WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES
DANIEL DUGGER AND DANIEL C. ISAKSEN
Abstract.Weak equivalences of simplicial presheaves are usually defined *
*in
terms of sheaves of homotopy groups. We give another characterization us-
ing relative-homotopy-liftings, and develop the tools necessary to prove*
* that
this agrees with the usual definition. From our lifting criteria we are *
*able to
prove some foundational (but new) results about the local homotopy theory
of simplicial presheaves.
1.Introduction
In developing the homotopy theory of simplicial sheaves or presheaves, the us*
*ual
way to define weak equivalences is to require that a map induce isomorphisms on
all sheaves of homotopy groups. This is a natural generalization of the situati*
*on
for topological spaces, but the `sheaves of homotopy groups' machinery (see Def-
inition 6.6) can feel like a bit of a mouthful. The purpose of this paper is to
unravel this definition, giving a fairly concrete characterization in terms of *
*lift-
ing properties_the kind of thing which feels more familiar and comfortable to t*
*he
ingenuous homotopy theorist.
The original idea came to us via a passing remark of Jeff Smith's: He pointed
out that a map of spaces X ! Y induces an isomorphism on homotopy groups if
and only if every diagram
(1.1) Sn-1 ___//_X66______
xx--fflffl__|__________
xxx___|_______|__
--xx___fflffl|_fflffl|_____
Dn Dn ____//_Y66______
fflfflx--x____________
| xxx____________
fflffl|--xx_______
Dn+1
admits liftings as shown (for every n 0, where by convention we set S-1 = ;).
Here the maps Sn-1 ,! Dn are both the boundary inclusion, whereas the two maps
Dn ,! Dn+1 in the diagram are the two inclusions of the surface hemispheres of
Dn+1. The map Dn+1 ! Y should be thought of as giving a homotopy between
the two maps Dn ! Y relative to Sn-1. In essence, the above lifting condition j*
*ust
guarantees the vanishing of the relative homotopy groups of X ! Y .
One advantage of this formulation is that one doesn't have to worry about bas*
*e-
points, but it also has other conveniences. If one looks back on the classical *
*lifting
theorems in [Sp], for instance, it is really the above property_rather than the*
* iso-
morphism on homotopy groups_which is being made use of over and over again.
In working with simplicial presheaves, it eventually became clear that a versio*
*n of
the above characterization was a useful thing to have around. It comes in at se*
*veral
points in [DHI ], where it is used to inductively produce liftings much like in*
* [Sp].
1
2 DANIEL DUGGER AND DANIEL C. ISAKSEN
Whereas for topological spaces the above characterization is `obvious', for s*
*im-
plicial presheaves it requires a little bit of work. Intuitively the result is *
*clear, but
to actually write down a proof one must (a) struggle with the combinatorics of
simplicial sets, and (b) deal with the `local homotopy theory' which demands th*
*at
everything be accomplished by a finite number of lifting arguments. The trouble
is that the modern way of avoiding (a) is to use the tools of model categories,*
* but
because of (b) we don't have these at our disposal. In this paper we develop so*
*me
basic machinery for handling this situation, so that in the end one can write o*
*ut
the proof fairly smoothly.
To describe the results more explicitly, we'll first give an analagous charac*
*teriza-
tion for weak equivalences X ! Y of simplicial sets. In this case, we must assu*
*me
that X and Y are fibrant. The (n - 1)-sphere is replaced by @ n; the n-disks are
replaced by n; and Dn+1 is replaced by the pushout RH( n, @ n) of the diagram
@ n @ n x 1 ! n x 1.
The simplicial set RH( n, @ n) is the domain of simplicial homotopies_relative
to @ n_between maps out of n. Once these substitutions are made into diagram
(1.1), one gets the same criterion for the map X ! Y to be a weak equivalence of
simplicial sets. See Proposition 4.1.
The generalization to the case of simplicial presheaves is now reasonably tra*
*ns-
parent. The main result of the paper is the following:
Theorem 1.2. Let C be a Grothendieck site, and let F ! G be a map between
locally fibrant simplicial presheaves. Then F ! G induces an isomorphism on all
sheaves of homotopy groups (for all choices of basepoint) if and only if it has*
* the
following property: for every solid-arrow diagram
(1.3) @ n X ____//F33__________
mmmmvv__fflffl___|______________
mmmm_______|________|__
vvmmm_________fflffl|__fflffl|
n X n X ____//_G33__________
fflffl mmmvvm__________________
| mmmmm_________________
fflffl|vvmm________
RH( n, @ n) X
in which X is representable, there exists a covering sieve of X such that for a*
*ny
U ! X in the sieve, the diagram obtained by pulling back to U has liftings as
shown.
(The dotted arrows in the above diagram are called `relative-homotopy-lifting*
*s',
and the fact that they only exist locally leads us to call this property the `l*
*ocal
RHLP'_see Defintion 6.11, as well as the general discussion in Section 3).
The advantage of this viewpoint on weak equivalences is that it provides a fr*
*ame-
work for using lifting arguments instead of computations of homotopy groups. For
questions in homotopy theory, lifting is sometimes a more convenient tool. In o*
*ur
case, this lifting characterization will be used to give elegant proofs of vari*
*ous results
(some old, some new) about local weak equivalences and local-fibrations. These *
*are
given in Section 7. To readers of [J1, J3], these come as no surprise. However,*
* our
proofs seem simpler and more conceptual than the ones involving sheaves of homo-
topy groups, in particular avoiding all references to stalks or Boolean localiz*
*ations.
