Waldhausen Cats 3: Exact Functors

We say that a functor $F:\Ccl\to\Ccl’$ between categories with cofibrations is exact if it satisfies:

  1. $F(*) = *$, where the right-hand side is the chosen zero object of $\Ccl’$.
  2. $F(f)$ is a cofibration in $\Ccl’$ whenever $f$ is a cofibration in $\Ccl$.
  3. $F$ takes pushouts along cofibrations to pushouts along cofibrations.

In short, an exact functor is one which preserves the additional structure defining a category with cofibrations. If $\Ccl$ and $\Ccl’$ happen to be abelian categories (cofibrations = monomorphisms), then an exact functor in the sense of a category with cofibrations is an exact functor in the usual sense (i.e. it preserves short exact sequences). This is because an exact functor as we have just defined it has to preserve the pushout diagram resulting from the pushout of $*\leftarrow A\rightarrow B$, the pushout of which is $B/A$ (together with the appropriate maps). And, the same reasoning shows that if $\Ccl$ and $\Ccl’$ are exact categories (cofibrations = admissible monics), then an exact functor as we have defined it is an exact functor there too.

There is another notion which is a little subtle: we say that $\Bcl\subseteq \Ccl$ is a subcategory with cofibrations if the inclusion functor is exact, and if whenever $f$ is an arrow in $\Bcl$ such that $f$ is a cofibration in $\Ccl$ and the quotient exists in $\Bcl$, then $f$ is a cofibration in $\Bcl$.

It might not be immediately apparent that this condition is not automatically satisfied for subcategory $\Bcl$ that is also a category with cofibrations. However, this condition is imposed on the definition to avoid some strange cases where the subcategory doesn’t have enough cofibrations. To make this phenomenon less mysterious, let’s look at a concrete example. Take $\Ccl$ to be the category of abelian groups. Then we shall define on $\Ccl$ two different structures of a category with cofibrations, which we will denote $\Ccl_1$ and $\Ccl_2$.

Once we have defined them, it will become apparent that the self-inclusion $\Ccl_1\subseteq\Ccl_2$ does not satisfy the definition of ‘subcategory with cofibrations’, because $\Ccl_1$ won’t have enough cofibrations. So, define $\Ccl_2$ to be the abelian category of abelian groups whose cofibrations are all monics. On the other hand, define $\Ccl_1$ to be the same category but whose cofibrations consist of all the monomorphisms appearing in split short exact sequences; i.e. those isomorphic to $0\to A\to A\oplus B\to B\to 0$, with the obvious maps. So in the language of exact categories, the admissible monics are inclusions $A\to A\oplus B$ given by $a\mapsto (a,0)$. Of course, $\Ccl_1$ still has the inclusion $\Z\mapsto \Z$ given by multiplication by $n > 1$, but this morphism is no longer a cofibration because the corresponding exact sequence $0\to \Z\to\Z\to\Z/n\to 0$ is not split!

The slightly strange structure we’ve put on $\Ccl_1$ in the language of exact categories is called the split exact structure, and is indeed an exact category. However, we don’t have to verify this because it’s trivial verify this class of morphisms also gives $\Ccl_1$ the structure of a category with cofibrations. Even though the inclusion functor $\Ccl_1\to\Ccl_2$ is exact, it’s not considered a subcategory with cofibrations, simply because $\Ccl_1$ doesn’t contain all the cofibrations you’d expect from something called a ‘subcategory-with-some-structure’!

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