COLIMITS FOR THE PRO CATEGORY OF TOWERS OF
SIMPLICIAL SETS
DAVID BLANC
Revised: January 18, 1995
Abstract.We describe a certain category of Indtowers into which the Pro*
* category
of towers of simplicial sets embeds, and in which all colimits (rather t*
*han just the
finite ones) may be constructed explicitly.
1.Introduction
The Pro category of towers of spaces (and of other categories) has been stud*
*ied
in several contexts, and used for a variety of applications in homotopy theory,*
* shape
theory, geometric topology, and algebraic geometry  see for example [AM , B*
*K , DF ,
EH , F, G , GV , H , HP , MS ]. Our interest in it first arose, in [BT ], in *
*the study of
vnperiodicity in unstable homotopy theory (cf. [Bo , D, Md , MT ]).
One problem in the usual version of the Pro category of towers is that, whil*
*e fi
nite limits and colimits exist, and may be constructed in a straightforward (le*
*velwise)
manner, the same does not hold for infinite colimits; and these were needed for*
* the
application we had in mind in [BT ]. It is the purpose of the present note to *
*improve
on the rather ad hoc solution to this difficulty presented in [BT , x3] (in ter*
*ms of of
what were there called "virtual towers"), by enlarging the Pro category of towe*
*rs in
such a way as to allow a straightforward construction of arbitrary colimits. On*
*e object
of this is to enable us to then provide a suitable framework for studying perio*
*dicity
in unstable homotopy theory in terms of a Quillen model category structure for *
*our
version of the Pro category of towers (see [Bc ]).
The construction we provide embeds (a suitable subcategory of) the Pro categ*
*ory
Tow of towers of simplicial sets in a certain category Net of strict Indtowe*
*rs, in
which we have explicit constructions for all colimits, as well as finite limits*
*. This
category Net can thus be thought of as a cocompletion of the Pro category of *
*towers
of spaces. We shall show in [Bc ] how this construction can also provide a "hom*
*otopy
theory of finite simplicial sets" (compare [Q , II, 4.10, remark 1]); it may be*
* of use in
other contexts, too.
There are other cocomplete categories in which Tow may be embedded  for
example, the category P roS? of all prosimplicial sets (cf. [AM , A.4.3 & *
*A.4.4]), or
the full category IndTow of all inductive systems of towers (cf. [J, VI, Thm*
*. 1.6] 
this is actually "the" cocompletion of Tow, in an appropriate sense: see [J, *
*VI, x1]
or [TT ]. One advantage of the approach described here is that one obtains a sm*
*aller,
_____________
1991 Mathematics Subject Classification. Primary 55U99; Secondary 18A30.
Key words and phrases. Prospace, Prosimplicial set, tower, net, Indtower, *
*colimit, cocompletion.
1
2 DAVID BLANC
and more mangeable, cocompletion, in this special case, and the construction of*
* the
colimits may be made quite explicitly.
A side effect of our approach is the elimination of certain "phantom phenome*
*na"
from the Pro category of towers (see x2.10(b) and x4.13 below).
1.1. conventions and notation. Let T? denote the category of pointed topological
spaces, S the category of simplical sets, and S? that of pointed simplicial s*
*ets (see
[My ]). We shall refer to the objects of S? simply as spaces. A finite simplic*
*ial set Xo
is one with only finitely many nondegenerate simplices (in all dimensions toge*
*ther).
(For technical convenience we prefer to work with simplicial sets, rather th*
*an topo
logical spaces. This makes no difference for our purposes, since T? and S? *
* have
equivalent homotopy theories, in the sense of Quillen  see [Q , I, x4].)
The category (ordered set) of natural numbers will be denoted by N, the ca*
*tegory
of abelian groups by AbGp, and the category of Rmodules (for a commutative r*
*ing
R) by RMod.
For any category C we shall denote by IndC the category of Indobjects ov*
*er C 
that is, diagrams F : J ! C, where J is a small filtered category (cf. [GV , *
*Defs. 2.7
& 8.2.1])  with the appropriate morphisms (see [GV , Def. 8.2.4]). Similarly,*
* P roC
denotes the category of Proobjects over C (i.e., diagrams F : J ! C where *
*Jop is
filtered  cf. [GV , Def. 8.10.1]).
For any functor F : I ! C we denote the (inverse) limit of F simply by li*
*mF or
lim IF , (rather than lim ), and the colimit (=direct limit) by colimF . The *
*(co)limit
is finite if the category I is such (finitely many objects and morphisms betwee*
*n them).
All limits and colimits in this paper are assumed to be small  i.e., Ob(I) *
* is a set.
A category C is called pointed if it has a zero object (i.e., one which is b*
*oth initial
and terminal). This object will be denoted by *C (or simply *).