And in some cases we don't know any proof other than via the lifting criterion.
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 3
We should remark that this approach via liftings is not at all meant to repla*
*ce
the definition involving sheaves of homotopy groups_in some situations that is
exactly the tool that is needed. But in general it is good to have both descrip*
*tions
at one's disposal.
The chief motivation for writing this paper was its application to our study *
*of lo-
calization for simplicial presheaves [DHI ]. But we've also found that the tech*
*niques
of homotopy-liftings are convenient tools to have around, and should be more we*
*ll-
known among abstract homotopy theorists. Reedy [R ] worked with a dual version
of these lifting criterion in the context of abstract model categories, and use*
*d them
to prove several key lemmas. We have reproduced a couple of his proofs here for
completeness, and with the goal of popularizing these ideas.
1.4. Organization of the paper.
Sections 2-5 deal only with simplicial sets. Section 2 has a few background r*
*e-
sults, and then in Section 3 we define relative-homotopy-liftings and develop t*
*heir
basic properties. In Section 4 these ideas are applied to get a lifting criteri*
*on for
weak equivalences of simplicial sets. Unfortunately, one of the key steps is ea*
*siest
to prove using model-category theoretic methods, and these do not generalize to
the simplicial presheaf setting. So Section 5 is devoted to giving a completely*
* com-
binatorial proof for this result. We finally get to simplicial presheaves in Se*
*ction 6.
We recall the traditional definition of weak equivalence using sheaves of homot*
*opy
groups, and then prove Theorem 1.2. Section 7 concludes with some applications
of this theorem.
We assume that the reader is familiar with standard results from the homotopy
theory of simplicial sets, including material to be found in [M ] or [GJ ]. Muc*
*h of
what we discuss, especially in Sections 3 and 4, can be easily generalized to a*
*bstract
model categories, but we do not treat this extra generality here. We'll also as*
*sume
a familiarity with sheaf theory and the homotopy theory of simplicial presheave*
*s,
for which we refer the reader to [J1]. It should be clear from our arguments how
indebted we are to that paper.
2. Background on simplicial sets
We start with some basic facts. Let Sn be the sphere n=@ n. If (K, x) is a
pointed simplicial set, then ßn(K, x) denotes the set of maps ( n, @ n) ! (K, x)
modulo the equivalence relation generated by simplicial homotopy relative to @ *
*n.
Of course this set has homotopical meaning only if K is fibrant.
In [K ], Kan constructed a fibrant-replacement functor called Ex1 . First, l*
*et
sdbe the barycentric subdivision functor [GJ , p. 183]. For any simplicial set *
*X,
ExX is the simplicial set whose k-simplices are elements of the set Hom (sd k, *
*X).
The functor Ex is right adjoint to sd. Now Exn is the n-fold composition of Ex,
and Ex1 is colimnExn. The functor Ex1 has some nice properties one wouldn't
expect from an arbitrary fibrant-replacement functor: It preserves fibre-produc*
*ts,
it preserves the set of 0-simplices, and it preserves fibrations. These propert*
*ies all
follow immediately from the definition.
The following two basic lemmas about simplicial sets will be used later. The
first is obvious, but its statement and proof becomes important when considering
simplicial presheaves later. The point is that the proof uses only basic lifti*
*ng
properties, not fancy model theoretic results.
4 DANIEL DUGGER AND DANIEL C. ISAKSEN
Lemma 2.1. Let i: K ,! L be a cofibration, and let X be a fibrant simplicial s*
*et. If
f :K ! X factors through any contractible simplicial set M, then f is simplicia*
*lly
null-homotopic and f extends over i.
Proof.For any simplicial set Y , let Cone Y be (Y x 1)=(Y x {1}). We have a
diagram
K _________//_M______//X::____
| | ________
| | _______
fflffl| fflffl|____
Cone K _____//ConeM.
The map M ! ConeM is an acyclic cofibration because M is contractible, so there
is a lift as shown. Composition with ConeK ! ConeM gives the desired simplicial
null-homotopy.
Now we have a diagram
K _____//ConeK_____//X::___
| | ________
| | _______
fflffl| fflffl|____
L _____//ConeL.
The map Cone K ! Cone L is an acyclic cofibration, so there is a lift as shown._
Composition with L ! ConeL gives the desired extension over i. |__|
Note that if K, L, and M are finite simplicial sets, then the desired lift ca*
*n be
produced using only finitely many applications of the Kan extension condition f*
*or
X. This will be important when we start generalizing to simplicial presheaves.
Lemma 2.2. Let i : K ! L be an acyclic cofibration between finite simplicial s*
*ets.
Then i can be built from the maps n,k,! n by a finite number of retracts, cob*
*ase
changes, and compositions.
Proof.We know that i is a retract of a relative J-cell complex j : M ! N (cf. [*
*H ,
Def. 12.5.8]), where J is the set of maps of the form n,k,! n. Since L is fin*
*ite,
its image in N belongs to a finite subcomplex. Thus i is actually a retract_of*
*_a
finite relative J-cell complex. |__|
3. Generalities about homotopy-liftings
This section establishes the definition and basic properties of what we call
`relative-homotopy-liftings'.
