TOWER TECHNIQUES FOR COFACIAL RESOLUTIONS
ASSAF LIBMAN
Abstract.Let J be a continuous coaugmented functor on spaces. For every
space X one constructs a cofacial resolution X ! JoX (namely a cosimpli-
cial resolution without its codegeneracy maps) in the usual way. Followi*
*ng
Bousfield and Kan, one defines JsX = totsJoX.
Suppose D is a small category and that X is a D-diagram of J-injective
spaces, namely X(d) ! JX(d) admits a left inverse for every object d in *
*D,
but in a way which need not be compatible, namely a map JX ! X cannot
be constructed out of this data. We show that for many free diagrams F,
the spaces homD(F, X) are Js-injective for s < 1. Thus, the functors Zs *
*of
Bousfield and Kan capture a large class of polyGEMs as their injective s*
*paces.
This generalises earlier results by the author. Our methods use pro-obj*
*ect
arguments, which are originally due to Farjoun.
Keywords: Homotopy limits, Towers, Coaugmented functors
1.Introduction
1.1.Recall that a cofacial space Xo is a cosimplicial space without its codegen*
*eracy
maps. A cofacial resolution (or simply a resolution) is an augmented cofacial s*
*pace
X-1 ! Xo (compare [2, pp. 271]). Such a resolution is called trivial if it admi*
*ts a
left contraction, namely maps Xn -s!Xn-1 which function as a codegeneracy map
s-1, namely sd0 = idand sdi= di-1s for all i > 0. Trivial or not, such a resolu*
*tion
admits maps
X-1 ! totnXo n 1
where totnis the cofacial analogue of the classical cosimplicial totnin [2]. It*
* is, in
fact, the homotopy limit of the truncation of Xo in codimension n, denoted (Xo)*
* n .
If D is a small category then the notion of resolution of D-diagrams is the o*
*bvious
extension of the above. Of interest are such resolutions which are termwise tri*
*vial,
namely X-1(d) ! Xo(d) is a trivial resolution of spaces for all objects d in D,*
* but
they do not combine to give a map Xo ! X-1.
Theorem A. [15, Theorem 3.2] Suppose that D is a small category such that
dim N(D) < 1. Let X-1 ! Xo be a termwise trivial resolution of (fibrant)
spaces. Then
holimDX-1 ! {totsholimDXo}s 0
is a pro-equivalence in the homotopy category of spaces. In particular holimDX-1
is a retract of totsholimDXo.
Suppose that J is a coaugmented functor on spaces which satisfies the mild
assumption of being simplicial (alternatively, continuous [3, pp. 20], [14, 3.2*
*],[13]).
____________
Date:
1991 Mathematics Subject Classification. 55U35,55T15,55P60,18A25.
1
The functor J can be used to construct cofacial resolutions X ! JoX and then,
following Bousfield and Kan, to define
JsX := totsJoX = holim(JoX) s.
A space X is called J-injective if X ! JX admits a left inverse. For such space*
*s,
the resolution X ! JoX are easily seen to be trivial. Examples of this situation
are described bellow. Theorem A was exploited to prove
Theorem B. [14, Theorem 4.7]. Suppose that D is a small category whose nerve is
finite dimensional, and that X :D ! Spc is a diagram of J-injective spaces. Then
there exists s < 1 such that holimDX is up to homotopy a retract of JsholimDX.
In fact s = dim(D).
1.2.The motivation for this discussion lies in the study of polyGEMs. Recall fr*
*om
[3, pp. 87,101] that a space is called a GEM (Generalised Eilenberg MacLane spa*
*ce)
if it is homotopy equivalent to a product of abelian Eilenberg MacLane spaces. *
*It
is a well known fact (due to J. Moore) that a space X is a GEM if and only if i*
*t is
SP1 -injective, namely if and only if X ! SP1 X admits a left homotopy inverse.
It is also known that a space is a GEM if and only if it is equivalent to an ab*
*elian
topological group.
A space X is called a polyGEM if it is homotopy equivalent to a space in a to*
*wer
{. .!.X2 ! X1 ! X0 = *} of principal fibrations, all of whose fibres are abelian
topological groups. Thus, the construction of a polyGEM is an iterative process*
*, in
which one starts with a GEM and repeatedly obtain the next space by taking the
total space in a principal fibration in which the fibre is abelian and the base*
* is the
given space.
When D is a finite dimensional category and X is a D-diagram of GEMs, it is
not hard to see that holimDX is a polyGEM by appropriately filtering the catego*
*ry
D (see [14, 5.4]). We call such spaces thin-polyGEMs. The simplicial analogue of
SP1 is Bousfield Kan's functor Z, and Theorem B implies
Theorem C. If X is a thin polyGEM then X is up to homotopy a retract of ZsX
for some s < 1.
1.3.The purpose of this note is to extend these results. The main tool that we *
*use
is the notion of strong pro-equivalence which is introduced and explored in Sec*
*tion
2. It is essentially due to Farjoun [5]. The main technical result of this pape*
*r is
4.11
Theorem D. Let X-1 ! Xo be a trivial resolution of (fibrant) spaces. Then the
tower map
X-1 ! {totsXo}s 0
is a strong pro-homotopy-equivalence.
The concept of strong pro-homotopy equivalence is much stronger than the more
familiar weak-pro-equivalence ,see [2, pp. 76]. Its power is in enabling indu*
*ctive
arguments. This enables to transform the painfully complicated proof of Theorem*
* A
given in [15], into a much softer and conceptual one. It also enables us to gen*
*eralise
it in 5.11 and 5.6 as follows.
Theorem E. Let D be a small category and that either
(a) D is finite dimensional and F :D ! S is a free diagram [4], or
2
(b) F :D ! S is a poly-free diagram (5.5).
Then if X-1 ! Xo is a termwise trivial resolution of fibrant D-diagrams, then
hom D(F, X-1) ! {totshomD(F, Xo)}s 0
is a strong pro-homotopy equivalence. In particular hom D(F, X-1) is up to homo-
topy, a retract of totshomD(F, Xo) for some s < 1.
