homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 below).
The definition is due to Grothendieck:
A basic localizer is a class $W$ of morphisms in Cat such that
$W$ contains all identities, satisfies the 2-out-of-3 property and, for any functor $i\colon A\to B$ such that there exists a functor $r\colon B\to A$ with $r i=1_A$ and $i r$ in $W$, we have $i$ in $W$ (in the literature this is sometimes called being weakly saturated),
If $A$ has a terminal object, then the functor $A\to 1$ is in $W$, and
Given a commutative triangle in $Cat$:
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.
One can show that basic localizers are stable under retracts; see Proposition 4.2.4 in (Cisinski 06). They are also closed under small filtered colimits; see Corollaire 4.2.22 in (Cisinski 06).
In Pursuing Stacks Grothendieck wrote about def. 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.
Grothendieck’s comments above refer in fact to a weaker definition, in which in Definition , we replace the third condition with an analogue of Quillen’s Theorem A, namely: any functor $f\colon A\to B$ such that, for any object $b$ of $B$, the induced functor $A/b\to 1$ is in $W$, is in $W$. This weaker notion is sometimes called a weak basic localizer.
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. (These are the weak equivalences in the Thomason model structure.)
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). (These are the weak equivalences of the canonical model structure.)
If $W$ is a basic localizer, we define the following related classes. We sometimes refer to functors in $W$ as weak equivalences.
If $W = \pi_0$-equivalences, then a category is aspherical iff it is connected, and a functor is aspherical iff it is initial.
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
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.
It is a non-obvious fact that the notion of basic localizer is self-dual.
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 $W$.
See Proposition 1.1.22 in (Maltsiniotis 05).
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.
The class of functors whose nerve is a weak homotopy equivalence is the smallest basic localizer.
See Théorème 2.2.11 in (Cisinski 04). A completely different proof is given in Corollaire 4.2.19 in (Cisinski 06)
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.
One can in fact prove that the class of functors whose nerve is a weak homotopy equivalence is the smallest weak basic localizer, as defined in Remark ; see Théorème 6.1.18 in (Cisinski 06). This refinement is related to the fact that weak homotopy equivalences are determined by the property that they induce equivalences of categories of locally constant presheaves; see Corollary 1.17 in (Cisinski 09). This is also related to the fact that cohomology with coefficients in local systems determine weak homotopy equivalences; see Théorème 6.5.11 and Scholie 6.5.13 in (Cisinski 06).
See also at Cisinski model structure.
Last revised on May 17, 2021 at 18:43:23. See the history of this page for a list of all contributions to it.