1.2. organization. In section 2 we give some background on towers of simplicial*
* sets,
their Pro category Tow, and the finite (co)limits in Tow. In section 3 we d*
*efine
"good" subcategories"  a concept which merely codifies those properties of *
*Tow
which are needed to construct its cocompletion. In section 4 we show that the *
*category
Net, consisting of certain strict Indobjects over such a good subcategory of *
*C, serves
as a cocompletion for C, in the sense of having all colimits (and all finite li*
*mits).
1.3. acknowledgements. I am grateful to the referee for his comments, and in pa*
*rtic
ular for providing references to previous work, and suggesting Remark 4.12. I *
*would
also like to thank Emmanuel DrorFarjoun, Bill Dwyer, Haynes Miller, and Brooke
Shipley for several useful conversations.
2. The category of towers
In order to fix notation, we recall the definition of the usual Pro category*
* of towers
of spaces:
2.1. towers of spaces. The objects we shall be studying are towers in S?  i.*
*e.,
sequences of pointed spaces and maps
pn pn1 p0
X = { : :X:[n + 1] ! X [n] ! X [n  1] ! : ::! X [0] };
COLIMITS FOR TOWERS OF SIMPLICIAL SETS 3
where the space X [n] is called the nth level of X (n 0), and the map pn*
* is
called the nth level map (or bonding map) of X. We denote such towers by Goth*
*ic
letters: X; Y; : :.:
For any n > m, the iterated level map pnm: X [n] ! X [m] is defined to b*
*e the
composite of
pn1 pm
X [n] ! X [n  1] ! : :!:X [m + 1] ! X [m]
(so pn+1n= pn). We set pnn= idX [n].
Definition 2.2. Such towers are simply objects in the functor category SN? of *
*dia
grams in S? indexed by the ordered set (N; ) of natural numbers. Thus a morphism
f : X ! Y between two towers
pn qn
X = {: :X:[n] ! X [n  1] : :}:and Y = {: :Y:[n] ! Y [n  1] ! : :}:
is just a sequence (f[k] : X [k] ! Y [k])1k=0of maps such that qkO f[k + 1] = f*
*[k] O pk
for k 0.
The category SN? has all limits and colimits, of course. However, we are in*
*terested
rather in the Pro category of towers of spaces:
Definition 2.3. Let N N! denote the set of sequences (ns)1s=0of natural number*
*s,
such that ns max {ns1; s} for all s > 0. We shall denote the elements of *
*N by
lower case Greek letters, with the convention that = (ms)1s=0, = (ns)1s=0, *
* and so
on. The set N is partially ordered by the relation , ms ns for all s 0
 in fact, is a lattice. N has a least element ! = (s)1s=0(though o*
*f course
no maximal elements). Moreover, N is a monoid under composition (where = O
is defined by ns = `ms), with ! as the unit and ; O .
Definition 2.4. Given a tower X and a sequence = (nk)1k=02 N , we define the
spaced tower over X, denoted X<>, to be:
pnk+1nk pn1n0
: :X:[nk+1] ! X [nk] ! : :!:X [n1] ! X [n0]:
In particular X = X. Note that ()<> is a functor on SN?.
If in N , there is an SN?map p : X<> ! X<> defined by pnkmk: X [n*
*k] !
X [mk ] for all k 0. Such a p will be called a selftower map (with respect*
* to X).
If = ! we write simply p for p , and call p a basic selftower map (*
*for X).
For = we have p = id, and the composite of two selftower maps (with res*
*pect
to the same X), when defined, is a selftower map.
Definition 2.5. We define now define the category of towers, denoted Tow, in wh*
*ich
the selftower maps have been inverted: the objects of Tow are towers of spa*
*ces (as
in x2.1) and its morphisms, called tower maps, are defined for any X, Z by:
Def
(2.6) HomTow(X; Y) = colim HomSN?(X<>; Y)
2 N
(compare [GV , x8.2.4]). Equivalently, one may define Pro maps of towers (cf. *
*[EH ,
2.1]) by:
HomTow(X; Y) ~= lim colim HomS?(X [n]; Y[s]):
s 2 N n 2 N
4 DAVID BLANC
(It is not hard to see this is equivalent to the above, since any Pro map of*
* towers
from X to Y is represented by a sequence of maps f
s : X [ns] ! Y [s] (s 0) *
*in S?,
which are compatible in the sense that for each s there is an ms ns; ns+1 suc*
*h that
qsO fs+1O pmsns+1= fsO pmsns: X [ms ] ! Y [s]. This defines an SN?map f : X*
*<> ! Y
for 2 N defined by k0 = n0 and ks+1 = max {ms; ks}. One readily verifies *
*this
correspondence between the two definitions of tower maps is bijective.)
Proposition 2.7. The category Tow has all finite limits and colimits.
