These conditions ensure in particular that a basic localizer always contains the weak equivalences of the Thomason model structure on Cat (see Maltsiniotis 11, 1.2.1), the localization at which is equivalent to the standard homotopy category. On the other hand, the standard equivalences of categories, which are the weak equivalences in the canonical model structure on Cat, do not form a basic localizer.
The definition is due to Grothendieck:
If has a terminal object, then the functor is in , and
Given a commutative triangle in :
if each induced functor between comma categories is in , then is also in .
The term in French is localisateur fondamental, which is sometimes translated as fundamental localizer.
These conditions are enough, I quickly checked this night, in order to validify all results developed so far on test categories, weak test categories?, strict test categories, weak test functors and test functors (with values in ) (of notably the review in par. 44, page 79–88), provided in the case of test functors we restrict to the case of loc. cit. when each of the categories has a final object. All this I believe is justification enough for the definition above.
The class of all functors between small categories is, of course, the maximal basic localizer.
For any derivator , the class of -equivalences is a basic localizer. This includes all the previous examples.
If is a basic localizer, we define the following related classes. We sometimes refer to functors in as weak equivalences.
If -equivalences, then a category is aspherical iff it is connected, and a functor is aspherical iff it is initial.
If nerve equivalences, then a category is aspherical iff its nerve is contractible, and a functor is aspherical iff it is homotopy initial.
We observe the following.
A category is aspherical iff the functor is aspherical, since the only comma category involved in the latter assertion is itself.
An aspherical functor is a weak equivalence. For if is aspherical, then consider the triangle
The third axiom tells us to consider, for a given , the functor . But is aspherical by assumption, while is aspherical by the second axiom since it has a terminal object. Thus, by 2-out-of-3, the functor is in , and thus by the third axiom is in .
If has a right adjoint, then it is aspherical. For in this case, each category has a terminal object, and thus is aspherical.
If denotes the interval category, then for any category the projection has a right adjoint, hence is aspherical and thus a weak equivalence. By 2-out-of-3, the two injections are also weak equivalences, so is a cylinder object for . It follows that if we have a natural transformation , then is in if and only if is. Moreover, if is a “homotopy equivalence” in the sense that it has an “inverse” such that and are connected to identities by arbitrary natural zigzags, then is a weak equivalence.
In particular, any left or right adjoint is a weak equivalence.
It is a non-obvious fact that the notion of basic localizer is self-dual.
A functor is in a basic localizer if and only if is in .
See Proposition 1.2.6 in (Cisinski 04).
Since the definition consists merely of closure conditions, the intersection of any family of basic localizers is again a basic localizer. It follows that there is a unique smallest basic localizer. The following was conjectured by Grothendieck and proven by Denis-Charles Cisinski.
See Théorème 2.2.11 in (Cisinski 04).
Note that this is a larger class than the class of “homotopy equivalences” considered above. For instance, the category generated by the graph
has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.
See also at Cisinski model structure.