nLab
basic localizer

Definition

As defined by Grothendieck, a basic localizer is a class $W$ of morphisms in Cat such that

1. $W$ contains all identities, satisfies the 2-out-of-3 property and is closed under retracts (in the literature this is sometimes called being weakly saturated),

2. If $A$ has a terminal object, then the functor $A\to 1$ is in $W$, and

3. Given a commutative triangle in $\mathrm{Cat}$:

   $$\begin{array}{ccccc}A& & \stackrel{u}{\to }& & B\\ & {}_v\searrow & & {\swarrow }_w\\ & & C\end{array}$$\array{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C }

   if each induced functor $v/c\to w/c$ between comma categories is in $W$, then $u$ is also in $W$.

The term in French is localisateur fondamental, which is sometimes translated as fundamental localizer.

Examples

• The class of all functors between small categories is, of course, the maximal basic localizer.

• The class of functors inducing an isomorphism on connected components is a basic localizer.

• The class of functors whose nerve is a weak homotopy equivalence is a basic localizer.

• For any derivator $D$, the class of $D$-equivalences is a basic localizer. This includes all the previous examples.

• The class of equivalences of categories is not a basic localizer (it fails the second condition).

Asphericity and local equivalences

If $W$ is a basic localizer, we define the following related classes. We sometimes refer to functors in $W$ as weak equivalences.

• A category $A$ is ($W$-)aspherical if $A\to 1$ is in $W$. Thus the second axiom says exactly that any category with a terminal object is aspherical.
• A functor $u:A\to B$ is ($W$-)aspherical if for all $b\in B$, the comma category $u/b$ is aspherical.
• When the hypotheses of the third axiom are satisfied, we say that $u$ is a local weak equivalence over $C$. Thus the third axiom says exactly that every local weak equivalence is a weak equivalence.

Example

If $W={\pi }_0$-equivalences, then a category is aspherical iff it is connected, and a functor is aspherical iff it is initial.

Example

If $W=$ 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 $A$ is aspherical iff the functor $A\to 1$ is aspherical, since the only comma category involved in the latter assertion is $A$ itself.

• An aspherical functor is a weak equivalence. For if $u:A\to B$ is aspherical, then consider the triangle

  $$\begin{array}{ccccc}A& & \stackrel{u}{\to }& & B\\ & {}_u\searrow & & \swarrow \\ & & B\end{array}$$\array{ A & & \overset{u}{\to} & & B\\ & _u \searrow & & \swarrow \\ & & B }

  The third axiom tells us to consider, for a given $b\in B$, the functor $u/b\to B/b$. But $u/b$ is aspherical by assumption, while $B/b$ is aspherical by the second axiom since it has a terminal object. Thus, by 2-out-of-3, the functor $u/b\to B/b$ is in $W$, and thus by the third axiom $u$ is in $W$.

• If $u$ has a right adjoint, then it is aspherical. For in this case, each category $u/b$ has a terminal object, and thus is aspherical.

• If $I$ denotes the interval category, then for any category $A$ the projection $A\times I\to A$ has a right adjoint, hence is aspherical and thus a weak equivalence. By 2-out-of-3, the two injections $A\rightrightarrows A\times I$ are also weak equivalences, so $A\times I$ is a cylinder object for $W$. It follows that if we have a natural transformation $f\to g$, then $f$ is in $W$ if and only if $g$ is. Moreover, if $f$ is a “homotopy equivalence” in the sense that it has an “inverse” $g$ such that $fg$ and $gf$ are connected to identities by arbitrary natural zigzags, then $f$ is a weak equivalence.

• In particular, any left or right adjoint is a weak equivalence.

Cisinski’s theorem

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 Cisinski.

Theorem (Cisinski)

The class of functors whose nerve is a weak homotopy equivalence is the smallest basic localizer.

Note that this is a larger class than the class of “homotopy equivalences” considered above. For instance, the poset $\mathbb{N}$ has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.

References

• Denis-Charles Cisinski, “Les préfaisceaux comme modèles des types d’homotopie”.