The following result is useful in algebraic K-theory.
Theorem 1 Let be a functor between categories. Suppose is contractible for each . Then 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 denotes the nerve of the category : it is the simplicial set whose -simplices consist of composable strings of morphisms of . The overcategory has objects consisting of pairs for , a morphism in ; morphisms in are morphisms in 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:
where ranges over the objects of . This expresses the nerve of as a colimit of simplicial sets arising as the nerves of . We will compare this with a similar expression for the nerve of , that is . Then, the point will be that is a weak equivalence for each ; 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 be a model category, and a small category. We are interested in obtaining a model structure on the category of functors . In fact, we are interested in showing that the colimit functor
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 .
Definition 2 The projective model structure (if it exists) on is the one defined such that the weak equivalences of functors are objectwise weak equivalences (i.e. those such that is a weak equivalence in for each ), 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 and let be a cofibration in . We consider the functors defined via
The obvious natural transformation
is a cofibration with respect to the projective model structure. In fact, if is any functor, then
From this, it follows that any lifting problem in
is equivalent to the lifting problem in :
From this, and from the pointwise definition of the weak equivalences and fibrations, it follows that must indeed be a cofibration. That is, if 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 is a cofibration.
Conversely, reversing the argument shows that any morphism in with the right lifting property with respect to all maps , as ranges over and ranges over cofibrations in , is a trivial fibration with respect to the projective model structure. That is, since has the right lifting property as in (1), the maps have the right lifting property with respect to all cofibrations as in (2).
Similarly, we find that the maps for a trivial cofibration in are trivial cofibrations in . 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 (for ranging over a collection of generating cofibrations in ) and the trivial cofibrations (as ranges over a collection of generating trivial cofibrations) form generating collections of cofibrations and trivial cofibrations in .
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 be a cofibrantly generated model category and a small category. Then the projective model structure exists on 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 as it is in . Also, is complete and cocomplete since is.
Let be a set of generating cofibrations and a set of generating trivial cofibrations, in , whose domains are small. Let be the set of all for a map in and , and let be the set of all for in . These are the versions of that will apply to the functor category. Note that is small if is small (from the Yoneda lemma, essentially).
We have seen above that any morphism in with the right lifting property with respect to is necessarily a trivial fibration. Similarly, every morphism in with the right lifting property with respect to 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 is a cofibration and a trivial fibration, then there is a lift by definition of the cofibrations. The subtle point is to show that if 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 for a generating trivial cofibration in (one can check easily that these are weak equivalences by our definition). By the small object, we can factor as a composite
where is a transfinite composite of push-outs of such maps for a generating trivial cofibration in (i.e., in ) and has the right lifting property with respect to all such maps, so is a fibration. But is a weak equivalence by the two-out-of-three property, so is a trivial fibration, and has the left lifting property with respect to the trivial fibration by what has already been seen. It follows by the “retract argument” that is a retract of , and consequently has the left lifting property with respect to fibrations (since does). In other words, one applies the lifting property to the diagram
to conclude that is a retract of .
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 be a cofibrantly generated model category, a small category. We are now going to use the projective model structure on to derive the colimit functor.
Proposition 5 The functor sending a diagram to its colimit is a left Quillen functor from to (where the former has the projective model structure).
Proof: By adjointness, we just need to check that the right adjoint sending an object to the constant functor at 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.
In particular, we see that preserves weak equivalences between cofibrant objects in (by Ken Brown’s lemma).
Definition 6 We define the homotopy colimit functor to be the left derived functor of the left Quillen functor (so factors through the homotopy category ).
Let’s recall how to compute . Given a functor , to compute , one finds a cofibrant replacement ; this is a cofibrant functor (i.e. object of ) together with a trivial fibration . Then, one takes the ordinary colimit of .
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 of functors which is a weak equivalence (i.e., such that is a weak equivalence for all ) is not necessarily one such that 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 be a functor. As varies, the categories varycovariantly; that is, given a map , there is a natural functor given by sending an object of to the object of . Note that each admits a forgetful functor to , and the following diagram always commutes:
As a result, for each , there is a commutative diagram of nerves:
Consequently there is induced a map .
Proposition 7 The canonical map is an isomorphism.
Proof: Let us determine the -simplices in the colimit; they should be the same as -tuples of composable arrows. Namely, note first that an -simplex in is given by the data
where is a tuple of composable arrows in , and is a morphism in . This is clear from the explicit description of : the last map determines all the other . The map from this -simplex of to the -simplex of just forgets the map .
As a result, must be a surjection. Given any tuple , it is the image of the tuple ; consequently every simplex in is hit by something in the colimit. Next, we should check that the map is injective. That is, the claim is that every tuple in the colimit such that the first part of the data (i.e., maps to a fixed -simplex in ) is identified. But every such tuple is identified with the data because the original tuple is obtained from this by push-forward (via ). This proves the result.
We are now almost there. We have an expression , and similarly , by applying the same result to the identity functor . The canonical map is the colimit map
Suppose now that each is weakly contractible; this is the hypothesis of Quillen’s Theorem A. Then each map is a weak equivalence; indeed, is contractible because it has a final object. To see that is a weak equivalence as a result, we only need to show that both diagrams are cofibrant in the functor category ; in this case, it will follow that the colimits for are actually homotopycolimits.
Proposition 8 The diagram is cofibrant in with the projective model structure.
Proof: Let us describe what a cofibrant object in should look like. The generating cofibrations of (with the usual model structure) are , so a set of generating cofibrations for this functor category is given by for (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 that can be described as follows. There is a filtration of sub-functors (-valued) , whose union is , such that is obtained from in the following way. There are a family of objects and -simplices with the following property: for any two morphisms , the images of are different. Moreover, the boundaries of lie in . In this case, is cofibrant. Indeed, what the condition states is that is obtained from by a push-out via various morphisms of the form .
The claim is that can be described in this way. Indeed, we take as the -skeleton, and the -simplices the family of where all the maps are non-identities. It is easy to see that two distinct maps will send two such simplices to different ones, and that the boundaries are in the -skeleton. As a result, our functor satisfies the above condition, and is cofibrant.
Now we can prove:
Theorem 9 (Theorem A) Let be a functor such that is contractible for each . Then is a weak equivalence.
Proof: Indeed, we have seen that the map can be described as
where both functors are cofibrant in the functor category 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.