Proof. This is well known; for completeness we recapitulate the proof:
To show that Tow has all finite limits, it suffices to show that Tow has a t*
*erminal
object and pullbacks (cf. [Bx1 , Prop. 2.8.2]). The tower * (with *[n] con*
*sisting of
a single point for each n) is clearly a terminal object in Tow. In order to de*
*fine the
pullback in Tow of two tower maps:
^f ^g
(2.8) X ! Z  Y ;
f g
choose any two SN?maps X ! Z  Y representing (2.8); their pullback P*
* (in
f[n] g*
*[n]
SN?) is defined levelwise (i.e., P [n] is the pullback of X [n] ! Z [n] *
* Y [n] in
S?), and similarly for the structure maps i : P ! X and j : P ! Y such that
f O i = g O j. The level maps of P are induced from those of X and Y by the na*
*turality
of the pullback in S?.
Now given a basic selftower map p : X<> ! X, denote by "P the pullback *
*(in
fOp g
SN?) of X<> ! Z  Y.
Again by the naturality of the pullback we can fit suitable spacings of P an*
*d "P
together as follows:
_________________
 nk 
 pk 
 ?
: :":P[nk]_P [nk]__"P[k]__P [k]
 n 6
 "pkk 
________________
implying that P and "Pare isomorphic; this shows that the pullback ^P of (2.8*
*) in
Tow is welldefined by taking the pullback P in SN? of any representatives of*
* ^f, ^g.
Next, given tower maps ^h: W ! X and ^k: W ! Y with ^gO ^k= ^fO ^h(in T*
*ow),
there are SN?representatives h : W ! X and k : W ! Y with g O k = f O h *
* in SN?
(for suitable spacings of X, Y and W). Thus by the universal property in S? *
* the
SN?maps h and k factor through the unique "universal" SN?map l : W ! P.
Conversely, given a tower map ^l2 HomTow(W; P) such that
(2.9) ^iO^l= ^h and ^jO^l= ^k in Tow ;
one can find SN?representatives for the maps in question such that i O l =*
* h and
j O l = k in SN?, so that the SN?map l : W ! P is the "universal map" as
above. Thus it suffices to check that any two universal SN?maps l : W ! P *
*and
l0 : W ! P<> represent the same tower map; but this follows readily from the
uniqueness of the universal maps in SN?.
COLIMITS FOR TOWERS OF SIMPLICIAL SETS 5
To show that Tow has all finite colimits, we show analogously that it has a*
*n initial
object and pushouts, again defined levelwise. _____
Remark 2.10. The category Tow may be embedded in a category with all colimits
(and all filtered limits)  namely, the category P roS? of all prosimplici*
*al sets (see
[AM , A.4.3 & A.4.4] or [GV , Props. 8.9.1 & 8.9.5]). The problem is that the*
* limit or
colimit of an infinite diagram of towers will not itself be a tower, and is rat*
*her difficult
to construct explicitly.
Note that the naive (levelwise) construction of colimits in Tow can fail i*
*n two
different ways:
(a) If `{Xff}ff2A is some (infinite) collection of towers, and we define a*
* tower Y
by Y [n] = ff2AXff[n], then Y is "too small"  in general, there will be *
*maps
^fff: Xff! Z in Tow such that for any choice of representatives fff: Xff ! Z in
SN?, the set of numbers {nff0}ff2A is unbounded. Thus there will be no way t*
*o define
a SN?representative of the putative corresponding map ^f: Y ! Z which restri*
*cts to
^fffon Xff.
(b) On the other hand, let (Ai)1i=0be some sequence of nontrivial spaces*
*, and
define towers (Xi)1i=0by letting X i[n] = Ai, `and (pi)n = IdAi, if n i, *
* and
X i[n] = * otherwise. If again we set Y [n] = 1i=0Xi[n], we see Y is now "to*
*o big":
For a given tower Z, any collection of maps (fi : Ai ! Z [i])1i=0yields a*
* unique
SN?map f : Y ! Z in the obvious way, and two such choices (fi)1i=0and (gi)*
*1i=0
yield equivalent tower maps (^f= ^gin Tow) if and only if there is an N such*
* that
fi= gi for i N (at least for suitable Z  e.g., if Z is constant) . Thus *
*there are
many such tower maps ^f: Y ! Z; but the corresponding maps ^fi: Xi! Z are all
trivial in Tow. (In some sense the maps ^fso defined may be thought of as "ph*
*antom
tower maps"  compare [GM ]).
3. Good subcategories
We now describe those properties of the category Tow which are needed to con*
*struct
the extension. Since this construction is also needed for [Bc ], we describe it*
* in greater
generality than required for our immediate purposes.
Definition 3.1. Let C be a pointed category, and F a small full subcategory. Fo*
*r each
A 2 C, let FA denote the subcategory of the over category C=A (cf. [Bx1 , *
*x1.2.7]),
whose objects are monomorphisms i : F ,! A with F 2 F and whose morphisms
are (necessarily monic) maps j : F ! F 0such that i0O j = i. Similarly, let*
* FA
denote the subcategory of the under category A=C, whose objects are epimorphi*
*sms
q : A!!F with F 2 F, and whose morphisms are (epic) maps p : F ! F 0with
p O q = q0.
