this in the colimit, so are identified. It follows that (1) is also injective.
The fact that this colimit is “canonical” follows from the fact that if is a morphism of presheaves, there is a functor between the categories associated to each of them. To clarify, if is a presheaf, then we have described a category and a functor such that is the colimit of . This association is functorial; if is a morphism of presheaves, then there is a functor that fits into an obvious commutative diagram.
Corollary 5 Any simplicial set is canonically a colimit of standard -simplices.
Proof: This follows from the previous result with .
Warning: Just because every element of is a colimit of representable presheaves does not mean that every element of is representable, even if is cocomplete. For instance, the empty presheaf (which assigns to each element of the category ) is not representable. I learned this from Todd Trimble’s answer here. The problem is that the Yoneda embedding does not commute with colimits.