Theorem A via general formalism

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.

1. The projective model structure

Let ${M}$ be a model category, and ${\mathcal{C}}$ a small category. We are interested in obtaining a model structure on the category ${\mathrm{Fun}(\mathcal{C}, M)}$ of functors ${F: \mathcal{C} \rightarrow M}$ . In fact, we are interested in showing that the colimit functor

$\displaystyle \mathrm{colim}: \mathrm{Fun}(\mathcal{C}, M) \rightarrow \mathcal{C}$

can be derived to a homotopy colimit functor; to derive it, the language of model categories is very powerful. To do this, we want to construct the cofibrations, fibrations, and weak equivalences directly from those of ${M}$ .

Definition 2 The projective model structure (if it exists) on ${\mathrm{Fun}(\mathcal{C}, M)}$ is the one defined such that the weak equivalences ${F \rightarrow G}$ of functors are objectwise weak equivalences (i.e. those such that ${F(c) \rightarrow G(c)}$ is a weak equivalence in ${M}$ for each ${c \in \mathcal{C}}$ ), and the fibrations are the objectwise fibrations.

It is not immediately obvious that there is such a model structure, but if there is one, then it is uniquely determined: the cofibrations are precisely those with the left lifting property with respect to the trivial fibrations. In fact, we can use this to describe certain examples of cofibrations. Let ${c \in \mathcal{C}}$ and let ${i: A \hookrightarrow B}$ be a cofibration in ${M}$ . We consider the functors ${F_{c, A}, F_{c, B}}$ defined via

$\displaystyle F_{c, A}(x) = \hom_{\mathcal{C}}(c, x) \times A, \quad F_{c, B}(x) = \hom_{\mathcal{C}}(c, x) \times B.$

The obvious natural transformation

$\displaystyle F_{c, A} \rightarrow F_{c, B}$

is a cofibration with respect to the projective model structure. In fact, if ${G: \mathcal{C} \rightarrow M}$ is any functor, then

$\displaystyle \hom(F_{c, A}, G) = \hom_M(A, G(c)), \quad \hom(F_{c, B}, G) = \hom_M(B, G(c)).$

From this, it follows that any lifting problem in ${\mathrm{Fun}(\mathcal{C}, M)}$

is equivalent to the lifting problem in ${M}$ :

From this, and from the pointwise definition of the weak equivalences and fibrations, it follows that ${F_{c, A} \rightarrow F_{c, B}}$ must indeed be a cofibration. That is, if ${G \rightarrow G'}$ is a trivial fibration of functors, then any lifting problem of the form (1) is equivalent to one of the form (2), which must have a solution if ${A \rightarrow B}$ is a cofibration.

Conversely, reversing the argument shows that any morphism ${G \rightarrow G'}$ in ${\mathrm{Fun}(\mathcal{C}, M)}$ with the right lifting property with respect to all maps ${F_{c, A } \rightarrow F_{c, B}}$ , as ${c}$ ranges over ${\mathcal{C}}$ and ${A \rightarrow B}$ ranges over cofibrations in ${M}$ , is a trivial fibration with respect to the projective model structure. That is, since ${G \rightarrow G'}$ has the right lifting property as in (1), the maps ${G(c) \rightarrow G'(c)}$ have the right lifting property with respect to all cofibrations as in (2).

Similarly, we find that the maps ${F_{c,A} \rightarrow F_{c, B}}$ for ${A \rightarrow B}$ a trivial cofibration in ${M}$ are trivial cofibrations in ${M}$ . Moreover, any map of functors with the right lifting property with respect to these maps must be a fibration in the projective model structure (if it exists).

It follows from the above discussion that:

Proposition 3 If the projective model structure exists, then the cofibrations ${F_{c, A} \rightarrow F_{c, B}}$ (for ${A \rightarrow B}$ ranging over a collection of generating cofibrations in ${M}$ ) and the trivial cofibrations ${F_{c , A } \rightarrow F_{c, B}}$ (as ${A \rightarrow B}$ ranges over a collection of generating trivial cofibrations) form generating collections of cofibrations and trivial cofibrations in ${\mathrm{Fun}(\mathcal{C}, M)}$ .

