Example of a weak basic localizer which is not a basic localizer? 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


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*$\mathcal W$ 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?
 A: As for basic localizers: they are all strongly saturated (Prop. 4.2.4 in Astérisque 308). I do not know any weak basic localizer which is not a basic localizer, but it is quite likely that there are such things. Here are some suggestions to determine where to look.
If $W$ is a basic localizer, we may consider the class $A$ of small categories $C$ such that the map to the final category $C\to e$ is in $W$.
There is then:


*

*the smallest basic localizer $W'$ containing maps of the form $C\to e$ for $C$ in $A$.

*the smallest weak basic localizer $W''$ containing maps of the form $C\to e$ for $C$ in $A$.


Here are several useful observations:


*

*Saying that $W$ is accessible is equivalent to the property that there is a cofibrantly generated model structure with the elements of $W$ as weak equivalences. Furthermore, accessible basic localizers correspond to accessible left Bousfield localizations of the homotopy theory of CW-complexes. This is what Thm. 4.2.15 in Astérisque 308 means. 

*If $W$ is accessible, then, with the notations introduced above, $W'=W''$ and $W'$ is accessible as well. Furthermore, the model structures associated to $W'$ (on the category of small categories or on the category of presheaves on any local test category) are proper, and the equality $W=W'$ holds if and only if the model category structures associated to $W$ are proper; i.e. the proper left Bousfield localizations of the homotopy theory of spaces are exactly the nullifications. This is Theorem 6.1.11 and Corollary 6.1.17 in Astérisque 308.


In particular, the smallest weak basic localizer is equal to the smallest basic localizer and thus corresponds to classical homotopy theory; see Thm. 6.1.18 in Astérisque 308.
There is another approach to this which I find enlightening.
Given any bicomplete $\infty$-category $C$ (or Grothendieck derivator $C$) there is an associated basic localizer $W_C$: the class of functors $f:X\to Y$ such that the induced functor
$$f^*:Fun(Y,C)\to Fun(X,C)$$
is fully faithful on constant functors.
There is also an associated weak basic localizer $W'_C$: the class of functors $f:X\to Y$ such that the induced functor
$$f^*:Fun_{lc}(Y,C)\to Fun_{lc}(X,C)$$
is an equivalence of categories, where $Fun_{lc}(Y,C)$ is the full subcategory of $Fun(Y,C)$ which consists of locally constant functors (i.e. functors $Y\to C$ sending all maps in $Y$ to invertible maps in $C$). This is proven in Prop. 1.12 and 1.16 in my paper Locally constant functors. In the case where $C$ is a proper left Bousfield localization of the Thomason model structure on the category of categories, we can see by descent that $Fun_{lc}(Y,C)$ the induced homotopy theory of $Cat/Y$ (i.e. is equivalent to an appropriate homotopy theory of spaces over $Y$) and thus that $W_C=W'_C$. But in general, there is no reason that $W_C$ and $W'_C$ will agree.
All this suggests that if we choose $C$ such that $W_C$ is not proper, there is a good chance that it will not agree with $W'_C$, which would be good starting point towards proving that $W'_C$ is not a basic localizer. I would first try with $C$ the homotopy theory of complexes of vectors spaces over $\mathbf Q$ (the fact that $W_C$ is not proper is Prop. 9.4.6 in Astérisque 308; more generally, Scholie 9.4.7 in loc. cit. is also a fun way to see that the properties of the +-construction show that any Bousfield localization by singular homology with any coefficients is not proper).
