I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences.
Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category $\mathsf{Cat}$ of small categories such that :
- (1) it contains the identities,
- (2) it has the 2-out-of-3 properties,
- (3) if $r \colon A \to B$ is a retract of $i \colon B \to A$ (i.e. $ri = \operatorname{id}_B$) and if $ir \in \mathcal W$, then $r \in \mathcal W$ (and so is $i$ also by the previous points),
- (4) if $A$ has a final object, then $A \to e \in \mathcal W$ ($e$ is the final category),
- (5) if $u \colon A \to B$ is a $C$-functor, and if for any $c$ object of $C$ the induced $C/c$-functor$u/c \colon A/c \to B/c$ is in $\mathcal W$, then so is $u$.
In this Cisinski's paper, it is shown that (and I perfectly understand it)
Proposition 1.1.9. Let $\mathcal W$ be a given basic localizer. If there is a natural transformation $u \Rightarrow v$, then $u \in \mathcal W$ if and only if $v \in\mathcal W$.
Then, it gives a corollary :
Corollary 1.1.10. Let $\mathcal W$ be a given basic localizer. For any adjunction $$ u : A \leftrightarrows B : v ,$$ $u$ and $v$ are in $\mathcal W$.
I'm trying to derive the corollary from the proposition without success... Of course, my first thought is to say that the unit and counit of the adjunction gives us (by the proposition and (1)) that $uv$ and $vu$ are in $\mathcal W$. And from there, I'm stuck. I'm pretty sure it is just some abstract non sense deriving from the 2-out-of-3 property but I can't see it.
The 2-out-of-3 property is a bit too weak for this. You can deduce it from the 2-out-of-6 property, but unfortunately that isn't given. Instead, you have to use the notion of asphericity (and axioms 4 and 5): see corollary 7.3.11 and proposition 7.3.12 in my notes.