Waldhausen Cats 4: Arrow Categories

Now and in the coming posts we will come to some crucial constructions that give new categories with cofibrations from old ones. The first one is the arrow category construction, that works with any type of category. So, we start out with any category $\mathcal{C}$. We define ${\rm Ar}\Ccl$ to be the category whose objects are morphisms of $\Ccl$ and whose arrows are the appropriate commuting squares. The question is: if $\Ccl$ is a category with cofibrations, how can we make ${\rm Ar}\Ccl$ into a category with cofibrations? There is one choice which seems pretty logical:

Theorem. Suppose $f:A\to A’$ and $g:B\to B’$ are objects of ${\rm Ar}\Ccl$. If we define a morphism $(\varphi:A\to B, \psi:A’\to B’)$ of ${\rm Ar}\Ccl$ from $f$ to $g$ to be a cofibration if $\varphi$ and $\psi$ are cofibrations in $\mathcal{C}$, then ${\rm Ar}\Ccl$ is a category with cofibrations.

Proof. Recall that cofibrations have to be a subcategory (perhaps we could call this the “zeroth axiom” of a category with cofibrations). This holds since composition of morphisms is done pointwise. In fact, this pointwise composition of morphisms also means that isomorphisms are cofibrations, because an isomorphism must necessarily be a pointwise isomorphism. And, if we observe that $*\to *$ is the zero object, then we see that $(*\to *)\to (A\to A’)$ is a cofibration for any $A\to A’$ in ${\rm Ar}\Ccl$.

Now, the last axiom: cofibrations must be stable under cobase change, and that cobase change under a cofibration actually exists!

Here’s what we have to prove: suppose that $(A\to A’)\to (B\to B’)$ is a cofibration, and $(A\to A’)\to (C\to C’)$ is any morphism in ${\rm Ar}\Ccl$. Then we need to prove that the pushout exists, and the morphism from $(C\to C’)$ to the pushout is also a cofibration. The natural guess for the pushout of $(C\to C’)\leftarrow (A\to A’)\to (B\to B’)$ is the pointwise pushout:


The dotted arrow comes from the universal property for the pushout $B\cup_A C$, and it makes the entire cube commute. It’s then easy to see that this is actually a pushout. Once we have done this, we see that the morphisms $C\to B\cup_A C$ and $C’\to B’\cup_{A’}C’$ are cofibrations since $\Ccl$ is a category with cofibrations. So, by definition, the map $(C\to C’)\to (B\cup_A C\to B’\cup_{A’}C’)$ is a cofibration in ${\rm Ar}\Ccl$.

However, that was just a warmup! As a preview for what’s to come, let $F_1\Ccl$ be the full subcategory of ${\rm Ar}\Ccl$ whose objects are cofibrations. We make $F_1\Ccl$ into a category with cofibrations as follows: we declare map $(A\to A’)\to (B\to B’)$ to be a cofibration if $A\to B$ is a cofibration, and $A’\cup_A B\to B’$ is a cofibration. Then $F_1\Ccl$ is a category with cofibrations. Proving this is the content of the next post!

(Note that I’ll always use the convention $A\to A’$ for an object of the arrow category rather than $A\to B$ as in Waldhausen’s paper. This is for good reason: it avoids the use of double primes, or even triple primes. Isn’t it easier to read $A\to A’, B\to B’, C\to C’,\dots$ instead of $A\to B, A’\to B’, A”\to B”,\dots$?)

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