simplicial group is, naturally enough, a simplicial object in the category of groups. It is equivalently a simplicial set ${G_\bullet}$ such that each ${G_n}$ has a group structure and all the face and degeneracy maps are group homomorphisms. By a well-known lemma, a simplicial group is automatically a Kan complex. We can thus think of a simplicial group in two ways. On the one hand, it is an ${\infty}$-groupoid, as a Kan complex. On the other hand, we can think of a simplicial group as a model for a “higher group,” or an ${\infty}$-group. This is the intuition that the nLab suggests.

Now it is well-known that in ordinary category theory, a group is the same as a groupoid with one object. So an ${\infty}$-groupoid with one object should be the same as an ${\infty}$-group. This is in fact true with the above notation. In other words, if we say that an ${\infty}$-groupoid is a Kan complex (as usual), and decide that an ${\infty}$-group is a simplicial group, then the ${\infty}$-groups are the “same” as the ${\infty}$-groupoids with one object.

Here the “same” means that the associated ${\infty}$-categories are equivalent. One way of expressing this is to say that there are natural model structures on ${\infty}$-groupoids with one object and simplicial groups, and that these are Quillen equivalent. This is a frequent way of expressing the idea that two ${\infty}$-categories (or at least, ${(\infty, 1)}$-categories) are equivalent. For instance, this is the way the ${\infty}$-categorical Grothendieck construction is stated in HTT, as a Quillen equivalence between model categories of “right fibrations” and simplicial presheaves. Let us express this formally.

Theorem 1 (Kan) There are natural model structures on the category ${\mathbf{SGrp}}$ of simplicial groups and on the category ${\mathbf{SSet}_0}$ of reduced simplicial sets (i.e. those with one vertex), and the two are Quillen equivalent.

To give this construction, we will first describe the classifying space of a simplicial group. One incarnation of this is going to give the functor from simplicial groups to reduced simplicial sets. This is a simplicial version of the usual topological classifying space. Recall that if ${G}$ is a topological group, then there is a classifying space ${BG}$ and a principal ${G}$-bundle ${EG \rightarrow BG}$ such that ${EG}$ is contractible. It follows from this that for any CW-complex ${X}$, the homotopy classes of maps ${X \rightarrow BG}$ are in bijection with the principal ${G}$-bundles on ${X}$ (in fact, ${EG \rightarrow BG}$ is a universal bundle).

We will need the appropriate notion of a principal bundle in the simplicial setting. Let ${G_\bullet}$ be a simplicial group.

Definition 2 A ${G_\bullet}$simplicial set is a simplicial set ${X_\bullet}$ together with an action ${G_\bullet \times X_\bullet \rightarrow X_\bullet}$ satisfying the usual axioms. Thus, each ${X_n}$ is a ${G_n}$-set. There is a category ${\mathbf{SSet}_G}$ of ${G_\bullet}$-simplicial sets and ${G_\bullet}$-equivariant maps.

We next give the definition of a principal bundle. Notice that because of the combinatorial nature of simplicial sets, we don’t make any local trivialty condition; that will fall out.

Definition 3 Let ${E_\bullet \rightarrow B_\bullet}$ be a map of ${G_\bullet}$-simplicial sets. We say that ${E_\bullet \rightarrow B_\bullet}$ is a principal ${G_\bullet}$-bundle if ${B_\bullet}$ has trivial action, each ${E_n}$ is a free ${G_n}$-set, and if ${E_\bullet/G_\bullet \simeq B_\bullet}$ under the natural map. (more…)

The following result is useful in algebraic K-theory.

Theorem 1 Let ${F: \mathcal{C} \rightarrow \mathcal{D}}$ be a functor between categories. Suppose ${\mathcal{C}/d}$ is contractible for each ${d \in \mathcal{D}}$. Then ${F: N\mathcal{C} \rightarrow N \mathcal{D}}$ is a weak homotopy equivalence.

I don’t really know enough to give a good justification for the usefulness, but in essence, what Quillen did in the 1970s was to show that the Grothendieck group of an “exact category” could be interpreted homotopically as the fundamental group of the nerve of the “Q-category” built from the exact category. As a result, Quillen was able to define higher K-groups as the higher homotopy groups of this space. He then proved a lot of results that were proved by ad hoc, homological means for the Grothendieck group of a category for the higher K-groups as well, by interpreting them in terms of homotopy theory. This result (together with the extension, “Theorem B”) is a key homotopical tool he used to analyze these nerves.

Here ${N \mathcal{C}}$ denotes the nerve of the category ${\mathcal{C}}$: it is the simplicial set whose ${n}$-simplices consist of composable strings of ${n+1}$ morphisms of ${\mathcal{C}}$. The overcategory ${\mathcal{C}/d}$ has objects consisting of pairs ${(c, f)}$ for ${c \in \mathcal{C}}$, ${f: Fc \rightarrow d}$ a morphism in ${\mathcal{D}}$; morphisms in ${\mathcal{C}/d}$ are morphisms in ${\mathcal{C}}$ making the natural diagram commute. We say that a category is contractible if its nerve is weakly contractible as a simplicial set.

There are other reasons to care. For instance, in higher category theory, the above condition on contractibility of over-categories is the analog of cofinality in ordinary category theory. Anyway, this result is pretty important.

But what I want to explain in this post is that “Theorem A” (and Theorem B, but I’ll defer that) is really purely formal. That is, it can be deduced from some standard and not-too-difficult manipulations with model categories (which weren’t all around when Quillen wrote “Higher algebraic K-theory I”).

To prove this, we shall obtain the following expression for a category:

$\displaystyle N \mathcal{C} = \mathrm{colim}_d N (\mathcal{C}/d),$

where ${d}$ ranges over the objects of ${\mathcal{D}}$. This expresses the nerve of ${\mathcal{C}}$ as a colimit of simplicial sets arising as the nerves of ${\mathcal{C}/d}$. We will compare this with a similar expression for the nerve of $\mathcal{D}$, that is ${N \mathcal{D} = \mathrm{colim}_d N(\mathcal{D}/d)}$. Then, the point will be that ${N(\mathcal{C}/d) \rightarrow N(\mathcal{D}/d)}$ is a weak equivalence for each ${d}$; this by itself does not imply that the induced map on colimits is a weak equivalence, but it will in this case because both the colimits will in fact turn out to be homotopy colimits. I’ll start by explaining what those are.

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 $F$ on an object $X$ in some abelian category, you replace $X$ by a projective resolution and evaluate the functor $F$ on this resolution. (And then, take its homology; in the setting of derived categories, though, one usually just takes $F$ of the projective resolution and leaves it at that.) Because $F$ on projective resolutions is much better behaved than $F$ 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 $A$ and over a fixed ring $B$, 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 $A$-algebras over $B$, but on the category of simplicial $A$-algebras over $B$. (more…)