Definition 3.1.A square of simplicial sets
(3.2) K _____//X
| |
| |
fflffl|fflffl|
L _____//Y
is said to have a relative-homotopy-lifting if there exists a map L ! X such
that the upper left triangle commutes and there is a simplicial homotopy relati*
*ve to
K from the composition L ! X ! Y to the given map L ! Y .
The map X ! Y has the relative-homotopy-lifting property (RHLP) with
respect to K ! L if every square (3.2) has a relative-homotopy-lifting.
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 5
Reedy [R , Lem. 2.1] used the dual to the above definition. Like him, we could
haved defined this property in an arbitrary simplicial model category (one prob*
*ably
doesn't even need the model category to be simplicial). All of our basic result*
*s go
through in that generality, but we won't ever need this.
Given K ! L, let RH(L, K) denote the pushout of the diagram
K -ß K x 1 -! L x 1
where the left map is the projection. The notation stands for `Relative-Homotop*
*y':
To give a map RH(L, K) ! X means precisely to give two maps L ! X which
agree on K, together with a simplicial homotopy between them relative to K. Note
that there is a canonical map L qK L ! RH(L, K). This map is a cobase change
of the map
(K x 1) [ (L x @ 1) ! L x 1,
so it is a cofibration if K ! L is.
We will sometimes use the fact that the existence of relative-homotopy-liftin*
*gs
can be rephrased as saying that the diagram
(3.3) uK ___//_X66________
uuu_|_____|______
uuu___|_____|_
zzuuu____fflffl|fflffl|_______
L vL ____//Y66_______
| vvv____________
| vvv_________
fflffl|--vv______
RH(L, K)
admits liftings as shown. While this diagram may seem somewhat awkward (espe-
cially when seeing it for the first time), it is often a very useful tool.
Here are some basic properties of relative-homotopy-liftings:
Lemma 3.4. Let f :X ! Y be a fixed map of simplicial sets. Consider the class
of all maps K ! L with respect to which f has the RHLP. This class is closed
under cobase changes and retracts. If Y is fibrant, then the composition of t*
*wo
cofibrations in the class is still in the class.
Proof.Closure under cobase changes follows from consideration of (3.3) and the *
*fact
that RH(L qK M, M) is isomorphic to RH(L, K) qK M. Closure under retracts
follows from the usual formal argument with lifting properties.
For composition, we start with two cofibrations i: K ! L and j :L ! M in the
class. Consider a lifting problem
g
K _____//X
ji|| |f|
fflffl|fflffl|
M _h__//_Y.
The first step is to produce a homotopy-lifting
g
K _____//X>>"
""
i||l"" |f|
fflffl|fflffl|""
L __hj_//Y
6 DANIEL DUGGER AND DANIEL C. ISAKSEN
relative to K. Let H : RH(L, K) ! Y be the relative-homotopy from fl to hj. So
now we look at the diagram
(M x {1}) qLx{1}RH(L, K) hqH_//_Y66______
fflffl _____________
~ | ____________
fflffl|__J______
RH(M, K)
and produce a lifting J using that Y is fibrant. Note that the vertical map abo*
*ve
is an acyclic cofibration because it is a cobase change of the acyclic cofibrat*
*ion
(M x {1}) [ (L x 1) ! M x 1.
Let m be the map J|Mx{0} . Note that m is simplicially homotopic to h relative *
*to
K.
At this point we have the square
L __l__//X
j || |f|
fflffl|fflffl|m
M _____//Y.
We produce a relative-homotopy-lifting n. A diagram chase shows that nji equals
g. On the other hand, fn is simplicially homotopic to m relative to K and hence*
* __
also to h; here we use that relative-homotopy is transitive because Y is fibran*
*t. |__|
Corollary 3.5.If f :X ! Y has the RHLP with respect to the maps @ n ,!
n for all n 0 and Y is fibrant, then p also has the RHLP with respect to all
cofibrations K ,! L of finite simplicial sets.
Proof.Every such cofibration K ,! L can be constructed by a finite number of __
compositions and cobase changes from the generating cofibrations @ n ,! n. |_*
*_|
Proposition 3.6.Suppose that K ,! L is a cofibration and we are given a square
S of the form
HK
K x 1 _____//X
| |
| |f
fflffl|HLfflffl|
L x 1_____//Y
in which X and Y are fibrant. Let S0 denote the square obtained by restricting *
*HK
and HL to time t = 0, and similarly for S1. Then S0 has a relative-homotopy-lif*
*ting
if and only if S1 does.
Proof.Suppose that S0 has a relative-homotopy-lifting l0. Then by gluing l0 to *
*HK
we get (L x {0}) [ (K x 1) ! X, and since X is fibrant this map extends over
L x 1. Let l1 denote the restriction of this map to L x {1}; we will show that*
* l1
is the desired relative-homotopy-lifting.
Pushing the homotopy L x 1 ! X down into Y , we can glue it to HL to get a
map defined on (L qK L) x 1. Together with the relative-homotopy from fl0 to
HL|t=0, we find that we actually have a map [RH(L, K)x{0}][[(LqK L)x 1] ! Y .
Since Y is fibrant, this extends over RH(L, K) x 1. Restricting to time t = 1_
gives the desired relative-homotopy of fl1 with HL|t=1. |_*
*_|
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 7
Proposition 3.7.Let f :X ! Y be a map between fibrant simplicial sets. Then
f has the RHLP with respect to every acyclic cofibration K ,! L. In particular,*
* f
has the RHLP with respect to the maps n,k,! n.