Theorem A is recovered by plugging in a weakly contractible F . Consequently,
we deduce (6.4)
Corollary F. Suppose that D and F are as in Theorem E, and J is a simplicial
coaugmented functor. If X : D ! Spc is a diagram of fibrant J-injective spaces,
then there exists s < 1 such that hom D(F, X) is a retract of Jshom D(F, X).
1.4.Applying Corollary F to the case when J is the functor Z, we see that we
have extended the class of polyGEMs which are detected by Zs in the sense that a
polyGEM X in this class is a retract of ZsX for some s < 1.
Firstly, if D is finite dimensional, F is any free D-diagram, and X a diagram*
* of
GEMs, then homD (F, X) is a retract of Zshom D(F, X) for some s < 1. The older
Theorem C is recovered by choosing F to be weakly contractible. Better, when D
is any small category (even as not homologically finite as a finite group) and *
*F is
poly-free (5.5), then hom D(F, X) is a retract of its Zs for some s < 1.
The class of`poly-free diagrams is the smallest class of diagram which contai*
*ns
the diagrams i2IAix D(di, -) for (cofibrant) spaces Ai, and is closed under
pushouts along cofibrations. It is, in some sense, a dual construction to polyG*
*EMs
which consist the smallest class of spaces which contain GEMs and is closed und*
*er
pullbacks along fibrations between GEMs.
We still don't know if all polyGEMs are retracts of their Zs.
2.Towers
2.1.Throughout, S denotes the category of simplicial sets. The standard n-
simplices are denoted n. These spaces are the objects in the standard cosimpli*
*cial
space o. We let di: n-1 ! n and si: n+1 ! n be the standard cosimplicial
coface and codegeneracy maps (0 i n).
2.2.Let C be a simplicial closed model category (cf. [9, Ch.II.2]). This, in pa*
*rtic-
ular means that there is a pairing bifunctor
: C x S ! C
which has partial right adjoints
map: Cop x C ! S and hom : Sop x C ! C.
Explicitly, the first functor is given by
map (X, Y )n = C(X n, Y ).
It is worthwhile noting that
(a) X 0 = X for all X 2 C.
(b) If f : A ! B is a (trivial) cofibration in S and if X 2 C is cofibrant, th*
*en
X f is a (trivial) cofibration in C (see [9, II.3.4]).
3
(c) For every X, Y 2 C and A 2 S, there is a map, natural in these objects
map (X, Y ) x A ! map(X, Y A)
which is the adjoint of
X (map (X, Y ) x A) = (X map(X, Y )) A ev-A--!Y A
2.3.Recall that a tower of objects in C is a functor Xo: Nop ! C where N is the*
*op
category {0 ! 1 ! 2 ! . .}.. These objects form the usual functor category CN
which we denote tow-C. It is, however, useful to treat towers as pro objects (c*
*f. [1,
Appendix]). Morphism sets of towers as pro-objects are by definition
pro-C(Xo, Yo) = limscolimtC(Xt, Ys).
The convention will be that towers are denoted by the symbols Xo, Yo, . .e.tc.
Morphisms of towers as diagrams over Nop are denoted by ff, fi, . .o.r f, g, . *
*...
Morphisms of towers though of as pro objects will be denoted by ~ff, ~fi, . .o.*
*r ~f, ~g, . . .
etc.
Evidently, every pro-morphism of towers can be represented by a sequence of
morphisms
{fs:Xt(s)! Ys}s 0, (t(s + 1) > t(s))
which render the following squares commutative (s 0).
Xt(s+1)-fs+1---!Ys+1
?? ?
y ?y
Xt(s) --fs--! Ys
Evidently, a level map {fs}s:Xo ! Yo, namely a morphism in tow-C, represents a
pro-map ~f.
2.4.Given a tower Xo and a simplicial set K, we denote by Xo K the obvious
tower which at level s is Xs K.
2.5.Given towers Xo and Yo, let
map(Xo, Yo) := limscolimtmap(Xt, Ys)
be the function complex (an object in S). Evidently,
map (Xo, Yo)n = pro-C(Xo n, Yo).
Moreover, this construction is natural in the sense that a morphism (of pro-obj*
*ects)
~f:Ao ! Bo defines simplicial maps
f~*
map(Bo, Xo) -! map (Ao, Xo) and
f~*
map(Xo, Ao) -! map (Xo, Bo).
This is done in an obvious way by choosing a representative for the morphism ~f
(2.3). We leave it to the reader to check that this is independent of the repre*
*sent-
ative, and remark that when ~fis represented by a level map {fs} then
map (f~, Xo) = limscolimtmap(ft, Xs) and
map (Xo, ~f) = limscolimtmap(Xt, fs).
4
2.6. Definition.(cf. [5].) A tower Xo is called fibrant if all the objects Xs a*
*re
fibrant and cofibrant and the maps Xs+1 ! Xs are fibrations for all s 0.
2.7. Remark. This is different than the terminology in [10] or [11]. The termi*
*no-
logy and results that will follow in this section, are very reminiscent of the *
*ones in
(simplicial) model categories. However, we do not know how to endow the category
tow-C with a model structure that correspond to our notion of fibrations etc.
2.8. Proposition.Let Xo be a fibrant tower. Then for every tower Wo of cofibrant
objects, the mapping space map (Wo, Xo) is a fibrant space (i.e. a Kan complex).
Proof.Observe that map (Wt, Xs) is a fibrant space for every s and t, and that *
*se-
quential colimits preserve fibrations in S. Hence {colimtmap(Wt, Xs)}s if a fib*
*rant_
tower in S , and its inverse limit is fibrant. |*
*__|
2.9. Definition.Tower maps ~f0, ~f1:Xo ! Yo are called homotopic, if as vertice*
*s,
they belong to the same component of map (Xo, Yo). We call them simplicially
homotopic if there exists a 1-simplex, namely a pro-map
~h:Xo 1 ! Yo
such that @i(~h) = ~fi.
By 2.8, when Yo is fibrant and Xo is levelwise cofibrant, then both notions
coincide.
