Waldhausen Cats, 2: Exact Categories and Cobase Change

In the last post, we introduced the notion of an exact category. Let’s recall that an exact category is an additive category $\Ccl$ with a distinguished class of sequences $\Ecl$ of the form $0\to A\to B\to C\to 0$ such that there is a fully faithful embedding of $\Ccl$ into an abelian category satisfying the two axioms:

  1. $\Ecl$ is exactly the set of all exact sequence in $\Acl$ that are contained in $\Ccl$
  2. (Closed Under Extensions) If $0\to A\to B\to C\to 0$ is an exact sequence in $\Acl$ with $A$ and $C$ in $\Ccl$ then $B$ is isomorphic to some object in $\Ccl$.

Moreover, an admissible monic in $\Ccl$ is a monic $A\to B$ appearing in some distinguished short exact sequence $0\to A\to B\to C\to 0$.

We also introduced the notion of a category with cofibrations (see the previous post for definitions). We claimed that an exact category becomes a category with cofibrations if we set the cofibrations as the admissible monics. We started a proof there, and left unfinished showing the following two facts: that admissible monics are closed under composition and the cobase change axiom. Let’s start with admissible monics:

Theorem. In an exact category $\mathcal{C}$, the admissible monics are closed under composition.
Proof. Let $A\to B$ and $B\to D$ be admissible monics, so that we have two sequences $0\to A\to B\to C\to 0$ and $0\to B\to D\to E\to 0$ in $\mathcal{E}$. We want to show that the composition $A\to B\to D$ is also in $\mathcal{E}$. In the ambient abelian category, there is an exact sequence $0\to A\to D\to D/A\to 0$. So, to finish the proof, we must show that $D/A$ is isomorphic to an object in $\mathcal{C}$. We do know that in $\mathcal{C}$, we have $C$, which is a representative for $B/A$. In the ambient abelian category, $0\to B/A\to D/A\to E\to 0$ is exact, and since both $B/A$ and $E$ are in $\mathcal{C}$, there is a representative for $D/A$ too, because $\mathcal{C}$ by definition is closed under extensions.

Theorem.Let $\Ccl$ be an exact category. If $A\rightarrowtail B$ is an admissible monomorphism and $A\to D$ is any map then the pushout $B\cup_AD$ exists in $\Ccl$ and the map $D\to B\cup_AD$ is an admissible monomorphism.
Proof. The arguments here are similar to the previous. Let $0\to A\to B\to C\to 0$ be a distinguished exact sequence, and $A\to D$ any map. We need to show that the pushout $B\cup_AD$ of $D\leftarrow A\to B$ exists in $\mathcal{C}$, and that $D\to B\cup_AD$ is an admissible monic. To do this, note that in the ambient abelian category, $0\to D\to B\cup_AD\to C\to 0$ is exact, where the map $B\cup_AD\to C$ is the map induced by $B\to C$. Since $D$ and $C$ are both in $\mathcal{C}$, we see that the pushout is there too, and thus $D\to B\cup_AD$ is an admissible monic.


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