Proof.Consider a square
g
K ____//_X
| |
| f|
fflffl|fflffl|
L __h_//_Y.
Using that X is fibrant, there is a map l :L ! X extending K ! X. We must give
a relative-homotopy from fl to h.
The cofibration L qK L ,! RH(L, K) is an acyclic cofibration because it is a
cobase change of the acyclic cofibration (L x @ 1) [ (K x 1) ,! L x 1. We may
extend the map fl q h : L qK L ! Y to RH(L, K) ! Y because Y is fibrant. This_
gives us the desired relative-homotopy. |__|
Observe that if K ! L is n,k! n then the proof only requires a finite number
of uses of the Kan extension condition. Combined with Lemma 2.2, this tells us
that the proposition applies to the simplicial presheaf setting whenever K and L
are finite simplicial sets. This won't be needed until Section 6.
4.Weak equivalences of simplicial sets
The following proposition now shows that weak equivalences between fibrant
simplicial sets can be detected using relative-homotopy-liftings. This is analo*
*gous
to the situation discussed in the introduction for topological spaces, where ev*
*ery
object is fibrant.
Proposition 4.1.A map f :X ! Y between fibrant simplicial sets is a weak
equivalence if and only if it has the RHLP with respect to the maps @ n ,! n,
for all n 0.
When the simplicial sets are not fibrant one has to allow oneself to subdivide
@ n and n, but we won't pursue this.
The following proof is similar to the proof of [R , Lem. 2.1]. The differenc*
*e is
that we only consider the RHLP with respect to the generating cofibrations, whi*
*le
Reedy considers the RHLP with respect to all cofibrations. We include the full
details for completeness.
Proof.First suppose that f has the RHLP. By Corollary 3.5, f has the RHLP with
respect to the cofibrations * ,! Sn for all n 1, as well as ; ! *. This shows*
* that
ßnX ! ßnY is surjective (for any choice of basepoint). Similarly, f has the RHLP
with respect to the cofibrations Sn _ Sn ,! RH(Sn, *) for all n. This shows that
ßnX ! ßnY is injective.
Conversely, we'll now suppose that f is a weak equivalence. Consider a square
@ n _____//X
fflffl |
| |
fflffl| fflffl|
n _____//Y.
8 DANIEL DUGGER AND DANIEL C. ISAKSEN
Factor f into an acyclic cofibration i: X ,! Z followed by an acyclic fibration
p: Z ! Y . Since X is fibrant, there is a map g :Z ! X making X a retract of Z.
Now the square
@ n ____//_X_i_//_Z
fflffl |
| p|
fflffl| fflffl|
n ____________//_Y
has a lift h because p is an acyclic fibration.
The composition gh is the desired homotopy-lift. Using that gi is the identit*
*y,
the upper left triangle commutes. Working in the undercategory @ n # sSet, we
see that ig represents the same map as idZ in the homotopy category_therefore
pigh represents the same map as ph in the homotopy category. But these latter
two maps have cofibrant domain and fibrant target, so they are actually simplic*
*ially
homotopic in @ n # sSet. The simplicial set RH( n, @ n) is precisely a cylinder
object for n in this undercategory, so pigh and ph are simplicially homotopic_
relative to @ n. |__|
Reedy [R , Th. B] showed that base changes along fibrations preserve weak equ*
*iv-
alences between fibrant objects. His proof used the criterion of Proposition 4*
*.1
(suitably generalized to arbitrary model categories) to detect weak equivalence*
*s.
We use this idea to obtain the following elementary proof of right properness f*
*or
simplicial sets_most standard references [GJ , H] use topological spaces to pro*
*ve
this. We include full details because this same proof will be applied to the ca*
*se of
simplicial presheaves.
Corollary 4.2 (Right properness).Let f : X ! Y be a weak equivalence of sim-
plicial sets, and let p : Z ! Y be a fibration. Then the map X xY Z ! Z is also*
* a
weak equivalence.
Proof.We need only show that Ex1 (XxY Z) ! Ex1 Z is a weak equivalence. Note
that Ex1 commutes with fibre-products and preserves fibrations, so this map is*
* a
base change of the weak equivalence Ex1 f along the fibration Ex1 p. Therefore,
we may assume that X, Y and Z are already fibrant.
The rest of the proof is the same as Reedy's argument. Suppose given a square
g
@ n ____//_X xY Z
fflffl |
| |
fflffl| fflffl|
n ___h____//Z.
We want to find a relative-homotopy-lifting for this square. First, take a rela*
*tive-
homotopy-lifting l for the composite square
g
@ n ____//_X xY Z____//X
fflffl |
| |f
fflffl| fflffl|
n ___h____//Z___p___//_Y,
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 9
which exists by Proposition 4.1 because X ! Y is a weak equivalence between
fibrant simplicial sets. Now consider the square
n x {1}___h___//Z
fflffl |
| |
fflffl| fflffl|
RH( n, @ n) _____//Y,
where the bottom horizontal map is the relative-homotopy from fl to ph. This
square has a lift H because the left vertical arrow is an acyclic cofibration. *
*Let H0
and H1 be the restrictions of H to n x {0} and n x {1} respectively. Note that
H1 = h and pH0 = fl.