2.10. Proposition.Let ~ff0, ~ff1:Xo ! Yo be simplicially homotopic pro-maps (2.*
*9)
between fibrant towers. Then
(a) For every fibrant tower To, the induced maps (~ffi)* = map(~ffi, To) are s*
*impli-
cially homotopic (as maps between Kan complexes).
(b) For every tower Wo of cofibrant objects, the induced maps
(~ffi)* = map(Wo, ~ffi)
are simplicially homotopic (as maps between Kan complexes).
Proof.Let ~hbe a simplicial homotopy (2.9). That is, the compositions
___d0//_ ~h
Xo _____//Xo 1____//Yo
d1
are precisely ~ff0and ~ff1.
To prove (a) it suffices to show that map (d0, To) ' map (d1, To). Consider t*
*he
projection map ß : Xo 1 ! Xo. Since ß . di = id, it suffices to show that
map(ß, To) is a homotopy equivalence. This will follow if we show that, for ins*
*tance,
map(d0, To) is one. Observe that d0 is a levelwise trivial cofibration, and sin*
*ce To
is fibrant, it follows that the maps
0)*
map (Xt 1, Ts) (d---!map(Xt, Ts)
are all trivial fibrations. Using the fact that sequential colimits preserve t*
*rivial
fibrations and fibrations in S, we see that the induced map
{colimtmap(Xt 1, Ts)}s ! {colimtmap(Xt, Ts)}s
is a levelwise homotopy equivalence of fibrant towers in S. Since the inverse l*
*imit of
a tower of fibrations is equivalent to its homotopy limit, it follows that map(*
*d0, To)
is an equivalence of Kan complexes, and the result follows.
5
Point (b) is even easier since the natural map (2.2(c))
map(Wo, Xo) x 1 ! map(Wo, Xo 1),
composed with map (Wo, ~h) gives the desired simplicial homotopy. |*
*___|
2.11. Definition.Let ~ff: Xo ! Yo be a morphism of fibrant towers. We call ~ffa
strong pro-homotopy equivalence, and write s.p.h.e. for short, if one of the fo*
*llowing
equivalent conditions hold.
(a) There exists a pro-map ~fi:Yo ! Xo such that ~ff. ~fiand ~fi. ~ffare simpl*
*icially
homotopic to the identity maps (2.9).
(b) For every fibrant tower To, the induced map
*
map (Yo, To) f~f-!map(Xo, To)
is a homotopy equivalence of fibrant spaces.
(c) For every tower Wo of cofibrant objects, the induced map
map (Wo, Xo) f~f*-!map(Wo, Yo)
is a homotopy equivalence of fibrant spaces.
Proof.Evidently (a) implies both (b) and (c) using Proposition 2.10. For the
converse we use the commutative diagram of fibrant objects in S.
*
map (Yo, Xo)-f~f---!map(Xo, Xo)
? ?
~ff*?y ?y~ff*
map (Yo, Yo)----!~ff*map(Xo, Yo).
If one assumes (b) then the horizontal arrows are homotopy equivalences between
Kan complexes, hence one can choose ~fi: Yo ! Xo such that ~ff*(f~i) = f~i~ffis
simplicially homotopic to the identity on Xo. Applying ~ff*to these vertices, a*
*nd
using the fact that the bottom arrow is a homotopy equivalence, it follows that
~ff. ~fi' idYo, so (a) holds.
If one assumes (c), then the vertical arrows are homotopy equivalences,_and t*
*he
argument is similar. |__|
2.12. Lemma. Let f : Xo ! Yo be a levelwise homotopy equivalence between
fibrant towers. Then f is a s.p.h.e.
Proof.If Wo is a tower of cofibrant objects, then for every t, s 0, the map
map (Wt, Xs) fs-!map(Wt, Ys)
is a homotopy equivalence of Kan complexes because Xs and Ys are fibrant and
Wt cofibrant. Since sequential colimits (indexed by t) carry equivalences betwe*
*en
fibrant spaces (resp. fibrations) to an equivalence of fibrant spaces (resp. *
*fibra-
tions), it follows that
{colimtmap(Wt, Xs)}s ! {colimtmap(Wt, Ys)}s
is a levelwise equivalence of fibrant towers in S. The result follows because i*
*n this_
case the inverse limit (indexed by s) is the same as the homotopy inverse limit*
*. |__|
6
2.13. Lemma. Consider the following pullback diagram in tow-C of fibrant towers
(and level maps).
0}s
Uo -{gs---!Yo
? ?
(2.13.1) {f0s}s?y ?y{fs}s
Zo ----! Xo.
{gs}s
Suppose that gs are fibrations in C. Then for every tower Wo of cofibrant objec*
*ts,
applying map (Wo, -) to (2.13.1)yields a homotopy cartesian square of fibrant o*
*b-
jects in S.
Proof.Let Ps denote the pullback square of (2.13.1)at level s. Since map (Wt, -)
and sequential colimits carry pullback squares (resp. fibration) to pullback sq*
*uares
(resp. fibrations), it follows that
(i)For every s, the diagram colimtmap(Wt, Ps) is a pullback square of fibrant
spaces, and colimtmap(Wt, gs) is a fibration.
(ii)The induced maps between these pullback squares
colimtmap(Wt, Ps+1) ! colimtmap(Wt, Ps)
are objectwise fibrations.
Since sequential inverse limits (indexed by s) of fibrations are equivalent to_*
*homo-
topy limits, the result follows. |__|
2.14. Corollary.Consider the diagram of fibrant towers commuting in pro-C
U0o_____________________//_AYo
|| AA~ffiA ____ ||
| AA __~ff |
| A__ ~~__ |
| Uo _____//Yo |
| |
| | | |
| | | |
| fflffl| fflffl| |
| Zo _____//Xo |
| ">> ``BB |
| ~fi"" BBB |
| "" ~flBB |
fflffl|"" B fflffl|
Z0o_____________________//X0o
Assume that both squares are cartesian in tow-C (so all vertical and horizontal
arrows are level map), and that the horizontal arrows are levelwise fibrations.*
* If
~ff, ~fiand ~flare s.p.h.e. then so is ~ffi.