Recall that a model structure is cofibrantly generated if there are sets of generating cofibrations and fibrations whose domaisn permit the small object argument. Many standard examples of model categories (i.e. spaces with the Quillen model structure, simplicial sets, chain complexes) are cofibrantly generated.

Proposition 4 Let ${M}$ be a cofibrantly generated model category and ${\mathcal{C}}$ a small category. Then the projective model structure exists on ${\mathrm{Fun}(\mathcal{C}, M)}$ and is cofibrantly generated.

Proof: As before, we define weak equivalences and fibrations pointwise. Say that a map is a cofibration if it has the left lifting property with respect to the trivial fibrations.

It is immediate that the retract axiom is satisfied in ${\mathrm{Fun}(\mathcal{C}, M)}$ as it is in ${M}$ . Also, ${\mathrm{Fun}(\mathcal{C}, M)}$ is complete and cocomplete since ${M}$ is.

Let ${I}$ be a set of generating cofibrations and ${J}$ a set of generating trivial cofibrations, in ${M}$ , whose domains are small. Let ${\mathbf{I}}$ be the set of all ${F_{c, A} \rightarrow F_{c, B}}$ for ${A \rightarrow B}$ a map in ${I}$ and ${c \in \mathcal{C}}$ , and let ${\mathbf{J}}$ be the set of all ${F_{c, A} \rightarrow F_{c, B}}$ for ${A \rightarrow B}$ in ${J}$ . These are the versions of ${I, J}$ that will apply to the functor category. Note that ${F_{c, A}}$ is small if ${A}$ is small (from the Yoneda lemma, essentially).

We have seen above that any morphism in ${\mathrm{Fun}(\mathcal{C}, M)}$ with the right lifting property with respect to ${\mathbf{I}}$ is necessarily a trivial fibration. Similarly, every morphism in ${\mathrm{Fun}(\mathcal{C}, M)}$ with the right lifting property with respect to ${\mathbf{J}}$ is a fibration. So, if the model structure exists, it will be cofibrantly generated (because small sets of cofibrations and trivial cofibrations will determine the model structure).

Now we need to show that the lifting property holds. This is the last part. Consider a diagram

If ${f}$ is a cofibration and ${g}$ a trivial fibration, then there is a lift by definition of the cofibrations. The subtle point is to show that if ${f}$ is a trivial cofibration (that is, a cofibration that happens to be a weak equivalence), then the lift exists. However, we have a candidate for trivial cofibrations for which the lift would exist: these are the maps ${F_{c, A} \rightarrow F_{c, B}}$ for ${A \rightarrow B}$ a generating trivial cofibration in ${M}$ (one can check easily that these are weak equivalences by our definition). By the small object, we can factor ${f}$ as a composite

$\displaystyle F \rightarrow R \rightarrow F'$

where ${F \rightarrow R}$ is a transfinite composite of push-outs of such maps ${F_{c, A} \rightarrow F_{c, B}}$ for ${A \rightarrow B}$ a generating trivial cofibration in ${M}$ (i.e., in ${J}$ ) and ${R \rightarrow F'}$ has the right lifting property with respect to all such maps, so is a fibration. But ${R \rightarrow F'}$ is a weak equivalence by the two-out-of-three property, so ${R \rightarrow F'}$ is a trivial fibration, and ${F \rightarrow F'}$ has the left lifting property with respect to the trivial fibration ${R \rightarrow F'}$ by what has already been seen. It follows by the “retract argument” that ${F \rightarrow F'}$ is a retract of ${F \rightarrow R}$ , and consequently ${F \rightarrow F'}$ has the left lifting property with respect to fibrations (since ${F \rightarrow R}$ does). In other words, one applies the lifting property to the diagram

to conclude that ${F \rightarrow F'}$ is a retract of ${F \rightarrow R}$ . $\Box$

The last subtlety in the proof (where one half of the lifting property leads to the other) is a common argument.

