category theory

# Basic localizers

## Idea

By a basic localizer one means a localizer on the category Cat of categories, hence a choice of a class of functors to be called the weak equivalences, subject to some conditions.

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.

Hence basic localizers are a tool for homotopy theory modeled on category theory. In fact, their introduction by Grothendieck was motivated from the study of test categories (see remark 2 below).

## Definition

The definition is due to Grothendieck:

###### Definition

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 $Cat$:

$\array{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C }$

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

###### Remark

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

###### Remark

In Pursuing Stacks Grothendieck wrote about def. 1 the following:

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 $(Cat)$) (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 $i(a)$ has a final object. All this I believe is justification enough for the definition above.

## Properties

### 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 $u\colon A\to B$ 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 $u\colon A\to B$ is aspherical, then consider the triangle

$\array{ A & & \overset{u}{\to} & & B\\ & _u \searrow & & \swarrow \\ & & B }$

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 $f g$ and $g f$ 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.

### Self-duality

It is a non-obvious fact that the notion of basic localizer is self-dual.

###### Theorem

A functor $u : A \to B$ is in a basic localizer $W$ if and only if $u^{op} : A^{op} \to B^{op}$ is in $W$.

###### Proof

See Proposition 1.2.6 in (Cisinski 04).

### 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 Denis-Charles Cisinski.

###### Theorem (Cisinski)

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

###### Proof

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

$\bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \cdots$

has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.