We say that F is a good subcategory of C if:
(a) F is closed under taking subobjects and quotient objects.
(b) F is finitecomplete and cocomplete, and the inclusion I : F ! C is @0(c*
*o)con
tinuous (i.e., any finite diagram (Fff)ff2A (A < 1) over F has a limit L a*
*nd a
colimit C in C, with L; C 2 F).
6 DAVID BLANC
(c) For any F 2 F the category FF is coartinian  that is, given a seque*
*nce of
quotient maps
q0 q1 q2 qn qn+1
F! ! G0!! G1!! : :G:n1!! Gn!! : :;:
there is an N such that qn is an isomorphism for n N (compare [GV , x8.12.*
*6]).
(d) Any morphism f : F ! C, with F 2 F and C 2 C, has an epimorphic image
Im(f) (see [Bx1 , Def. 4.4.4])  which is necessarily in FC.
The inclusions iG : G ! C thus induce a natural bijection:
(3.2) F;C: colim HomF(F; G) =! HomC(F; C):
G 2 FC
Definition 3.3. Let Towst denote the category of (essentially) strict towers o*
*f sim
plicial sets (cf. [GV , x8.12.1])  that is, the full subcategory of Tow wh*
*ose objects
are towers X for which there is an N such that all level maps pn : X [n + 1] !*
* X [n]
are epimorphisms for n N. (We think of these as being "good" towers, because
they avoid the pathologies mentioned in 2.10(b)). Note that Towst has all fi*
*nite
colimits and products, but not all pullbacks.
Let F = FTow denote the full subcategory of Towst whose objects are tower*
*s X
such that each X [n] is a finite simplicial set (x1.1), and there is an N such *
*that pn is
an isomorphism for n N. We denote by Sp(X) the finite simplicial set limk*
*X [k]
(which is naturally isomorphic to X [n] for n N).
Proposition 3.4. F = FTow is a good subcategory of Towst.
Proof. (1) Given Y 2 F, let ^f: X ! Y be a monomorphism in Towst, with
a representative f : X<> ! Y. For simplicity of notation let X = X<>. Now let
Z = {: :Z:[n] sn!Z[n  1] ! : :}: be the (levelwise) pullback (in SN?) of
f f
X ! Y  X ;
with h1; h2 : Z ! X the two projections.
Since f O h1 = f O h2 and ^fis a monomorphism in Tow, there is a 2 N *
*such
that h1 O s = h2 O s : Z<> ! X in SN?.
Now let (x0; x1) 2 Z [k]t X [k]tx X [k]t be a pair of kth level tsimpli*
*ces of
X. Since the level maps pn : X [n + 1] ! X [n] of X are epimorphisms, ther*
*e are
tsimplices x0; x12 X [nk]t such that pnkk(xi) = xi (i = 0; 1).
Thus qnkk(f(x0)) = f(pnkk(x0)) = f(x0) = f(x1) = f(pnkk(x1)) = qnkk(f(x1)),*
* and
since Y 2 F, each level map qn of Y is monic, and so (f(x0)) = (f(x1)), *
* i.e.,
(x0; x1) 2 Z [nk]t, with snkk(x0; x1) 2 Z [nk]t = (x0; x1). But then x0 = x*
*1, since
v1 O s = v2 O s . Thus f is levelwise monic.
But this implies that each qn O f[n + 1] = f[n] O pn is monic, so pn is,*
* too, and
since Y 2 F we see X 2 F, too.
(2) If ^f: X ! Y is any epimorphism in Towst, we shall show more generall*
*y that
^fmay be represented by a levelwise epimorphism: without loss of generality, ^f*
*has an
SN?representative f : X ! Y; by factoring f via its (levelwise) image, Im(f), *
*we may
COLIMITS FOR TOWERS OF SIMPLICIAL SETS 7
assume that f is levelwise monic. Now set Z = Y=X to be the (levelwise) push*
*out
(in SN
? ) of
f
*  X ! Y ;
with two SN?maps: g : Y!!X the quotient map, and * = h : Y!!X the
trivial map. Clearly g O f = * = h O h (in SN?), so there is a 2 N such*
* that
g O q = h O q : Y<> ! Z, since ^fis an epimorphism in Tow. But then g O q*
* = *,
so q factors as f O q (for q : Y<> ! X), with q O f<> = p , so that f;*
* q are
inverese to each other in Tow, and thus f is an isomorphism. As before we con*
*clude
that if X 2 F then also Y 2 F.
(3) Given a finite diagram over F, its limit and colimit in Tow may be de*
*fined
levelwise by Proposition 2.7, so are in F.
(4) Since epimorphisms in Towst are actually levelwise surjections, the cat*
*egory
FY is equivalent to a finite category for any Y 2 F  so in particular it is c*
*oartinian.
(5) Given ^f: X ! Y in Towst, with X 2 F, one can define Im(f) = Z
to be the levelwise image tower for some SN?representative f : X<> ! Y of ^*
*f.