2. The homotopy colimit functor

Let ${M}$ be a cofibrantly generated model category, ${\mathcal{C}}$ a small category. We are now going to use the projective model structure on ${\mathrm{Fun}(\mathcal{C}, M)}$ to derive the colimit functor.

Proposition 5 The functor ${\mathrm{colim}: \mathrm{Fun}(\mathcal{C}, M) \rightarrow M}$ sending a diagram to its colimit is a left Quillen functor from ${\mathrm{Fun}(\mathcal{C}, M)}$ to ${M}$ (where the former has the projective model structure).

Proof: By adjointness, we just need to check that the right adjoint ${M \rightarrow \mathrm{Fun}(\mathcal{C}, M)}$ sending an object ${X \in M}$ to the constant functor at ${X}$ is a right Quillen functor. In other words, we need to check that it preserves fibrations and trivial fibrations. But this is immediate from the definition of fibrations and weak equivalences in the projective model structure. $\Box$

In particular, we see that ${\mathrm{colim}}$ preserves weak equivalences between cofibrant objects in ${\mathrm{Fun}(\mathcal{C}, M)}$ (by Ken Brown’s lemma).

Definition 6 We define the homotopy colimit functor ${\mathrm{hocolim}: \mathrm{Fun}(\mathcal{C}, M) \rightarrow \mathrm{Ho} M}$ to be the left derived functor of the left Quillen functor ${\mathrm{colim}: \mathrm{Fun}(\mathcal{C}, M) \rightarrow M}$ (so ${\mathrm{hocolim}}$ factors through the homotopy category ${\mathrm{Ho} \mathrm{Fun}(\mathcal{C}, M)}$ ).

Let’s recall how to compute ${\mathrm{hocolim}}$ . Given a functor ${F: \mathcal{C} \rightarrow M}$ , to compute ${\mathrm{hocolim} F}$ , one finds a cofibrant replacement ${F' \rightarrow F}$ ; this is a cofibrant functor (i.e. object of ${\mathrm{Fun}(\mathcal{C}, M}$ ) ${F}$ together with a trivial fibration ${F' \rightarrow F}$ . Then, one takes the ordinary colimit of ${F'}$ .

In other words, one observes that the colimit functor is only really well-behaved on cofibrant objects on the functor category, so one uses them. Here the failure to be “well-behaved” means that a morphism ${F \rightarrow G}$ of functors which is a weak equivalence (i.e., such that ${Fc \rightarrow Gc}$ is a weak equivalence for all ${c \in \mathcal{C}}$ ) is not necessarily one such that ${\mathrm{colim} F \rightarrow \mathrm{colim} G}$ is a weak equivalence.

There is a pretty geometric way to think of the homotopy colimit of simplicial sets or topological spaces, but that deserves another post. For the purposes of this one, we can approach it purely formally.

3. Theorem A

With the machinery of homotopy colimits established, it will be straightforward to prove Theorem A. In order to do this, we shall express the nerve of a category using a colimit of the nerves of overcategories.

Let ${F: \mathcal{C} \rightarrow \mathcal{D}}$ be a functor. As ${d \in \mathcal{D}}$ varies, the categories ${\mathcal{C}/d}$ varycovariantly; that is, given a map ${d \rightarrow d'}$ , there is a natural functor ${\mathcal{C}/d \rightarrow \mathcal{C}/ d'}$ given by sending an object ${(c, f: Fc \rightarrow d)}$ of ${\mathcal{C}/d}$ to the object ${(c, Fc \stackrel{f}{\rightarrow} d\rightarrow d')}$ of ${\mathcal{C}/ d'}$ . Note that each ${\mathcal{C}/d}$ admits a forgetful functor to ${\mathcal{C}}$ , and the following diagram always commutes:

As a result, for each ${d \rightarrow d'}$ , there is a commutative diagram of nerves:

Consequently there is induced a map ${\mathrm{colim}_{\mathcal{D}} N(\mathcal{C}/d) \rightarrow N(\mathcal{C})}$ .