Proof.Immediate from 2.13 and 2.11(c). |___|
2.15.(Loop objects and fibres.) Suppose now that C is also pointed, namely it h*
*as
an object * which is both initial and terminal.
Given a fibration f :X ! B in C, we define its fibre F as the pullback of
X ---f-! B ---- *.
Clearly, when B is fibrant, then so is the fibre of f. We call the resulting se*
*quence
F -i!X -f!B
7
a fibre sequence in C.
The loop object of every X 2 C, denoted X, is the pullback of
hom( 1, X) ----! hom(@ 1, X) ---- hom(@ 1, *) = *.
Observe that given objects W and X in C, the function complex map(W, X) is nat-
urally a pointed simplicial set. Using this basepoint, there is a natural isomo*
*rphism
(2.15.1) map (W, X) ~=map(W, X)
which follows from the fact that map(W, -) preserve pullback squares and from t*
*he
isomorphism (cf. [9, II.2.3])
map C(W, homC(A, X)) ~=mapS (A, mapC(W, X)) = homS (A, mapC(W, X)).
2.16. Lemma. Suppose that Fo -i!Xo p-!Bo is a level map of fibrant towers in
a pointed simplicial closed model category C. Assume further that at every leve*
*l s
the maps form a fibre sequence. Then for every levelwise cofibrant tower Wo, the
sequence
map (Wo, Fo) ! map(Wo, Xo) ! map(Wo, Bo)
is a homotopy fibre sequence of Kan complexes in S*.
Proof.Follows immediately from Lemma 2.13. |___|
2.17. Corollary.Let C be a pointed simplicial closed model category in which all
objects are cofibrant. Consider the following diagram of fibrant towers which c*
*om-
mutes in tow-C.
Fo ----! Eo ----! Bo
? ? ?
f?y ?yg ?yh
Fo0----! E0o----! B0o.
Suppose that the vertical arrows form levelwise fibre sequences. Then
(a) If g, h are s.p.h.e. , then so is f.
(b) If f, g are s.p.h.e. then so is h.
(c) If f, h are s.p.h.e. then so is g.
Proof.(a) is immediate from 2.14. Point (b) follows by looping the resulting ho*
*mo-
topy fibre sequences in 2.16, and using (2.15.1), and the fact that h is a lev*
*el map
of fibrant towers because Bo and B0oare. Point (c) follows by looping this_sequ*
*ence
once more. |__|
3.Coherent functors
3.1.The concept of öc herent functors" is devised to give a uniform proof for o*
*ur
main results in both the pointed and unpointed categories. The unpointed case
is more fundamental, and the reader who is happy with this case only may safely
ignore the term öc herent" throughout the paper.
In this section we restrict attention to the categories of unpointed and poin*
*ted
simplicial sets, denoted S and S* denoted by Spc.
These categories are related by the pair of adjoint functors u - the forgetful
functor, and (-)+ - adjoining a disjoint basepoint functor.
8
3.2.Recall that Spc is a closed symmetric monoidal category (cf. [12]), and is,
thus, enriched over itself. We lat äM p " denote the internal hom-spaces. It fo*
*llows
that if D is a small category, then the functor category SpcD is also an Spc-ca*
*tegory
with hom-spaces given by the end formula
Z
Map D(X, Y ) = Map (X(d), Y (d)).
d2D
Similarly, the pairing on Spc (see 2.2)has an obvious prolongation to a pairi*
*ng
: Spcx SD ! SpcD, and the latter has an adjoint hom :(SD )opx SpcD ! Spc
given by Z
hom D(K, X) = hom (K(d), X(d)).
d2D
It is easy to see that
æ
(3.2.1) homD (K, X) = MapMD(K,aX)p Spc = S
D(K+ , X) Spc = S*
3.3.Given a small functor f : C ! D, there is an induced restriction functor
f* : SpcD ! SpcC. By [16, pp. 229], this functor has a left adjoint, the left K*
*an
extension of f, denoted Lanf. It is not hard to verify that this adjunction pro*
*longs
to a natural isomorphism of spaces
Map C(X, f*Y ) ~=Map D(LanfX, Y ).
Moreover, since Lanf(K+ ) = (LanfK)+ for every K 2 SC , then
hom C(X, f*Y ) ~=homD (LanfX, Y ).
3.4. Definition.A coherent functor F : SpcD ! Spc is comprised of a pair of
functors - one for each of the categories S and S*, which are related by
uF = F u
3.5. Example. Our prime examples of coherent functors are given by
FA (-) := homD (A, -)
where A 2 SD . Indeed, if X 2 SD*, then
uFA (X) = uMap *D(A+ , X) = Map D(A, uX) = FA (u(X)).
3.6. Definition.Let F, G:Spc D ! Spc be coherent functors. A coherent natural
transformation t:F ! G is a pair of natural transformations between the unpoint*
*ed
and pointed functors, which are subject to the condition
tu = ut.
The set of coherent natural maps is denoted CNat(F, G).
3.7. Example. If A and B are D-diagrams in S, then any map ':B ! A induces
a coherent map (3.5)
hom D (', -): FA (-) ! FB (-).
Indeed, for all pointed diagrams of spaces,
uhom D(', -) = uMap D('+ , -) = Map D(', u(-)) = homD (', u(-)).
This is, in fact, the only example:
9
3.8. Proposition.Consider the functors FA and FB above (3.5). Then t : FA !
FB is a coherent map if and only if t = homD (', -) for some ':B ! A. That is,
there is a natural bijection
~=
SD (B, A) -! CNat(FA , FB ).
Proof.The "if" part was proved in 3.7. Conversely, assume that t is coherent.
Consider the unpointed t. Since SD is an S-category, and hom = Map (3.2.1),
Yoneda's Lemma in enriched categories (see [12, pp. 45]) implies that the unpoi*
*nted
t is given by
t = Map D(', -) = homD (', -)
for some unique ' : B ! A. Similarly, when Spc = S*, then by (3.2.1), and by
Yoneda's Lemma again, the pointed t, denoted t* is induced by a map
_ :B+ ! A+ .
By assumption t is coherent, so for all objects in SD*,
uMap D('+ , -) = Map D(', u(-)) = tu(-) = ut*(-) = uMap D(_, -).