This is independent of the representative f chosen, since given another represe*
*ntative
f0: X<> ! Y, there is a ; such that f0O s = f O s , and thus both maps
have the same (levelwise) image, because Z is in F by (2), and thus the level m*
*aps sn
of Z are epimorphic. We write i : Z ! Y for the inclusion, with f : X!!Z such *
*that
f = i O f.
Now if ^j: W ! Y is another monomorphism in Towst, equipped with a tower map
^g: X ! W such that ^jO^g= ^fin Tow, we may assume without loss of generality t*
*hat
^gis represented by g : X<> :! W with j O g = f = i O f, and moreover by fa*
*ctoring
g itself through its image we may assume g is levelwise epimorphic, so W 2 F *
* by
(2), being a quotient of X.
Finally, factoring our chosen SN?representative j : W ! Y through Im(j) *
* (which
is in F, by (2)), we find that the SN?map W!!Im(j) is an isomorphism, as in*
* (a);
but since Im(j) ,! Y is a levelwise monomorphism, by the universal property of *
*Im
in S? (and thus in SN?) there is a (levelwise) monomorphism k : Z ! W thro*
*ugh
which g and i factor, showing that f = i O f is indeed initial in Towst among*
* the
factorizations ^f= ^jO ^gof ^fwith ^jmonic. _____
3.5. good generating subcategories. We shall in fact be interested in good sub
categories F C which generate C (cf. [Bx1 , Def. 4.5.1] or [Me , V, x7])  *
* that is,
such that for any object C 2 C, {f : F ! C}f:F!C;F2F is an epimorphic family
([GV , x10.3]).
Note that because of 3.1(d) and (3.2), this is equivalent to requiring that,*
* for all
C; D 2 C, there be a canonical natural inclusion of sets:
(3.6) JC;D : HomC(C; D) ,! lim colim HomF(F; G);
F 2 FC G 2 FD
induced by the restrictions fF for any f : C ! D and the correspondences *
*1F;D
of (3.2).
Example 3.7. (i) The category of pointed sets is generated by the good subcate*
*gory
of finite pointed sets.
8 DAVID BLANC
(ii) The category S? of pointed simplicial sets is generated by the good su*
*bcategory
Sf
? of finite pointed simplicial sets (x1.1).
(iii) The category of torsion groups is generated by the good subcategory o*
*f finite
groups.
(iv) The category AbGp is generated by the good subcategory of finitely gen*
*erated
abelian groups, and more generally RMod is generated by the good subcategory*
* of
f.g. Rmodules for any noetherian ring R.
In these cases the natural inclusion JC;D of (3.6) is actually bijective.
Proposition 3.8. The category Towst is generated by the subcategory FTow.
Proof. Let ^f6= ^g: X ! Y be two different tower maps, with X 2 Towst; with*
*out
loss of generality we may assume they have SN?representatives f; g : X<> ! Y
respectively. By definition 2.5, there is a k 0 such that for all n k we*
* have
f[n] O pnk6= g[n] O pnk, so in particular for each n k there is a tsimplex x*
*n 2 X [n]t
(t independent of n) such that f[n](xn) 6= g[n](xn). Since X 2 Towst, the leve*
*l maps
pn of X are surjective, and we may evidently assume pn(xn+1) = xn for all n*
* N.
To each xn 2 X [n]t there corresponds a map 'xn : [t] ! X [n]t, and let Z *
*[n] 2
S? denote the simplicial set Im('xn). Then Z = {: :Z:[n] sn!Z[n  1] ! : :}*
*: is
in fact a subtower of X, with sn = pnZ [n+1], and because each Z [n] is a quo*
*tient of
both [t] and Z [n + 1], for sufficiently large n the maps sn must be isomorph*
*isms
(since [t] has only finitely many nonisomorphic quotients), so that Z 2 FX.
Clearly fZ 6= gZ, and both have images in F by Definition 3.1(d) and Pro*
*position
3.4  which proves JX;Y is indeed onetoone. _____
It may be useful to think of the finite subtowers Z ,! X (Z 2 F) as the a*
*nalogue
of the stable cells of a CW spectrum  compare [A , III, x3].
Remark 3.9. Note that in general our category C will not be locally generated b*
*y the
subcategory F, in the sense of [GU , xx7,9], because C need not be cocomplete *
* and
we are interested precisely in such cases, because only then will the cocomplet*
*ion of
C be of interest. C need not even be @0accessible in the sense of [Bx2 , Def.*
* 5.3.1],
because we do not assume that FC has all colimits for arbitrary C 2 C.
4. Nets and cocompletion
When C is generated by a good subcategory F, it embeds in the category Inds*
*tF
of strict Indobjects over F; by constructing all colimits for IndstF, (or r*
*ather, for
an equivalent subcategory Net), we show that this can serve as a cocompletion *
*for C.
In analogy with the completion of a metric space, the objects of Net are the*
*mselves
directed systems of suitable towers; one should think of these as representing *
*their
colimit (which may not exist in Tow).