Proposition 7 The canonical map ${\mathrm{colim}_{\mathcal{D}} N(\mathcal{C}/d) \rightarrow N(\mathcal{C})}$ is an isomorphism.

Proof: Let us determine the ${n}$ -simplices in the colimit; they should be the same as ${n+1}$ -tuples of composable arrows. Namely, note first that an ${n}$ -simplex in ${N(\mathcal{C}/d)}$ is given by the data

$\displaystyle c_0 \rightarrow \dots \rightarrow c_n , \quad F(c_n) \rightarrow d$

where ${c_0 \rightarrow \dots \rightarrow c_n}$ is a tuple of composable arrows in ${\mathcal{C}}$ , and ${F(c_n) \rightarrow d}$ is a morphism in ${\mathcal{D}}$ . This is clear from the explicit description of ${\mathcal{C}/d}$ : the last map ${F(c_n) \rightarrow d}$ determines all the other ${F(c_i) \rightarrow d}$ . The map from this ${n}$ -simplex of ${N(\mathcal{C}/d)}$ to the ${n}$ -simplex of ${N(\mathcal{C})}$ just forgets the map ${F(c_n) \rightarrow d}$ .

As a result, ${\mathrm{colim}_\mathcal{D} N(\mathcal{C}/d) \rightarrow N(\mathcal{C})}$ must be a surjection. Given any tuple ${c_0 \rightarrow \dots \rightarrow c_n}$ , it is the image of the tuple ${c_0 \rightarrow \dots \rightarrow c_n , F(c_n) \stackrel{1}{\rightarrow} F(c_n)}$ ; consequently every simplex in ${N(\mathcal{C})}$ is hit by something in the colimit. Next, we should check that the map is injective. That is, the claim is that every tuple ${c_0 \rightarrow \dots \rightarrow c_n, F(c_n) \rightarrow d}$ in the colimit ${\mathrm{colim}_\mathcal{D} N(\mathcal{C}/d)}$ such that the first part of the data ${c_0 \rightarrow \dots \rightarrow c_n}$ (i.e., maps to a fixed ${n}$ -simplex in ${N(\mathcal{C})}$ ) is identified. But every such tuple is identified with the data ${c_0 \rightarrow \dots \rightarrow c_n, F(c_n) \stackrel{1}{\rightarrow} F(c_n)}$ because the original tuple is obtained from this by push-forward (via ${F(c_n) \rightarrow d}$ ). This proves the result. $\Box$

We are now almost there. We have an expression ${N(\mathcal{C}) = \mathrm{colim} N(\mathcal{C}/d)}$ , and similarly ${N(\mathcal{D}) = \mathrm{colim} N(\mathcal{D}/d)}$ , by applying the same result to the identity functor ${\mathcal{D} \rightarrow \mathcal{D}}$ . The canonical map ${N(\mathcal{C}) \rightarrow N(\mathcal{D})}$ is the colimit map

$\displaystyle \mathrm{colim} N(\mathcal{C}/d) \rightarrow \mathrm{colim} N(\mathcal{D}/d).$

Suppose now that each ${N(\mathcal{C}/d)}$ is weakly contractible; this is the hypothesis of Quillen’s Theorem A. Then each map ${N(\mathcal{C}/d) \rightarrow N(\mathcal{D}/d)}$ is a weak equivalence; indeed, ${N(\mathcal{D}/d)}$ is contractible because it has a final object. To see that ${\mathrm{colim} N(\mathcal{C}/d) \rightarrow \mathrm{colim} N(\mathcal{D}/d)}$ is a weak equivalence as a result, we only need to show that both diagrams are cofibrant in the functor category ${\mathrm{Fun}(\mathcal{D}, \mathbf{SSet})}$ ; in this case, it will follow that the colimits for ${N(\mathcal{C}), N(\mathcal{D})}$ are actually homotopycolimits.