The maps l and H0 together define a map m : n ! X xY Z. A diagram
chase shows that g is the restriction of m to @ n, and H is the necessary_relat*
*ive-
homotopy. |__|
5.A combinatorial proof
In the previous section, Proposition 4.1 compared weak equivalences between
fibrant simplicial sets to maps that have the RHLP with respect to the cofibrat*
*ions
@ n ,! n. Unfortunately, the proof of one implication of the proposition relied
on model category theoretic methods. When we generalize to simplicial presheaves
later on, these methods are not at our disposal. Thus, our goal in this section*
* is to
show by purely combinatorial methods that if f is a weak equivalence then it has
the RHLP with respect to the maps @ n ,! n.
Throughout this section f :X ! Y denotes a map between fibrant simplicial
sets.
First note that surjectivity on homotopy groups says precisely that f has the
RHLP with respect to the maps * ! Sn for all n 1 as well as the map ; ! *.
Using this, we have:
Lemma 5.1. If f :X ! Y is a map between fibrant simplicial sets that induces
surjections on homotopy groups, then f has the RHLP with respect to the maps
n,k,! @ n, for any n 1.
Proof.Suppose given a square
n,k_____//X
fflffl |
| |
fflffl| fflffl|
@ n _____//Y.
Since n,kis contractible and X is fibrant, the map n,k! X is simplicially nul*
*l-
homotopic (by Lemma 2.1)_choose a null-homotopy. By composing with X ! Y ,
we also get a null-homotopy for the composition n,k ! Y ; so we have a map
(@ n x {0}) [ ( n,kx 1) ! Y that is constant on n,kx {1}. Because Y is
fibrant, this map extends to a map @ n x 1 ! Y that is constant on n,kx {1}.
10 DANIEL DUGGER AND DANIEL C. ISAKSEN
We have constructed a homotopy (in the sense of Proposition 3.6) between the
original square and a square of the form
n,k________//_*_______//_X
fflffl fflffl |
| | |
fflffl| fflffl| fflffl|
@ n _____//@ n= n,k____//Y.
By Proposition 3.6, we need only construct a relative-homotopy-lifting for this*
* new
square. The left square is a pushout, so we need only construct a relative-homo*
*topy-
lifting for the right-hand square by Lemma 3.4.
Note that @ n= n,kis isomorphic to Sn-1. Therefore, f has the RHLP with
respect to * ! @ n= n,kbecause f induces a surjection on (n - 1)st homotopy_
groups. |__|
Theorem 5.2. If f :X ! Y is a weak equivalence between fibrant simplicial sets,
then it has the RHLP with respect to the maps @ n ,! n, for all n 0.
Proof.Surjectivity on ß0 immediately gives the result for n = 0. So suppose n *
* 1
and we have a lifting diagram
g
@ n _____//X
fflffl |
| |
fflffl|hfflffl|
n _____//Y.
Routine lifting arguments show that there is a simplicial homotopy @ n x 1 ! X
between g and a map that factors through @ n= n,n. As in the proof of Lemma
5.1, we can extend this to a simplicial homotopy n x 1 ! Y , and we are reduced
to producing a relative-homotopy-lifting for a square of the form
@ n= n,n __g_//_X
fflffl |
| |
fflffl|h fflffl|
n= n,n _____//Y.
Note that @ n= n,nis isomorphic to Sn-1 and n= n,nis contractible. Lemma
2.1 shows that g :Sn-1 ! X becomes null in ßn-1(Y ). Since f is injective on
homotopy groups, g is simplicially null-homotopic. Therefore, g extends to a map
l : n= n,n! X by Lemma 2.1.
Define a map H :@ n+1 ! Y by making the (n + 1)st face equal to h, the nth
face equal to fl, and all the other faces equal to the basepoint *. Similarly, *
*define
J : n+1,n+1! X by making the nth face equal to l and all the other faces equal
to the basepoint. So we have a square
_J___//
n+1,n+1 X
fflffl |
| |
fflffl| fflffl|
@ n+1 _H____//Y,
and by Lemma 5.1 this has a relative-homotopy-lifting m. The (n + 1)st face of *
*m __
gives a map n ! X that is a relative-homotopy-lifting for our original square.*
* |__|
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 11
6.Local weak equivalences of simplicial presheaves
In this section we prove the main theorem (stated here as Theorem 6.15). We
start by recalling some of the tools from [J1]: the use of local lifting proper*
*ties
and sheaves of homotopy groups. Then we set up the local version of relative-
homotopy-liftings, and observe that everything we've done so far still works in*
* this
setting.
6.1. Local-liftings. Fix a Grothendieck site C. Recall that a map of simplicial
presheaves F ! G is a local-fibration if it has the following property: given a*
*ny
square
(6.2) n,k X ____//_F
fflffl |
| |
fflffl| fflffl|
n X _____//G
in which X is representable, there exists a covering sieve of X such that for a*
*ny
map U ! X in the sieve, the induced diagram
(6.3) n,k U ____//_ n,k X____//F44__________
fflffl ___________|______
| _______________ |
fflffl|_________ fflffl|
n U ______////_ n __X__//G
has a lifting as shown. We are not requiring that the liftings for different U'*
*s be
compatible in any way, only that they exist. This kind of `local lifting proper*
*ty'
will appear often in the course of the paper, so we adopt the following convent*
*ion:
Convention 6.4. Suppose given a lifting diagram like (6.2), in which a repre-
sentable presheaf X appears. We say this diagram has local liftings if there ex*
*ists
a covering sieve R of X such that for any U ! X in R the diagram obtained by
pulling back to U admits liftings. For instance, using this language, a map F !*
* G
is a local-fibration provided that every diagram (6.2) admits local liftings.
