Waldhausen Cats 7: Some Exact Functors

In the last post we proved that $F_1\Ccl$ is a category. Consider now the category $F_1^+\Ccl$ whose objects are pairs $(A\to A’, A’/A)$ where $A\to A’$ is an object of $F_1\Ccl$ and $A’/A$ is a choice of pushout of $*\leftarrow A\to A’$. In other words, we can think of $F_1^+\Ccl$ to be the category of sequences $A\to A’\to A’/A$ where $A’/A$ is the pushout. Perhaps even better, we can think of $F_1^+\Ccl$ as the category whose objects are all commutative squares

that are pushout squares. Here I used the symbol $B$ to denote an arbitrary pushout as there may be more than one choice of pushout. In general I’ll just write $A’/A$ as I’ll just be talking about one choice at a time.

Notice that a morphism $(A\to A’)\to (B\to B’)$ in $F_1\Ccl$ determines a morphism of pushout squares an vice-versa: indeed, given such a morphism, there is a unique morphism $A’/A\to B’/B$. Hence, the functor $F_1^+\Ccl\to F_1\Ccl$ given by sending $A\to A’\to A’/A$ to $A\to A’$ is an equivalence of categories because it is fully faithful and essentially surjective. This equivalence naturally gives $F_1^+\Ccl$ the structure of a category with cofibrations.

There are also functors $s,t,q:F_1^+\Ccl\to\Ccl$ given by:

• $s(A\to A’\to A’/A) = A$
• $t(A\to A’\to A’/A) = A’$
• $q(A\to A’\to A’/A) = A’/A$

We will now see in the rest of the post that these three functors $s,t,q$ are exact functors; in other words, they preserve the structure of a category with cofibrations. In the following proofs, we won’t say anything about the zero object, since that part is trivial. Also, that $s$ is exact is pretty straightforward, but it will be a good warmup for the harder ones.

Finally, the nontrivial and tricky part!

Proof. Suppose $(A\to A’\to A’/A)\to (B\to B’\to B’/B)$ is a cofibration in $F_1^+\Ccl$; in other words, $(A\to A’)\to (B\to B’)$ is a cofibration in $F_1\Ccl$. We need to show that the natural map $A’/A\to B’/B$ is a cofibration back in $\Ccl$. This follows from considering three pushout squares:

The three inner squares are pushouts, so the two big rectangles give the isomorphisms and the required map, being a pushout along $A’\cup_AB\to B’$. (One should compare this proof to the diagram-chasing proof for abelian groups and cofibrations=injections.)

Now we have to show that $q$ takes pushouts to pushouts. Let us again recall the big pushout diagram

Now we already mentioned that this is actually a pushout square, though we didn’t prove it. This comes down to proving that $B’/B\cup_{A’/A}C’/C$ is actually a choice for the quotient $(B’\cup_{A’}C’)/(B\cup_AC)$. To do so, first assume we have a commutative diagram

We would like to show that there exists a unique arrow $(B’\cup_A’ C’)/(B\cup_A C)\to X$ making the entire diagram commute. Do do so, first note that this diagram gives a commutative diagram

which in turn, together with the first diagram gives the commutative diagram

Which gives the desired arrow (I’ve left out a few steps, here, but these are straightforward diagram manipulations. Check them!)

In a few more posts we will be coming to the point where we can define a Waldhausen category, or a category with cofibrations and weak equivalences. It’s probably helpful to look at a rough roadmap of our immediate goals:

1. Define a Waldhausen category
2. Define the $K$-theory of a Waldhausen category
3. Show that this agrees with the “obvious” definition for $K_0$.
4. State the additivity theorem and explain its importance
5. Prove the additivity theorem
6. Deduce various corollaries from the additivity theorem

After these goals, I shall probably go through the rest of Section 1, including explaining how $K$-theory from Waldhausen’s construction gives the same thing as Quillen’s Q-construction for exact categories.