Waldhausen S-construction for exact categories

Question

Let $\mathcal{C}$ be an exact category. Then, we can consider $\mathcal{C}$ as a Waldhausen category, where the cofibrations are admissible monomorphisms. By Waldhausen $S$-construction we know that $S_n(\mathcal{C})$ is also a Waldhausen category. Does the Waldhausen construction imply that $S_n\mathcal{C}$ is also an exact category? If yes, what are admissible monomorphisms and admissible epimorphisms in $S_n(\mathcal{C})$? If not, is there any known structure on $S_n(\mathcal{C})$ to make it an exact category?

Answer 1

Let me add to Tim's answer that this also holds when suitably defining all of those terms in the higher categorical context, as does Barwick in https://arxiv.org/pdf/1212.5232. Remark also that there is nothing to construct: it is a property of a Waldhausen category to be an exact category by Corollary 4.8.1 of Barwick's paper.

In fact, in 4.9, Barwick gives the conditions for the Waldhausen structure to be exact, and in this case, they should not be too hard to check, using notably the description of the Waldhausen structure of $S_n\mathcal{C}$ given in 5.15, though I reckon Tim's way is probably faster.

Answer 2

Yes, if $\mathcal C$ is a Quillen-exact category, then the Waldhausen category $S_n(\mathcal C)$ is also a Quillen-exact category. The admissible monos are the Waldhausen cofibrations, and the admissible epis are the admissible monos of $S_n(\mathcal C^{op})$ as you'd expect.

To see this, recall that a morphism $f_\bullet : (X_0 \rightarrowtail \cdots \rightarrowtail X_n) \to (Y_0 \rightarrowtail \cdots \rightarrowtail Y_n)$ in $S_n(\mathcal C)$ is a cofibration if $f_1, \dots, f_n$ are cofibrations and every naturality square $(X_n \to X_{n+1}) \to (Y_n \to Y_{n+1})$ has a cofibration comparison map $Y_n \cup_{X_n} X_{n+1} \to Y_{n+1}$. If $\mathcal C$ is Quillen exact, and hence an extension-closed subcategory of an abelian category $\mathcal D$, then this is equivalent to saying that each of these naturality squares is a pullback square in $\mathcal D$, or equivalently that the maps between the cokernels $X_{n+1}/X_n \to Y_{n+1} /Y_n$ are monomorphisms in $\mathcal D$. From this we can see that $f$ is a cofibration iff $f_{ij}: X_{ij} \to Y_{ij}$ is a monomorphism in $\mathcal D$ for each $i\leq j$ (using the full staircase diagram now rather than just the filtration). Dually, $f$ is a cofibration in $S_n (\mathcal C^{op})$ iff $f_{ij}$ is an epimorphism in $\mathcal D$ for each $i \leq j$. Therefore, to see that this gives a Quillen exact structure on $S_n(\mathcal C)$, it will suffice to show that $S_n(\mathcal C)$ is closed under extensions in the abelian category $Fun(Fun([n], [n]) , \mathcal D)$ of all staircase diagrams. If $X_\bullet , Y_\bullet \in S_n(\mathcal C)$ and $Z_\bullet \in Fun(Fun([n], [n]), \mathcal D)$ is an extension, then $Z_{ij} \in \mathcal C$ for all $i \leq j$ simply because $\mathcal C$ is extension-closed in $\mathcal D$. To see that $Z_\bullet$ satisfies the requisite exactness properties to lie in $S_n(\mathcal C)$, I think it suffices to repeatedly use the fact that if you have a 3 x 3 grid in an abelian category and 5 of the 6 sequences appearing are short exact, then so is the remaining one (that's the 9 lemma, right?).