Because of Lemma 2.2, a map is a local-fibration if and only if it has the lo*
*cal
right lifting property with respect to all maps K X ,! L X for every acyclic
cofibration K ,! L between finite simplicial sets.
6.5. Sheaves of homotopy groups. Let F be a simplicial presheaf on C. Given
an object X of C and a 0-simplex x in F (X), we define presheaves ßn(F, x) on t*
*he
site C # X by the formula U 7! ßn(F (U), x|U ).
Definition 6.6.A map of simplicial presheaves f :F ! G is a local weak equiv-
alence if
(1)The induced map ß0F ! ß0G yields an isomorphism upon sheafification, and
(2)For every X in C and every basepoint x in F0(X), the map of presheaves on
C # X given by ßn(Ex 1 F, x) ! ßn(Ex 1 G, fx) also becomes an isomorphism
upon sheafification. (Here Ex1 F is the presheaf U 7! Ex1 (F (U)), of course*
*).
Local weak equivalences are called `topological weak equivalences' in [J1]. O*
*ne
can also use the presheaf ßlocn(F, x), whose value on an object U ! X is the
set of based maps Sn ! F (U) modulo the equivalence relation generated by local
simplicial homotopy (see [J1, p. 44]). Two maps Sn ! F (U) are locally simplici*
*ally
12 DANIEL DUGGER AND DANIEL C. ISAKSEN
homotopic if there exists a covering sieve of U such that for every V ! U in
the sieve, the two restrictions Sn ! F (V ) are simplicially homotopic as based
maps. The following result appears in [J1, Prop. 1.18], except for an unnecessa*
*ry
hypothesis.
Lemma 6.7. The map ßn(F, x) ! ßlocn(F, x) is an isomorphism after sheafifica-
tion, for any simplicial presheaf F .
Before proving Lemma 6.7, we recall the following property of sheafifications.
Lemma 6.8. A map f :F ! G between presheaves of sets induces an isomorphism
on sheafifications if and only if the following two conditions are satisfied:
(1)Given any X in C and any s in G(X), there is a covering sieve R of X such
that the restriction s|U belongs to the image of F (U) in G(U) for any eleme*
*nt
U ! X of R;
(2)Given any X in C and any two sections s and t in F (X) such that f(s) = f(t),
there exists a covering sieve R of X such that s |U = t |U in F (U) for every
element U ! X of R.
Proof.Condition (2) is equivalent to F + ! G+ being an objectwise monomor-
phism, which in turn is equivalent to the same property for F ++ ! G++ . If
F ++ ! G++ is an objectwise surjection then property (1) is easily seen to hold.
Finally, properties (1) and (2) together imply that im(G(X) ! G++ (X))
im(F ++(X) ,! G++ (X)). From this one deduces that F ++ ! G++ is an ob- *
*__
jectwise surjection (using that the domain and codomain are sheaves). |*
*__|
We will make use of the above two conditions in studying sheaves of homotopy
groups.
Proof of Lemma 6.7.Since local simplicial homotopy is a larger equivalence rela-
tion than simplicial homotopy, the map is an objectwise surjection. This verifi*
*es
condition (1) of Lemma 6.8.
For condition (2), suppose that s and t are two maps Sn ! F (U) that are
related by a finite chain of local simplicially homotopies. There is a finite s*
*equence
s = s0, s1, . .,.sn = t of maps Sn ! F (U) such that si and si+1 are simplicial*
*ly
homotopic after restricting to a sieve Ri. Taking R to be a common refinement of
each Ri, we conclude that s and t are related by a chain of simplicial homotopi*
*es__
after restricting to R. This verifies condition (2) of Lemma 6.8. *
* |__|
In general, ßlocn(F, x) does not carry homotopical information unless F is lo*
*cally-
fibrant. At first glance, the sheafification of ßn(F, x) seems not to be homoto*
*pically
meaningful unless F is objectwise fibrant, but Lemma 6.7 shows that we only need
F to be locally-fibrant. In other words, for locally-fibrant simplicial preshea*
*ves one
can ignore the presence of Ex1 in Definition 6.6:
Proposition 6.9.If F and G are locally-fibrant, then a map f : F ! G is a local
weak equivalence if and only if
(1)The induced map ß0F ! ß0G yields an isomorphism upon sheafification, and
(2)For every X in C and every basepoint x in F0(X), the map ßn(F, x) !
ßn(G, fx) is an isomorphism upon sheafification.
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 13
Proof.Consider the square
F _____//Ex1F
| |
| |
fflffl| fflffl|
G ____//_Ex1G.
By [J1, Prop. 1.17] and Lemma 6.8, the horizontal maps satisfy the above condi-
tions. Therefore the left vertical map satisfies the conditions if and only if *
*the right
vertical map does. The usual complications with choosing basepoints do not_aris*
*e_
because Ex1 preserves 0-simplices. |__|
6.10. Local relative-homotopy-liftings. The relative-homotopy-lifting criterion
for weak equivalences of simplicial sets (given in Proposition 4.1) has an obvi*
*ous
extension to the presheaf category in which we only require local liftings.