Definition 4.1. A strict Ind object over a category G is a diagram X : I ! G
indexed by a small filtered partially ordered category I, such that all bonding*
* maps
X(f) : Xff! Xfi (for f : ff ! fi in A) are monomorphisms (cf. [GV , Def. 8.1*
*2.1]).
The full subcategory of IndG whose objects are strict will be denoted by In*
*dstG.
In order to simplify our constuctions, it is convenient to consider the subc*
*ategory
Net IndstG defined as follows (this is actually equivalent to IndstG, under *
*suitable
assumptions  see Fact 4.4 below):
COLIMITS FOR TOWERS OF SIMPLICIAL SETS 9
Definition 4.2. If G is a pointed category, a net over G to be a strict Indobj*
*ect
(X
ff)ff2A indexed by a lattice with least element 0, such X0 = *
**, and
for each ff; fi 2 A, the square:
iff^fi;ff
Xff^fi_____Xff

 
iff^fi;fi iff;ff_fi
 
? ifi;ff_f?i
Xfi_______Xff_fi_
Figure 1
is both cartesian and cocartesian. (Since we required the bonding maps of the *
*net to
be monic, this simply means that Xff^fiis the intersection Xff\ Xfiof Xffand Xfi
and Xff_fiis their union Xff[ Xfi (cf. [Bx1 , Def. 4.2.1 & Prop. 4.2.3])
Definition 4.3. If (Xff)ff2A and (Yfi)fi2B are two nets over G, a proper net *
*map
between them is a pair , where OE : A ! B is a orderpreserv*
*ing map
with OE(0) = 0, and for each ff 2 A, fff: Xff! YOE(ff)is a morphism in G. *
*We
require that for all ff fi in A, the diagram:
fff
Xff______YOE(ff)

 
iff;fi iOE(ff);OE(fi)
 
? ffi ?
Xfi_____YOE(fi)_
commutes. If YOE(ff)= Im(fff) for all ff 2 A  in other words, each fff i*
*s epic 
we say is a minimal proper net map.
Two proper net maps ; < ; (gfi)fi2B> : (Xff)ff2A! (Yfi)fi2Bar*
*e equivalent
 written ' < ; (gfi)fi2B>  if for each ff 2 A there is*
* an ae(ff) such
that OE(ff) _ (ff) ae(ff) (in the lattice B) and the diagram
fff
Xff___________YOE(ff)

 
gff iOE(ff);ae(ff)
 
? i (ff);ae(?ff)
Y (ff)________Yae(ff)_
commutes. Note that if G is a category with images, then each equivalence class*
* of
proper net maps will have a unique minimal representative.
The category of nets over G, with equivalence classes of proper net maps as *
*mor
phisms, will be denoted NetG. We shall sometimes use the notation f : (Xff)f*
*f2A!
(Yfi)fi2B to denote a morphism of nets (i.e., an equivalence class of proper ma*
*ps) 
cf. [GV , x8.2.45].
Fact 4.4. If G has finite unions and intersections, every object in IndstG i*
*s isomor
phic to one in NetG.
Proof. By the dual of [MS , I, x1, Thm. 4] every (strict) Indobject over G is*
* Ind
isomorphic to a (strict) Indobject (X )2 indexed by a directed ordered set <*
*; >
10 DAVID BLANC
which is closure finite  i.e., the set of predecessors of every 2 is fin*
*ite. Now
let A be the free lattice generated by , and set X S
ff^fi= flff;fiXfl, with *
* Xff_fi
defined by Figure 1. Since is cofinal in A, we actually have an Indisomorphi*
*sm
(X )2 ,! (Xff)ff2A. _____
This shows that we could assume, if we wish, that our nets are always indexe*
*d by
closure finite lattices (and this will in fact be the case for FTow, of cours*
*e, because
in this case FF will be a finite category for each F 2 F), but this is not *
*needed for
our constructions.
Proposition 4.5. If F C is good (so in particular has all finite colimits), *
*then
N etF has all colimits.
Proof. It suffices to show N etF has coproducts and pushouts (cf. [P , x2.6, P*
*rop. 1 &
2]):
I. Given any`collection {(Xiff)ff2Ai}i2I of nets over F (indexed by an arbi*
*trary set
I), let B = i2IAi denote the coproduct lattice  so that the elements of B *
*are of
the form fi = ffi1_ : :_:ffin for ffij2 Aij (and ij 6= ik for j 6= k).
The coproduct net is then defined to be
n_
( Xijffij)ffi1_:::_ffin2B;
j=1
and the universal property for the coproduct evidently holds.
II. Given two net maps with minimal proper representatives:
(Xff)ff2A____________(Yfi)fi2B


< ; (gff)ff2A>

?