But since the forgetful functor u is faithful, it follows that, '+ and _ induce*
* the
same representable functor on SD*, hence _ = '+ , and we have established that,*
*_as
a coherent natural map, t = homD (', -). |__|
3.9.The set CNat(F, G) is naturally the vertex set of a simplicial set CNat (F,*
* G),
called the space of coherent maps. By definition
CNat (F, G)n = CNat(F n, G)
Observe that 1-simplices of CNat (F, G) correspond to simplicial homotopy of co-
herent natural maps F ! G.
3.10. Proposition.Given A, B 2 SD , there is a natural isomorphism of simpli-
cial sets (3.5,3.9)
~=
Map D(B, A) -! CNat (FA , FB ).
Proof.Immediate from the following natural isomorphisms and 3.8
Spc (hom (A, -) n, hom(B, -)) ~=
Spc (hom (A, -), hom( n, hom(B, -)) ~=
Spc (hom (A, -), hom(B x n, -)).
|___|
4. The tower associated to a resolution of spaces
4.1.We let Spc denote either the unpointed or pointed categories of simplicial
sets. See 3.1. Objects in these categories are called spaces.
4.2.For every n -1 we let [n] = {0, . .,.n} be linearly ordered in the usual *
*way.
By convention [-1] is the empty set.
Let denote the category whose objects are the sets [0], [1], [2], . .a.nd w*
*hose
morphisms are the strictly monotone maps between these sets. Obviously, is a
subcategory of the usual cosimplicial category , and is generated by the morph*
*isms
di:[n] ! [n + 1] where 0 i n + 1, subject to the relation djdi= di-1dj if i*
* > j.
We further let 0 denote the category adjoined with the initial object [-1].
10
Obviously, is the category underlying "reduced cosimplicial objects" namely
cosimplicial objects without their codegeneracy maps. Likewise, 0 is the categ*
*ory
underlying augmented such objects (cf. [2, pp. 271]). See bellow.
4.3. Definition.A cofacial space is a -diagram of spaces. We shall denote cofa-
cial spaces by Xo, Y oetc. An extremely important cofacial space is o, which is
the restriction of the standard cosimplicial space o to . Other useful variat*
*ions
are skn o which is the simplicial n-skeleton of o.
4.4.We note that the maps (-1 n m 1)
skn o ! skm o
are free, and hence cofibrations in S equipped with the usual model category
structure (see [7]). In particular, skn o are cofibrant.
4.5. Definition.A cofacial resolution (or merely ä resolution") of a space X, *
*is
0
a cofacial space Xo together with a map X -d! X0 such that d0d0 = d1d0. We
usually write X-1 for X, and a resolution is thus noting but a functor X: 0 ! S*
*pc.
We shall use boldface capitals X, Y etc. to denote resolutions. Given such X, we
let Xo be the cofacial part, and X-1 its augmentation.
4.6. Definition.Given a cofacial space Xo, we let
totsXo = hom (sks o, Xo).
Clearly, if X is a resolution, then there is a natural map
0
X-1 -d!totsXo.
Furthermore, from 4.4 it follows that if Xo is a fibrant cofacial space, namely
objectwise fibrant, then totsXo is a fibrant space, and when s = 1, it is homot*
*opy
equivalent to holimXo.
0
4.7. Definition.A left contraction for a resolution X-1 -d!Xo is a collection of
maps s:Xn ! Xn-1 for all n 0, such that sd0 = idand sdi= di-1s for all i > 0.
Compare this with [8].
A resolution which admits a left contraction is called trivial. Notice that a
resolution may be trivialised in more than one way.
4.8.It is useful to define a category of trivial resolutions. More precisely, w*
*e let +
be the category whose objects are Obj( 0), and whose morphism sets are generated
by the (abstract) morphisms
di:[n - 1] ! [n] 0 n, 0 i n
s:[n] ! [n - 1] n 0
subject to the relations
djdi= di-1dj i > j
(4.8.1) sd0 = id
sdi= di-1s i > 0.
One can, in fact, embed + as the subcategory of which consists of the objects
[1], [2], [3], . .a.nd morphisms are those order preserving maps ' : [n] ! [k] *
*such
that 0 2 '-1(0) and ' is strictly monotone outside '-1(0). We shall, however, n*
*ot
need this description.
11
It is evident that trivial resolution are precisely diagrams over + .
4.9.Consider the small functors
__k_//_o+jo_*
where * denotes the trivial category,
j(*) = [-1]
and k is the inclusion functor, namely (4.8)
k([n]) = [n], k(di) = di.
Given a trivial resolution (a + -diagram) X, then Xo = k*(X) and X-1 = j*(X).
4.10.For every space Y we let coY denote the constant cofacial space.
0
Suppose that X is a trivial resolution. Then the augmentation map X-1 -d!X0
and the contractions s induce natural cofacial space maps
d0: coX-1 ! Xo and oe :Xo ! coX-1
given by
(d0)n+1: X-1 ! Xn and sn+1: Xn ! X-1.
These maps give rise to natural maps (see 4.9)
d0:j*(X) ! totnk*(X) and oe :totnk*(X) ! j*(X).
Explicitly,
0
d0: X-1 = hom(*, coX-1) ! totncoX-1 -d!totnXo
oe :totnXo -s!totncoX-1 ! tot0coX-1 = X-1.
We make the observation that these maps are coherent maps between coherent
functors Spc + ! Spc (3.4,3.6). The rest of this section is dedicated to the pr*
*oof
of
4.11. Theorem. Let X be a fibrant trivial resolution, i.e. a + -diagram of fi*
*brant
spaces. Then the natural map of fibrant towers
d0: {X-1}s 0 ! {totsXo}s 0
is a s.p.h.e. (2.11). In fact, oe defined in 4.10, is a pro-homotopy inverse.
4.12. Remark. The point of this theorem is in the strength of the relation bet*
*ween
the towers involved. It is immediate that X-1 is a retract of totnXo for all n,*
* and
that X-1 ' tot1Xo. But these facts are very weak statements compared to 4.11.
