In Grothendieck's homotopy theory, the category $Cat$ of small categories is used to model spaces, or some localization thereof. Grothendieck gives two sets of axioms which the relevant weak equivalences $\mathcal W \subseteq Mor(Cat)$ might be required to satisfy. They are as follows:
Definition 1: $\mathcal W \subseteq Mor(Cat)$ is a weak basic localizer if
- $\mathcal W$ is weakly saturated;
- For every $C \in Cat$ with a terminal object, the functor $C \to 1$ is in $\mathcal W$;
- If $u: A \to B$ is a functor and for every $b \in B$, the induced functor of slice categories $u/b: A/b \to B/b$ is in $\mathcal W$, then $u \in \mathcal W$.
Definition 2: $\mathcal W \subseteq Mor(Cat)$ is a basic localizer if it is a weak basic localizer and in addition satisfies
3'. If $A \xrightarrow u B \xrightarrow v C$ are functors and for every $c \in C$ the induced functor of slice categories $u/c: A/c \to B/c$ is in $\mathcal W$, then $u \in \mathcal W$.
Notes:
In condition (1), "weakly saturated" means that $\mathcal W$ contains the identity functors, is closed under 2/3, and has the property that if an idempotent $e$ is in $\mathcal W$, then so are the maps splitting the idempotent. In all the important examples, $\mathcal W$ is actually strongly saturated in the sense that it consists of exactly those functors inverted upon passage to the homotopy category $Cat \to Cat[\mathcal W^{-1}]$.
Condition (3) is the case of Condition (3') where $v$ is the identity functor.
There are many examples of basic localizers. The minimal one is the class $\mathcal W$ of weak homotopy equivalences (i.e. functors inducing a weak homotopy equivalence of geometric realizations), and they also include cohomological localizations, a localization coming from the plus construction, etc.
For much of the theory of test categories, one only needs a weak basic localizer, but e.g. to construct the Cisinski model structure on a presheaf category one needs a basic localizer.
Questions:
What is an example of a weak basic localizer which is not a basic localizer?
What is an example of a (weak) basic localizer for which $\mathcal W$ is not strongly saturated?