(Zfl)fl2C
Figure 2
For each fi 2 B and fl 2 C, let A(fi;fl)= {(ff 2 A  OE(ff) fi & (ff) f*
*l} (a
sublattice of A), and for each ff 2 A(fi;fl), let W ff= Wfffi_fldenote the p*
*ushout in:
fff iOE(ff);fi
Xff_______YOE(ff)________Yfi
g  pp
ff pp
? pp
Z (ff) pp
i  pp
(ff);fl pp
? ?
Zflpppppppppppppppppppppppppp_Wfffi_fl
Now for each fi0 _ fl0 in the coproduct lattice B q C let
[ [
Lfi0_fl0= {(ff; fi; fl) 2 A x B x C  ff 2 A(fi;fl)}:
fi0fifl0fl
COLIMITS FOR TOWERS OF SIMPLICIAL SETS 11
For any (ff; fi; fl) 2 Lfi0_fl0, the bonding maps ifi0;fiand ifi0;fiinduce *
*a map
qfi0_fl0(ff;fi;fl): Yfi0q Zfl0! Wfffi_fl:
We let U(ff;fi;fl)= Ufi0_fl0(ff;fi;fl)denote Im(qfi0_fl0(ff;fi;fl)) Wfffi_fl.
Note that, for fixed fi0 _ fl0 2 B q C, the objects U(ff;fi;fl)form a dia*
*gram in F
indexed by the (possibly infinite) filtered set Lfi0_fl0, and set
Def fi0_fl0
(4.6) Wfi0_fl0= colim U(ff;fi;fl):
(ff; fi; fl) 2 Lfi0_fl0
This limit exists in F  in fact, in F0 = FYfi0qZfl0, by [Bx1 , Prop. 2.16.*
*3]  since
F0 is coartinian by Def. 3.1(c), and thus has all filtered colimits. The natur*
*al map
ifi0_fl0;fi1_fl1: Wfi0_fl0! Wfi1_fl1
(induced by the fact that each qfi0_fl0(ff;fi;fl)factors through qfi1_fl1(ff;fi*
*;fl)) is always a monomor
phism. Thus we have defined as net (Wfi_fl)fi_fl2BqC) over F. (Had we not req*
*uired
that our nets be strict Indobjects, we could have defined Wfi0_fl0more simply*
* as the
colimit of the objects Wfffi0_fl0for ff 2 A(fi0;fl0)).
We claim that this net is the pushout for the diagram in Figure 2: given a *
*commu
tative diagram in NetF
(Xff)ff2A____________(Yfi)fi2B
 
 
< ; (gff)ff2A> 
 
(Z?fl)______________l2C?(V")"2E
(where we may assume the proper representatives indicated make it commute on the
nose), we define a net map : (Wfi_fl)fi_fl2BqC! (V")"2E*
* as follows:
set
Def _ 0 0
o(fi _ fl) = ae(fi ) _ oe(fl ):
(ff0;fi0;fl0)2L^fi_fl
We then have `fi_fl: Yfiq Zfl! Vo(fi_fl)induced by the appropriate bonding ma*
*ps,
and if OE(ff) fi and (ff) fl, the diagram
fff iOE(ff);fi
Xff_______YOE(ff)______Yfi
g   
ff  
? ? 
Z (ff)_____Wfffi_flpppffi O hfi
pppp`fi_fl
i (ff);fl pppp 
 pppp
?__________________s_iO?kfl
Zfl Vo(fi_fl)
commutes, so `fi_flinduces a map `fffi_fl: Wfffi_fl! Vo(fi_fl), and thus `*
*fi_fl: Wfi_fl!
Vo(fi_fl). One may also verify that has the appropria*
*te universal
property. _____
12 DAVID BLANC
Proposition 4.7. If F C is good (so in particular has all finite limits), the*
*n N etF
has all finite limits, too.
Proof. It suffices to show that N etF has pullbacks (it clearly has a terminal*
* object 
namely, the zero net indexed by the zero lattice). Thus, given two net maps:
(Yfi)fi2B




< ; (gfl)fl2C?>
(Zfl)fl2C____________(Xff)ff2A
where we assume the indicated representatives are minimal, for each (fi; fl) 2*
* B x C,
set W(fi; fl) to be the pullback of
gfl ffi
Zfl! X (fl)_OE(fi)Yfi:
It is readily verified that this defines a pullback net with the required unive*
*rsal prop
erty. _____
Proposition 4.8. If C is generated by a good subcategory F C, then there is *
*an
embedding of categories I : C ! NetF, defined I(C) = FC = (F )F2FC.
(Note that FC is both the lattice indexing the net I(C) 2 NetF, and the net*
* itself)
Proof. For any f : C ! D in C and F 2 FC a subobject of C which is in F, the
image Im(fF) is in FD by Def. 3.1(e). Thus we may define a proper net map
: FC ! FD by OEf(F ) = Im(fF) 2 FD and fF = fF : F !
