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 1Let 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 thesimplex 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 2Asimplicial setis a contravariant functor from to the category of sets. In other words, it is a presheaf on the simplex category. Amorphismof simplicial sets is a natural transformation of functors. The class of simplicial sets thus becomes a category . Asimplicial objectin 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 . (more…)