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 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 ${X_\bullet}$.

Example 1 Let ${X}$ be a topological space. Then we define its singular simplicial set ${\mathrm{Sing} X_\bullet}$ as follows. We let ${(\mathrm{Sing} X)_n = \hom_{\mathbf{Top}}(\Delta_n, X)}$. Using the functoriality of ${\Delta_n}$ discussed above, it is clear that there are maps ${(\mathrm{Sing} X)_n \rightarrow (\mathrm{Sing} X)_m}$ for each ${[m] \rightarrow [n]}$.

Example 2 Given ${n \in \mathbb{Z}_{\geq 0}}$, we define the standard ${n}$-simplex ${\Delta[n]_\bullet}$ via

$\displaystyle \Delta[n]_m = \hom_{\Delta}([m], [n]).$

Given a category ${\mathcal{C}}$, we know that there is a way of generating presheaves on ${\mathcal{C}}$. For each ${X \in \mathcal{C}}$, we consider the presheaf ${h_Y}$ defined as ${Y \mapsto \hom_{\mathcal{C}}(Y, X)}$; the presheaves obtained are the representable presheaves. The standard simplices are a special case of that.

Proposition 3 Let ${X_\bullet}$ be a simplicial set. Then there is a natural bijection

$\displaystyle X_n \simeq \hom_{\mathbf{SSet}}(\Delta[n]_\bullet, X_\bullet).$

In other words, mapping from a standard ${n}$-simplex into ${X_\bullet}$ is equivalent to giving an ${n}$-simplex of ${X}$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 ${n}$-simplices ${\Delta[n]_\bullet}$. In general, this will be sufficient to characterize the functor. We are going to see that the values on the standard ${n}$-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 ${\mathcal{C}}$ be any small category. We shall, most often, take ${\mathcal{C}}$ to be ${\Delta}$. Let ${\hat{\mathcal{C}}}$ be the category of presheaves on ${\mathcal{C}}$.

Proposition 4 Any presheaf on ${\mathcal{C}}$ is canonically the colimit of representable presheaves.

Proof: Let ${F \in \hat{\mathcal{C}}}$ be a presheaf on ${\mathcal{C}}$. For each ${ X \in \mathcal{C}}$, we let ${h_X}$ be the representable presheaf defined, as above, by ${h_X(Y) = \hom_{\mathcal{C}}(Y, X)}$. Now form the category ${\mathcal{D}}$ whose objects are morphisms of presheaves ${h_X \rightarrow F}$, such that the morphisms between ${h_X \rightarrow F}$ and ${h_Y \rightarrow F}$ are given by commutative diagrams

Note that these commutative diagrams depend on nothing fancy: a morphism ${h_X \rightarrow h_Y}$ is just a map ${X \rightarrow Y}$, in view of Yoneda’s lemma. There is a functor ${\phi: \mathcal{D} \rightarrow \hat{\mathcal{C}}}$ sending ${h_X \rightarrow F}$ to ${h_X}$. The image of this functor consists of representable presheaves (clear) and, by definition of the category ${\mathcal{D}}$, there is a map

$\displaystyle \phi( c) \rightarrow F, \quad \forall c \in \mathcal{D}$

that commutes with the diagrams. So there is induced a map

$\displaystyle \varinjlim_{\mathcal{D}} \phi(c) \rightarrow F. \ \ \ \ \ (1)$

This is a map from a colimit of representable presheaves to ${F}$. The claim is that it is an isomorphism.

But by the Yoneda lemma, if ${X \in \mathcal{C}}$ and ${\alpha \in F(X)}$, then there is a map ${h_X \rightarrow F}$ in ${\hat{\mathcal{C}}}$ such that the identity in ${h_X(X)}$ is sent to ${\alpha}$ in ${F(X)}$. It follows that we can hit any element in any part of the presheaf ${F}$ by a representable presheaf. Thus the map (1) is surjective.

Now let ${X \in \mathcal{C}}$ be a fixed object. We want to show that the map

$\displaystyle \varinjlim_{\mathcal{D}} \phi(c)(X) \rightarrow F(X)$

is injective. Note that we can calculate direct limits in ${\hat{C}}$ “piecewise.”

Suppose two elements ${\alpha_1 \in \phi(c_1)(X)}$ and ${\alpha_2 \in \phi(c_2)(X)}$ are mapped to the same element of ${F(X)}$. Then ${c_1, c_2}$ correspond to maps ${h_{Y_1} \rightarrow F, h_{Y_2} \rightarrow F}$ given by elements ${\beta_1 \in F(Y_1), \beta_2 \in F(Y_2)}$, and ${\alpha_1, \alpha_2}$ correspond to maps ${f_1: X \rightarrow Y_1, f_2: X \rightarrow Y_2}$. The fact that they map to the same thing in ${F(X)}$ means that ${f_1^*(\beta_1) = f_2^*(\beta_2)}$, where the star denotes pulling back. Call ${\gamma = f_1^*(\beta_1) = f_2^*(\beta_2)}$.

We are now going to show that ${\alpha_1, \alpha_2}$ are identified in the colimit. To see this, we construct diagrams

and

The first comes from the map ${f_1: X \rightarrow Y_1}$ and the map ${h_X \rightarrow F}$ given by ${\gamma}$; the map ${h_{Y_1} \rightarrow F}$, given by ${\beta_1}$, is just the object ${c_1}$. The second diagram is similar. The first shows that the object ${\alpha_1 \in h_{Y_1}(X)}$ of the colimit is identified with the identity of ${h_X(X)}$ by ${f_1}$ in the diagram (where ${h_X}$ is an element of ${\mathcal{D}}$ by the map ${h_X \rightarrow F}$ given by ${\gamma}$). Similarly ${\alpha_2}$ is identified with this in the colimit, so ${\alpha_1, \alpha_2}$ are identified. It follows that (1) is also injective.

The fact that this colimit is “canonical” follows from the fact that if ${F \rightarrow F'}$ is a morphism of presheaves, there is a functor between the categories ${\mathcal{D}}$ associated to each of them. To clarify, if ${F}$ is a presheaf, then we have described a category ${\mathcal{D}_F}$ and a functor ${\mathcal{D}_F \rightarrow \mathcal{C}}$ such that ${F}$ is the colimit of ${\mathcal{D}_F \rightarrow \mathcal{C} \rightarrow \hat{\mathcal{C}}}$. This association is functorial; if ${F \rightarrow F'}$ is a morphism of presheaves, then there is a functor ${\mathcal{D}_F \rightarrow \mathcal{D}_{F'}}$ that fits into an obvious commutative diagram.

Corollary 5 Any simplicial set is canonically a colimit of standard ${n}$-simplices.

Proof: This follows from the previous result with ${\mathcal{C} = \Delta}$.

Warning: Just because every element of ${\hat{\mathcal{C}}}$ is a colimit of representable presheaves does not mean that every element of ${\hat{\mathcal{C}}}$ is representable, even if ${\mathcal{C}}$ is cocomplete. For instance, the empty presheaf (which assigns to each element of the category ${\emptyset}$) is not representable. I learned this from Todd Trimble’s answer here. The problem is that the Yoneda embedding does not commute with colimits.