Im(fF) D, for any F 2 FC. This defines I on morphisms. Definition 3.1(e) a*
*lso
implies that I : HomC(C; D) ! HomNetF(FC; FD ) is monic, since if f; g : C !*
* D
satisfy fF = gF for all F 2 FC, then f = g. _____
We may summarize our results for the Pro category of towers of spaces in the*
* fol
lowing
Theorem 4.9. The functor I : Tow ! NetF, defined by I(X) = FX restricts to
an embedding of Towst in the cocomplete and finite complete category of nets o*
*ver
FTow. I preserves all finite limits, and the functor ITowst preserves all co*
*limits.
Proof . If W is the pullback in Tow of
^f ^g
(4.10) Z ! X  Z ;
(which may not be in Towst, even if (4.10) is), then (as in the proof of Propos*
*ition 2.7)
W may be constructed as the levelwise pullback of any SN?representatives of (*
*4.10),
so W [n] is a subobject of Y [n] x Z[n] (by the usual construction in S?). Th*
*us any
finite subtower of W is just a finite subobject of Y x Z, satisfying the ap*
*propriate
(levelwise) compatibility condition  so that FW is isomorphic to the pullb*
*ack net
for
F^f F^g
FZ ! FX  FZ
constructed in the proof of Proposition 4.7.
COLIMITS FOR TOWERS OF SIMPLICIAL SETS 13
Similarly, if W is the pushout in Towst of
^f ^g
(4.11) Z  X  ! Z ;
then W may be may be constructed as the levelwise pushout of any representative*
*s of
(4.11) and W [n] is thus a quotient of Y [n] q Z[n]. Note that the structur*
*e maps
of the pushout induce an epimorphism ^h: Y q Z!!W.
Now if U is a finite subobject of W, then it is in fact a quotient of some f*
*inite
subobject V0q V00,! Y q Z, with V0 2 FY and V002 FZ, as in the proof of
Proposition 3.8. But since W [n] ~=(Y [n] q Z[n])= ~, where the equivalence r*
*elation
~ is generated by f[n](x) ~ g[n](x), we see that any finite subspace U[n] *
*W [n]
(and thus U W) is obtained form a finite subspace V0q V 00 Y [n] q Z[n] by*
* a
finite colimit as in the proof of Proposition 4.5. This shows that FW is iso*
*morphic
to the pushout net for
F^f F^g
FZ  FX ! FZ:
Remark 4.12. The fact that NetF serves as a cocompletion of C, when F is a g*
*ood
subcategory generating C, follows directly from more general results:
By [J, VI, Thms. 1.6 & 1.8] we know that IndC is the cocompletion of C (ass*
*uming
C itself is finitecocomplete), and it is easy to see that C embeds in IndF *
*(as in the
proof of Proposition 4.8), so that IndC embeds cocontinuously in Ind(IndF*
*),
which is equivalent to IndF (see [GV , Cor. 8.9.8]). Because F is coartinia*
*n (Def.
3.1(c)), IndF is equivalent to IndstF (see [GV , x8.12.6]), which is equ*
*ivalent in
turn to NetF by Fact 4.4 and Def. 3.1(b).
However, we believe that the explicit description fo the colimits in NetF gi*
*ven above
may be more useful than that obtained form unwinding the above chain of equival*
*ences.
The results relating specifically to towers of simplicial sets  Proposition*
*s 3.4 & 3.8
 may also be extended to other Pro categories of towers over categories C gen*
*erated
by a good subcategory F, such as towers of sets (cf. x3.7).
Remark 4.13. The example in 2.10(b) shows that the functor I : Tow ! NetF of t*
*he
Theorem fails to be an embedding, since the tower Y defined there has no nontr*
*ivial
finite subobjects, and thus I(Y) = FY = *, even though there are nontrivial *
*maps
Y ! Z in Tow.
This is not a serious flaw, since one often chooses to work with the "good" *
*towers
of Towst in applications. In fact, there is a certain advantage to this fact,*
* from our
point of view, since it yields a version of the Pro category of towers from whi*
*ch we have
eliminated the phantom phenomena (as in the case of Towst), but still have fi*
*nite
limits (and have actually added infinite colimits).
Question 4.14.Although only colimits were needed for our application in [BT ], *
*one can
obviously ask the same question regarding the completion of Tow  that is, embe*
*dding
the Pro category of towers in one where arbitrary limits (ideally: both limits*
* and
colimits) may be constructed. While the categorical part of our construction c*
*ould
presumably be dualized, it is not clear that the category Tow, or any other ver*
*sion of
the Pro category of towers, will indeed satisfy the required assumptions, since*
* specific
14 DAVID BLANC
properties of Tow and S? were used in the proof of Proposition 3.8 and Theo*
*rem
4.9.
Note however that for any small finitecomplete category C, the category Ind*
*C has
all limits (as well as all colimits), and the inclusion C ,! IndC preserves *
*all limits
which exist in C, by [J, VI, Prop. 1.7].
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University of Haifa, 31905 Haifa, Israel
Email address: blanc@mathcs.haifa.ac.il