It is also much stronger than the pro-equivalence in [15, Theorem 3.2].
4.13.The strategy of the proof is to show that both composition oed0 and d0oe a*
*re
s.p.h.e. Clearly, oed0 = id, and so we are only left with d0oe. The naturality *
*in X of
these maps, enables us to show that idand d0oe belong to the same component of
the function complex map ({totsXo}s, {totsXo}s).
4.14. Proposition.For every fibrant trivial resolution X, oed0 = id.
Proof.This is true on the level of cofacial maps (see 4.10). *
*|___|
To establish the fact that d0oe is a s.p.h.e. we need some preparation.
12
4.15. Definition.For every n 0, let vn denote the collapsing map of n onto
the vertex 0 of n+1. Specifically, vn is the composition
0)n dn+1...d2d1
vn : n (s---! 0 -------! n+1.
4.16. Proposition.There exist maps (n 0)
ßn : n x 1 ! n+1
such that
(a) ßn . (1 x d0) = d0: n ! n+1.
(b) ßn . (1 x d1) = vn (see 4.15)
(c) The following diagram commutes for every n 0 and 0 i n + 1.
n x 1 --ßn--! n+1
? ?
dix1?y ?ydi+1
n+1 x 1 ----!ß n+2.
n+1
Proof.Consider the linearly ordered sets [n] = {0, 1, . .,.n} (see 4.2) as cate*
*gories.
Let cn : [n] ! [n + 1] be the functor (s0)n, and let jn : [n] ! [n + 1] be the *
*functor
d0. Upon taking nerves, cn induces vn and jn induces d0. There is a unique natu*
*ral
transformation 'n : cn ! jn which induces the map ßn. We leave the rest_to the
reader. |__|
4.17.Given s, q 0 we let s,qdenote the composition of natural maps (4.10,4.9)
0
s,q:totsk*(-) oes-!j*(-) d-!totqk*(-).
Observe that the tower map
d0 . oe :{totsk*(X)}s ! {totsk*(X)}s
is in fact represented by the level map { s,s(X)}.
We remark that s,qis natural in X as well as in s and q and that it is also
coherent 3.6 (as a composition of coherent maps).
4.18. Proposition.Let X be a trivial resolution, namely a + -diagram of spaces.
There is a map of simplicial sets, natural in X
Map ( o, k*Lank o) ! map({totsk*X}s, {totsk*X}s)
whose image contain the vertices ~idand d0oe (see 4.10). In fact
(a) The unit of adjunction j : o ! k*Lank o is a preimage of ~idand
(b) A preimage for d0oe is given by
0)n j(0) (d0)n+1s
n : n (s---! 0 --! (k*Lank o)(0) = (Lank o)(0) -----! (Lank o)(n).
Proof.Observe that (3.3)
(4.18.1) totnk*(-) = hom(skn o, k*(-)) ~=hom(Lankskn o, -)
is a coherent functor (3.4,3.5). If X is a trivial resolution, then there is an*
* obvious
map
CNat (totpk*(-), totqk*(-)) ! map(totpk*(X), totqk*(X)).
13
From 3.10, 3.3 and (4.18.1)
(4.18.2) CNat (totpk*(-), totqk*(-)) ~=Map (Lank skq o, Lankskp o) ~=
Map (skq o, k*Lank skp o).
We thus obtain natural maps (in S)
Map (skq o, k*Lank skp o) ! map(totpk*(X), totqk*(X)).
Now, sequential colimits (indexed by p) commute with map (skq o, -) (because
skq o is finite) and with k* and Lank. Therefore, we obtain the desired map
(4.18.3) :Map ( o, k*Lank o) ! map({totpk*X}p, {totqk*X}q).
Since idand d0oe = { s,s} (4.17) are levelwise coherent maps, they correspond to
vertices in the LHS of (4.18.3). In fact there is an effective way to compute t*
*hese
vertices by setting p = q, plugging Lankskp o into (4.18.2), computing the image
under idand d0oe of the vertex
j 2 totpk*Lankskp o = Map (skp o, k*Lank skp o)
where j is the unit of adjunction, and letting p ! 1.
Evidently, idcorresponds to the unit map j : skp o ! k*Lank(skp o), and when
p tends to 1, we obtain the unit j of o, which is point (a).
By the same method, the vertex := d0oe(jskp o) in the L.H.S of (4.18.3)
corresponds to d0oe. Using the explicit description of these maps (4.10), we se*
*e that
n is given by the composition
0)n j(0) (d0)n+1s
n (s---! 0 --! (k*Lankskp o)(0) = (Lankskp o)(0) -----! (Lankskp o)(n)
Letting p ! 1 we obtain point (b). |___|
4.19. Proposition.The vertices j and defined in 4.18 belong to the same com-
ponent of Map ( o, k*Lank o). In fact, they are the vertices of a 1-simplex in *
*this
space.
Proof.We shall construct a 1-simplex h such that @0(h) = j and j = @1(h) = .
Curiously, the data for this homotopy is contained in j.
We first define maps øn for n 0,
(4.19.1) øn : n+1 ! (Lank o)(n)
by the composition
øn : n+1 j(n+1)----!(k*Lank o)(n + 1)
o)(s)
= (Lank o)(n + 1) (Lank--------!(Lank o)(n).
Observe that
(4.19.2) øn . d0 = j(n)
by the commutativity of the following diagram (use 4.8; the bottom row is øn)
j(n) * o ________ o
n ________//(k Lank )(n)______ (Lank )(n)R
o 0| RRRR
d0|| (k*Lank o)(d0)|| (Lank )(d|) RidRRRRR
fflffl| fflffl| fflffl| RR((
n+1 j(n+1)//_(k*Lank o)(n +_1)__(Lank o)(n +(1)Lank/o)(s)/_(Lank o)(n).
14
Furthermore,
(4.19.3) øn . vn = n
due to the commutativity of (again use 4.8)
j(0) * o _________ o
0 _______//_(k Lank )(0)______(Lank )(0)R
| R(LankRo)(dn...d0s)RRRR
dn+1...d1|| dn+1...d1|| (Lank o)(dn+1...d1)| RRRR
fflffl| fflffl| fflffl| RR((
n+1 j(n+1)//_(k*Lank o)(n +_1)__(Lank o)(n +(1)Lank/o)(s)/_(Lank o)(n).