Definition 6.11.Let K ! L be a map of simplicial sets. A map f :F ! G of
simplicial presheaves is said to have the local RHLP with respect to K ! L if
every diagram
(6.12) K X ___//_F44___________
ooooo_|________|_______
oooo______|_______ |
wwoo_______fflffl|__fflffl|_
L X L X ___//_G44___________
| ppppp________________
| ppp____________
fflffl|wwpp_______
RH(L, K) X
admits local liftings.
In other words, the definition requires that there exists a covering sieve of*
* X
such that for any map U ! X in the sieve, the induced diagram
K U ____//_K X____//F44________
fflffl _________|______
| ______________|
fflffl|_________ fflffl|
L U_____//L X_____//G
has a relative-homotopy-lifting. The liftings and simplicial homotopies one get*
*s as
U varies need not be compatible in any way.
The basic results about relative-homotopy-liftings from Section 3 all go thro*
*ugh
in the present context. One only has to observe that the arguments require fini*
*tely
many uses of the lifting conditions. The following result will be especially us*
*eful to
us:
Lemma 6.13. Let f :F ! G be a fixed map of simplicial presheaves. Consider
the class of all maps K ! L of simplicial sets with respect to which f has the *
*local
RHLP. This class is closed under cobase changes and retracts. If G is locally-f*
*ibrant,
then the cofibrations in this class are also closed under composition.
Proof.The proof is the same as that of Lemma 3.4, except that the relative- *
* __
homotopy-liftings are replaced by local relative-homotopy-liftings. *
* |__|
Corollary 6.14.If f :F ! G has the local RHLP with respect to the maps @ n ,!
n for all n 0 and G is locally-fibrant, then f also has the local RHLP with *
*respect
to all cofibrations K ,! L of finite simplicial sets.
14 DANIEL DUGGER AND DANIEL C. ISAKSEN
Proof.Every cofibration K ,! L can be constructed by a finite number of compo-__
sitions and cobase changes from the generating cofibrations @ n ,! n. |_*
*_|
Here is the main theorem of the paper:
Theorem 6.15.
(a)If F and G are locally-fibrant, then a map F ! G is a local weak equivalence
if and only if it has the local RHLP with respect to the maps @ n ,! n.
(b)If F and G are arbitrary and R is any fibrant-replacement functor for sSet,
then a map F ! G is a local weak equivalence if and only if RF ! RG has the
local RHLP with respect to the maps @ n ,! n.
Proof.For (a), we begin by assuming that F and G are locally-fibrant and that
the map f :F ! G has the local RHLP. By Corollary 6.14, it has the local RHLP
with respect to all cofibrations between finite simplicial sets. In particular,*
* it has
the local RHLP with respect to * ,! Sn; this proves condition (1) of Lemma 6.8 *
*for
ßlocn(F, x) ! ßlocn(G, fx) (for n = 0 one uses the RHLP with respect to ; ! *).*
* On
the other hand, f also has the local RHLP with respect to Sn _ Sn ,! RH(Sn, *);
this proves condition (2) of Lemma 6.8 for ßlocn(F, x) ! ßlocn(G, fx). Here we *
*use
that local simplicial homotopy is an equivalence relation for locally-fibrant s*
*implicial
presheaves [J1, Lem. 1.9]. Now Lemma 6.7 and Proposition 6.9 tell us that we ha*
*ve
a local weak equivalence.
We now assume that f :F ! G is a local weak equivalence. To prove that it has
the local RHLP with respect to the maps @ n ,! n we follow exactly the same
argument as in Section 5, observing that there are only finitely many applicati*
*ons
of the various lifting properties.
To prove (b), let R be any fibrant-replacement functor for sSet. We need only
observe that F ! G is a local weak equivalence if and only if RF ! RG is one.
Since RF and RG are objectwise-fibrant (hence locally-fibrant as well),_part_(a)
applies. |__|
7. Applications
Both the lifting characterization of local weak equivalences and the definiti*
*on
involving sheaves of homotopy groups are useful to have around. For instance,
up to some technical difficulties in choosing basepoints, it is transparent fro*
*m the
homotopy group definition that the local weak equivalences have the two-out-of-
three property. This is awkward to show using the lifting characterization, how*
*ever.
We now give some results which are easy consequences of our lifting criterion.
Proposition 7.1 (Local right properness).Let F ! G be a local weak equivalence
between simplicial presheaves, and let J ! G be a local-fibration. Then the map
J xG F ! J is also a local weak equivalence.
Proof.The proof is the same as the proof of Corollary 4.2, except that liftings*
* are
replaced by local liftings. Note that Ex1 commutes with fibre-products of simpl*
*icial
presheaves since both Ex1 and fibre-products are defined objectwise. Also, simi*
*lar __
to observations in the proof of [J1, Prop. 1.17], Ex1 preserves local-fibration*
*s. |__|
Recall that a local acyclic fibration of simplicial presheaves is a map that
is both a local weak equivalence and a local-fibration. The (=)) direction of t*
*he
following proposition was proved in [J3, Lemma 7]_we can now prove the other
one (see [J3, Lemma 11] for a weaker version).
