In this post I want to describe the following result, which I think is pretty neat and should be more widely known:
Theorem: On the category of (small) categories there is a unique model structure in which the weak equivalences are the equivalences of categories.
This unique model structure is of course the so-called “canonical” model structure of André Joyal and Myles Tierney. (The fact that it is the unique one with these weak equivalences lends credence, I think, to using the name “canonical”). It is proper, cartesian, simplicial, combinatorial, and every object is both fibrant and cofibrant. I first learned of this uniqueness result from Steve Lack’s comments on this MathOverflow question, though there were some details left to fill in. I hope to do that here.
Below I will give an elementary proof of the above theorem, partly so I have it written down somewhere for future reference. Charles Rezk has a nice write-up of this model structure, and I will start by describing it. It consists of:
- the canonical cofibrations, which are those functors of categories which are injective on objects,
- the canonical acyclic cofibrations, which are those equivalences of categories which are injective on objects (these are necessarily injective on morphisms too),
- the canonical acyclic fibrations, which are those equivalences which are surjective on objects, and
- the canonical fibrations, which are the “iso-fibrations“. These are the functors such that for any and any isomorphism in Y, then there exists an isomorphism in X which maps to under f.
Let’s fix some notation which will be useful later.
- Let E be the free walking isomorphism. This is the contractible category with two objects. (A category is contractible if it is equivalent to the terminal category pt).
There is an equivalence , which includes the terminal category as one of the objects. The canonical fibrations are precisely those maps which have the right lifting property with respect to this functor.
Now let us suppose that we have a model category structure on the category of categories in which the weak equivalences are the equivalences of categories. Such a structure consists of certain classes of fibrations and cofibrations. Our goal is to show that these must be the canonical fibrations and canonical cofibrations above.
The proof will use some basic properties about model categories:
- The cofibrations and acyclic cofibrations are closed under retracts, compositions, and pushouts along arbitrary maps.
- The acyclic cofibrations are precisely those cofibrations which are also (weak) equivalences. The acyclic fibrations are also weak equivalences.
- The fibrations have the right lifting property with respect to the acyclic cofibrations and the acyclic fibrations have the right lifting property with respect to the cofibrations.
It also rests on a
Key Fact: every equivalence class of objects contains a fibrant representative and a cofibrant representative.
(Recall that an object is cofibrant if the unique map from the initial object (the empty category, in this case) is a cofibration. Dually, and object is fibrant if the unique map to the terminal object is a fibration.)
What do these facts tell us?
Trivial Lemma: The inclusion is a cofibration.
Proof: Since each equivalence class of categories contains a cofibrant representative, we know that there exists some cofibration for a non-empty category A. The desired map is a retract of this, hence also a cofibration. ◊
The acyclic fibrations must have the right lifting property with resect to all cofibrations, hence with respect to this map . This means the acyclic fibrations must be surjective on objects. Since they are equivalences too, this implies the following consequences.
- the acyclic fibrations are a subset of the canonical acyclic fibrations, hence
- the cofibrations contain the canonical cofibrations, hence
- the acyclic cofibrations contain the canonical acyclic cofibrations, and hence
- the fibrations are a subset of the canonical fibrations.
This means we are half-way there. We must rule out the possibility that there could be more cofibrations. (This includes the case there are more acyclic cofibrations).
A Somewhat Less Trivial Lemma: If the cofibrations contain a map which is not a canonical cofibration (i.e. fails to be injective on objects), then the following map is also a cofibration (hence an acyclic cofibration as it is an equivalence):
Proof: Suppose that we have a functor which is a cofibration but not injective on objects. Then there exists at least one pair of objects in the source category which map to the same object in the target category. Call these objects x and y, and their image p.
The cofibrations are closed under pushouts along arbitrary maps and this allows us to alter this map to make a new cofibration. First note that a functor from a category to E is the same as a partition of its objects into two disjoint sets. Thus we may choose a functor which separates x and y. We may form the pushout along this map to get a new cofibration:
At this point we would like to form a retract onto the desired morphism. The problem is that this might not be possible as the image in X of the non-trivial isomorphism in E might fail to be an identity. If that is the case we will not be able to retract onto the desired map.
However cofibrations are also closed under composition. Let be the contractible category with the same objects as X. There is a unique functor
which is the identity on objects. Since it is injective on objects it is a canonical cofibration, hence this map must also be a cofibration. Composing gives us a new cofibration:
and now this retracts onto the desired map. ◊
Whew! That was the hardest part of the proof. Glad that’s over.
So we have learned that if the cofibrations contain more than just the canonical cofibrations, then they also contain , which is then necessarily an acyclic cofibration. This leads us to define the following:
Definition: A category is gaunt if every isomorphism is an identity.
Gaunt categories are what you get when you take category theory and strip away the fleshy meat of topology (in this case 1-types or groupoids). We also have this:
Another Trivial Lemma: If the acyclic cofibrations contain the map , then the fibrant objects are necessarily gaunt.
The proof is just unraveling definitions. We also have a
Trivial Observation: Not every category is equivalent to a gaunt category (e.g. non-trivial groupoids).
But now we see a contradiction emerge. For a model structure, every equivalence class of objects must contain a fibrant representative. If the cofibrations contain more that the canonical cofibrations, then is a cofibration and hence the fibrant objects are gaunt. The equivalence class of, say, a non-trivial groupoid cannot be thus represented. We are thus led to conclude:
Theorem: There is precisely one model structure on the category of categories in which the weak equivalences are the equivalences of categories. It is the canonical model structure.