We now define the 1-simplex h = {hn} in Map ( o, k*Lank o) by the composition
hn : n x 1 ßn-! n+1 øn-!(Lank o)(n).
First, we claim that {hn} is indeed a cofacial space map (namely, an 1-simplex).
This follows from the commutativity of the following diagram for every n 0 and
every 0 i n + 1.
ßn øn o
n x 1 ______// n+1______//_(Lank )(n)
dix1|| di+1|| |(Lank|o)(di)
fflffl| fflffl| fflffl|
n+1 x 1 ßn+1_// n+2øn+1//_(Lank o)(n + 1).
The left square commutes by 4.16 while the right one by
o)(s)
n+1 -j(n+1)---!(Lank o)(n +(1)Lank--------!(Lank o)(n)
?? ?? ??
di+1y (Lank o)(di+1)y y(Lank o)(d0)
o)(s)
n+2 ----! (Lank o)(n + 2)-(Lank-------!(Lank o)(n + 1)
j(n+2)
which commutes by the identities in (4.8.1).
It follows that {hn} is a 1-simplex in Map ( o, k*Lank o) and moreover, using
4.16 and (4.19.2)
@0(hn) = hn . (1 x d0) = ønßn(1 x d0) = ønd0 = j(n).
Using (4.19.3)
@1(hn) = hn . (1 x d1) = ønßn(1 x d1) = ønvn = n.
|___|
Proof of Theorem 4.11.First, notice that, since X is fibrant and skp o are all *
*free
diagrams then the tower {totsX}s is fibrant (2.6).
Since oed0 = id(4.14), it suffices to show, by 2.11, that idand d0oe belong t*
*o the
same component of map (totok*X, totok*X). Using 4.18 it suffices to show that j
and belong to the same component of Map ( o, k*Lank o), which is precisely_the
assertion of 4.19. |__|
15
5. Resolutions of diagrams
5.1. Definition.Let D be a small category. A cofacial D-diagram Xo is a cofacial
object in the functor category SpcD , namely a functor Xo : ! SpcD. A cofacial
resolution, or simply a resolution, of D-diagrams is a functor X : 0 ! Spc D.
Compare this with 4.3,4.5. This data is the same as a functor X : D x 0 ! Spc,
and we will usually prefer this description.
As before Xo, Y o, . .d.enote cofacial D-diagrams (D should be understood from
the context) and boldface letters X, Y, . .d.enote resolutions of D-diagrams.
5.2. Definition.We call a resolution X of D-diagrams termwise trivial, if for
every object d in D, the resolution of spaces X(d) is trivial (4.7).
Observe that the trivialisations for every X(d) are not assumed to be compati*
*ble,
namely, a termwise trivial resolution X is not a diagram X: + ! SpcD and, in
general, cannot be turned into one by a different choice of the trivialisations.
5.3.Given a cofacial resolution X as in 5.1, and a diagram F : D ! S, we will
consider the obvious cofacial resolution hom D(F, X) which in codimension i -1
is hom D(F, Xi).
5.4. Definition.Let D be a small category. A diagram Y : D ! S is called a
relative free extension of a diagram X if there is a family {dff}ff2Aof objects*
* in
D and a family {Aff! Bff}ff2Aof cofibration in S which fit into the following
pushout square `
ff2AAffx?D(dff, -)----! X?
?y ?y
`
ff2ABffx D(dff, -)----! Y
Here D(dff, -) is the free diagram at dff(cf. [4]) considered as a diagram of d*
*iscrete
spaces.
Observe that the vertical arrows are cofibrations.
5.5. Definition.Let D be a small category. A diagram X : D ! S is called
poly-free if
X = Xn Xn-1 . . .X0 = ;
where Xi Xi+1is a relative free extension (5.4).
Clearly X is cofibrant.
5.6. Theorem. Let D be a small category and F : D ! S a poly-free diagram.
Then for every termwise trivial (objectwise) fibrant resolution X of D-diagrams,
the tower map
hom D(F, X-1) ! {totshomD(F, Xo)}s 0
is a s.p.h.e. In particular there exists s < 1 such that hom D(F, X-1) is up to
homotopy a retract of totshomD(F, Xo).
Proof.By definition there exists a filtration
F = Fn Fn-1 . . .F0 = ;
where Fi Fi+1is a free relative extension (5.4). We prove the theorem inductiv*
*ely
for the Fi's.
16
To start with, the case i = 0 is trivial because hom D(;, -) = *. By definiti*
*on
(5.4) of relative free extension we obtain a pullback square of fibrant resolut*
*ions
Q
homD (Fi+1, X)----! ff2Ahom D(Bff, X(dff))
?? ?
y ?y
Q
homD (Fi, X) ----! ff2Ahom D(Aff, X(dff))
where the vertical arrows are fibrations. Since X is termwise trivial, then the
resolutions of spaces in the right corners of the square are also trivial, so b*
*y Theorem
4.11 they induce a s.p.h.e. upon taking the tototower. By induction hypothesis *
*the
same happens with the resolution in the bottom left corner of the square. By_2.*
*14
the induction step is complete. |__|
5.7. Remark. The same proof works for diagrams F which`are elements in the
smallest class of diagrams which contain the diagrams i2IAix D(di, -) and is
closed under pushouts along cofibrations.
5.8. Corollary.Suppose that F ! F 0is a cofibration between poly-free diagrams
over a small category D. Let X be a termwise trivial fibrant resolution of poin*
*ted
spaces. Then
Map D(F 0=F, X-1) ! {totsMapD (F 0=F, Xo)}s 0
is a s.p.h.e. where F 0=F is pointed in the obvious way.
Proof.Use the Theorem 5.6, 2.17(a) and the fibre sequence
Map (F 0=F, X) ! homD (F 0, X) ! homD (F, X).
|___|
5.9.Examples of poly-free diagrams include
(a) Classifying spaces ED = D=- (see [2, pp. 292]) of finite dimensional categ*
*or-
ies D, i.e. ones whose nerve is finite dimensional.