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 15
Proposition 7.2.A map p: F ! G of simplicial presheaves admits local liftings
in every square
(7.3) @ n X ____//_F::___
fflffl____|____
| _______|__
fflffl|___fflffl|_
n X _____//G
if and only if it is a local acyclic fibration.
Proof.First suppose that the local liftings exist. Then p also has the local-li*
*fting
property with respect to the maps n,k,! n, since these can be built from the
maps @ r ! r by finitely many cobase-changes and compositions. Therefore p is
a local-fibration. Similar to observations in the proof of [J1, Prop. 1.17], th*
*e map
Ex1 p: Ex1 F ! Ex1 G has the local lifting property with respect to all maps
@ n X ,! n X. In particular, Ex1 p has the local RHLP with respect to the
maps @ n ,! n; we use constant relative-homotopies. Using Theorem 6.15(b), p
is a local weak equivalence. This finishes one impliciation.
For the other direction, first assume that F and G are locally-fibrant. Since
F ! G is a local weak equivalence one gets local relative-homotopy-liftings by
Theorem 6.15. Similar to the proof of Corollary 4.2, the fact that F ! G is a l*
*ocal
fibration allows one to homotope the local homotopy-liftings to get actual local
liftings.
Now suppose that p is an arbitrary local acyclic fibration, and suppose given
a lifting square as in (7.3). As we have already observed, Ex1 p is also a loc*
*al
acyclic fibration, but with locally-fibrant domain and codomain. So by the prev*
*ious
paragraph there are local liftings for the composite square
@ n X ____//_F___//_Ex1F55____
| |_________|_____
| ____|________ |
fflffl|___fflffl|___fflffl|
n X _____//G____//_Ex1G.
This translates to saying that for a sufficiently large k there are local lifti*
*ngs in the
square
sdk@ n X ____//_@ n X____//_F44______________
| __|_________|______
| _______|______ |
fflffl|________fflffl| |fflffl
sdk n X _____//_ n X____//G,
where the left horizontal maps are the `last vertex maps' [GJ , p. 183].
Let C be the mapping cylinder of sdk@ n ! @ n, and let D be the mapping
cylinder of sdk n ! n. Notice that C is a subcomplex of D. Since the map
sdk@ n X ! F factors through @ n X, the constant homotopy (sdk@ n x
1) X ! F factors through C X. Now, we have squares
[C [ (sdk n x {0})] U____//F
fflffl |
| |
fflffl| fflffl|
D U _____________//G
16 DANIEL DUGGER AND DANIEL C. ISAKSEN
for all U ! X in a covering sieve of X, where the maps C U ! F and D
U ! G are these `constant homotopies'. The map C [ (sdk n x {0}) ! D is a
trivial cofibration between finite simplicial sets (both the domain and codomai*
*n are
contractible), so the square has a local lifting. By precomposing these lifting*
*s with_
the inclusion n ,! D, one obtains local liftings for the original square. *
* |__|
Corollary 7.4 (cf. [J3, Lemma 19]).Let p: F ! G be a local-fibration, and let
i: K ,! L be a cofibration of finite simplicial sets. If p is a local weak equi*
*valence
or i is a weak equivalence, then the induced map
F L! F KxGK GL
is a local acyclic fibration.
Proof.To see that the map is a local acyclic fibration, it is enough by Proposi*
*tion 7.2
to check that it has the local lifting property with respect to the maps @ n ! *
* n.
By adjointness, one need only check that F ! G has the local lifting property w*
*ith
respect to the map
j :(L x @ n) [ (K x n) ! L x n.
If K ! L is an acyclic cofibration then so is j, and therefore the result follo*
*ws from
Lemma 2.2 and the definition of local fibration. If F ! G was an acyclic fibrat*
*ion,
then the result follows from Proposition 7.2 because j is obtained by a finite_*
*number
of cobase changes and compositions from the inclusions @ n ! n. |__|
Finally, we end with the following result. It is needed in [DHI ], and we kno*
*w of
no proof that avoids the local lifting techniques we've just developed.
Corollary 7.5.Let F ! G be a map between locally-fibrant simplicial presheaves,
and let K ,! L be a cofibration of finite simplicial sets. If either map is a *
*weak
equivalence then the induced map F L! F KxGK GL is a weak equivalence.
Proof.First, we know from [J1, Cor. 1.5] that both F L and F K xGK GL are
locally-fibrant. A lifting square
@ n X ________//_F L
fflffl |
| |
fflffl| fflffl|
n X _____//F KxGK GL
may be rewritten via adjointness as
M X ________//F
fflffl |
| |
fflffl| fflffl|
(L x n) X_____//G,
where M = (Lx@ n)[(K x n). The map M ,! Lx n is a cofibration between
finite simplicial sets. When F ! G is a local weak equivalence, Corollary 6.14
and Theorem 6.15(a) tell us that the above square has a relative-homotopy-lifti*
*ng.
Using adjointness once again, we get a relative-homotopy-lifting for the origin*
*al
square.
The other case is similar. If K ,! L is a weak equivalence then M ,! L x n is
also one. So by the local version of Proposition 3.7 (which only works for acyc*
*lic
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES 17
cofibrations between finite simplicial sets), we have a relative-homotopy-lifti*
*ng_since
F and G are locally-fibrant. |__|
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Department of Mathematics, Purdue University, West Lafayette, IN 47907
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
E-mail address: ddugger@math.purdue.edu
E-mail address: isaksen.1@nd.edu