Proposition 8 The diagram ${d \mapsto N(\mathrm{Fun}(\mathcal{C}/d))}$ is cofibrant in ${\mathrm{Fun}(\mathcal{D}, \mathbf{SSet})}$ with the projective model structure.

Proof: Let us describe what a cofibrant object in ${\mathrm{Fun}(\mathcal{D}, \mathbf{SSet})}$ should look like. The generating cofibrations of ${\mathbf{SSet}}$ (with the usual model structure) are ${\partial \Delta[n] \rightarrow \Delta[n]}$ , so a set of generating cofibrations for this functor category is given by ${\hom(d, \cdot) \times \partial \Delta[n] \rightarrow \hom(d, \cdot) \times \Delta[n]}$ for ${d \in \mathcal{D}, n \in \mathbb{Z}_{\geq 0}}$ (by the construction of the projective model structure). In particular, we see that the cofibrations can be described as retracts of transfinite compositions of push-outs of such maps (by the small object argument).

Suppose that we have a functor ${F \in \mathrm{Fun}(\mathcal{D}, \mathbf{SSet})}$ that can be described as follows. There is a filtration of sub-functors ( ${\mathbf{SSet}}$ -valued) ${F_0 \subset F_1 \subset F_2 \subset \dots \subset F}$ , whose union is ${F}$ , such that ${F_i}$ is obtained from ${F_{i-1}}$ in the following way. There are a family of objects ${\left\{d_{ij}\right\} \subset \mathcal{D}}$ and ${n}$ -simplices ${\sigma_{ij} \in F_i(d_{ij})}$ with the following property: for any two morphisms ${f, g: d_{ij} \rightarrow d}$ , the images of ${\sigma_{ij} \in d}$ are different. Moreover, the boundaries of ${d_{ij}}$ lie in ${F_{i-1}}$ . In this case, ${F}$ is cofibrant. Indeed, what the condition states is that ${F_i}$ is obtained from ${F_{i-1}}$ by a push-out via various morphisms of the form ${\hom(d_{ij}, \cdot) \times \partial\Delta[n] \rightarrow \hom(d_{ij}, \cdot) \times \Delta[n]}$ .

The claim is that ${d \mapsto N(\mathrm{Fun}(\mathcal{C}/d))}$ can be described in this way. Indeed, we take as ${F_n}$ the ${n}$ -skeleton, and the ${n}$ -simplices the family of ${c_0 \rightarrow c_1 \rightarrow \dots \rightarrow c_n, F(c_n) \stackrel{1}{\rightarrow} F(c_n)}$ where all the maps ${c_j \rightarrow c_{j+1}}$ are non-identities. It is easy to see that two distinct maps ${d \rightarrow d'}$ will send two such simplices to different ones, and that the boundaries are in the ${n-1}$ -skeleton. As a result, our functor satisfies the above condition, and is cofibrant. $\Box$

Now we can prove:

Theorem 9 (Theorem A) Let ${F: \mathcal{C} \rightarrow \mathcal{D}}$ be a functor such that ${\mathcal{C}/d}$ is contractible for each ${d \in \mathcal{D}}$ . Then ${F: N(\mathcal{C}) \rightarrow N(\mathcal{D})}$ is a weak equivalence.

Proof: Indeed, we have seen that the map ${F}$ can be described as

$\displaystyle \mathrm{colim} N(\mathcal{C}/d) \rightarrow \mathrm{colim} N(\mathcal{D} /d),$

where both functors ${d \mapsto N(\mathcal{C}/d), d \mapsto N(\mathcal{D}/d)}$ are cofibrant in the functor category ${\mathrm{Fun}(\mathcal{D}, \mathbf{SSet})}$ by the previous result (with respect to the projective model structure). But by hypothesis, the map of functors is a weak equivalence, and we know that taking colimits (as a left Quillen functor) preserves weak equivalences between cofibrant objects. $\Box$