(b) R with the usual Z action. More generally, if G is a group which has a
finite dimensional classifying space, then its universal cover is an examp*
*le of
a poly-free diagram.
For (a) we can do even better:
5.10.Suppose that D is a finite dimensional small category, namely the nerve of
D has dimension n < 1. Then D is obviously a direct category. More explicitly,
define a height function | . |:Obj (D) ! N by the assignment (cf. [2, pp. 292])
|d| = dim N(D=d).
Then obviously, if d '-!e is a morphism in D, then h(d) h(e), and equality ho*
*lds
if and only if ' = id.
5.11. Theorem. Let D be a finite dimensional small category. If X is a termwise
trivial fibrant resolution of D-diagrams, then for every cofibrant diagram F :D*
* ! S,
the tower map
hom D(F, X-1) ! {totshomD(F, Xo)}s
is a s.p.h.e. In particular, there exists s < 1 such that the natural map
homD (F, X-1) ! totshomD(F, Xo)
17
admits a left homotopy inverse, and holimDX-1 is, up to homotopy, a retract of
totsholimDXo for some s < 1.
Proof.Let Dk denote the full subcategory of D consisting of the objects d, such
that |d| k, and let j : Dk ! D denote the inclusion. By inspection, the natur*
*al
map Lanjj*(F ) ! F is injective (for this use the description of left Kan exten*
*sions
in [16, pp. 236], and the definition of free diagrams in [4] or [7]). Let F kde*
*note its
image. It is easy to see that F kis the subdiagram of F generated by the simpli*
*ces
of F (d) for all objects d such that |d| k.
Our proof proceeds by induction to show that for all k -1
(5.11.1) homD (F k, X-1) ! {totshomD (F k, Xo))}s
is a s.p.h.e. The conclusion of the theorem is obtained from k = dim(D).
The case k = -1 is trivial as F k= ;. For the induction step we consider the
following commutative diagram in Spc 0.
homD (F k, X) ----! hom D(F k-1, X)
?? ?
(5.11.2) y ?y
Q Q k-1
|d|=khom(F (d), X(d))----! |d|=khom(F (d), X(d))
This square is cartesian because D is direct and homD (F k, X) ~=homDk (j*F, j**
*X).
All the resolutions in the vertices of the squares are fibrant because X is fib*
*rant
and F jare cofibrant for all j. Moreover, the horizontal maps are clearly (term*
*wise)
fibrations because F k-1(d) ! F k(d) are cofibrations for all d 2 D. Now use the
induction hypothesis for the top right vertex of the square, and Theorem 4.11 f*
*or_
the resolutions at the bottom vertices, together with 2.14. |*
*__|
6.Applications to simplicial functors and polyGEMs
6.1.Recall that a coaugmented functor J on spaces (pointed or not) is one with
a natural map X ! JX for every space X. Coaugmented functors J give rise to
cofacial resolutions
X ! JoX
for every space X, in the usual way (cf. [14, Section 2]).
We say that a space X is J-injective if the map X ! JX admits a left homotopy
inverse. Our main example is when J is the Bousfield Kan [2] functor R : S ! S
where R is a unitary ring. Then the R-injective spaces are those which are homo*
*topy
equivalent to the product of Eilenberg-MacLane spaces of R-modules, known as
GEMs (see [3, p. 87]).
For a J-injective space, the resolution X ! JoX is easily seen to be trivial.
6.2.We remark that by a simple manipulation which does not alter the homotopy
types, J can be assumed to satisfy
(a) the coaugmentation X ! JX is a cofibration
(b) JX is fibrant (Kan complex) for all X. This will be assumed throughout. In
that case, the coaugmentation X ! JX admits a left homotopy inverse if and only
if it admits a strict left inverse.
6.3.We say that J is simplicial if for every A 2 S there is a natural map
(JX) A ! J(X A).
18
It is easy to show that a continuous coaugmented functor J, namely one which
induces a continuous map for all spaces X and Y
Map (X, Y ) ! Map (JX, JY )
is simplicial.
The force of simplicial functors is in the existence of a natural map of reso*
*lutions
(compare [15])
Johom D(F, X) -c!homD (F, JoX).
So in particular the composition
homD (F, X) ! Johom D(F, X) -c!homD (F, JoX)
is induced by X ! JoX.
6.4. Theorem. Let D be a small category and assume that either
(a) F :D ! S is poly-free (5.5) or
(b) D is finite dimensional and F :D ! S is cofibrant.
If X :D ! Spc is a diagram of fibrant J-injective spaces, then
homD (F, X) ! Jshom D(F, X)
admits a left homotopy inverse for some s < 1. In particular, when F is chosen
weakly contractible (always possible in case (b)), then it follows that holimDX*
* is up
to homotopy a retract of JsholimDX.
Proof.The resolution X ! JoX is fibrant and termwise trivial (6.1). The result
follows from Theorems 5.11, 5.6 using the composition (see 6.3)
homD (F, X) ! Jshom D(F, X) = totsJohom D(F, X) -c!totshomD(F, JoX).
|___|
Recall from the introduction the concepts of GEMs and polyGEMs. Recall also
that X is a GEM if and only if it is up to homotopy Z-injective where Z is the
Bousfield Kan functor [2].
6.5. Corollary.Assume that D and F are as in 6.4. If X : D ! Spc is diagram
of fibrant GEMs, then hom D(F, X) is a polyGEM and is up to homotopy a retract
of Zshom D(F, X) for some s < 1.
Proof.The first assertion is implicit in the proofs of 5.6 and 5.11. The secon*
*d_
follows from 6.4 and the remarks in 6.2. |__|
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*hematics,
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[17]C. L. Reedy, Homotopy theory of model categories. Preprint, 1973. Available*
* from http://
math.mit.edu/ psh.
Department of Mathematical Sciences, King's College, University of Aberdeen, *
*Ab-
erdeen AB24 3UE , Scotland, U.K.
E-mail address: assaf@maths.abdn.ac.uk
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