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 {\Delta} be the category of finite (nonempty) ordered sets and order-preserving morphisms. The object {[n]} will denote the set {\left\{0, 1, \dots, n\right\}} with the usual ordering. Thus {\Delta} is equivalent to the subcategory consisting of the {[n]}. This is called the simplex category.

There is a functor from {\Delta} to the category {\mathbf{Top}} of topological spaces. Given {[n]}, we send it to the standard topological {n}-simplex {\Delta_n} that consists of points {(t_0, \dots, t_{n}) \in \mathbb{R}^{n+1}} such that each {t_i \in [0, 1]} and {\sum t_i = 1}. Given a morphism {\phi: [m] \rightarrow [n]} of ordered sets, we define {\Delta_m \rightarrow \Delta_n} by sending

\displaystyle (t_0, \dots, t_m) \mapsto (u_j), \quad u_j = \sum_{\phi(i) = j} t_i.

Here the empty sum is to be regarded as zero.

1.2. Simplicial sets

Definition 2 A simplicial set {X_{\bullet}} is a contravariant functor from {\Delta} 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 {\mathbf{SSet}}. Asimplicial object in a category {\mathcal{C}} is a contravariant functor {\Delta \rightarrow \mathcal{C}}.

We have just seen that the category {\Delta} is equivalent to the subcategory consisting of the {[n]}. As a result, a simplicial set {X_{\bullet}} is given by specifying sets {X_n} for each {n \in \mathbb{Z}_{\geq 0}}, together with maps

\displaystyle X_n \rightarrow X_m

for each map {[m]\rightarrow [n]} in {\Delta}. The set {X_n} is called the set of {n}-simplices of {X_\bullet}. (more…)