THREE MODELS FOR THE HOMOTOPY THEORY OF
HOMOTOPY THEORIES
JULIA E. BERGNER
Abstract.Given any model category, or more generally any category with
weak equivalences, its simplicial localization is a simplicial category *
*which can
rightfully be called the "homotopy theory" of the model category. There *
*is
a model category structure on the category of simplicial categories, so *
*taking
its simplicial localization yields a "homotopy theory of homotopy theori*
*es." In
this paper we show that there are two different categories of diagrams o*
*f sim
plicial sets, each equipped with an appropriate definition of weak equiv*
*alence,
such that the resulting homotopy theories are each equivalent to the hom*
*o
topy theory arising from the model category structure on simplicial cate*
*gories.
Thus, any of these three categories with their respective weak equivalen*
*ces
could be considered a model for the homotopy theory of homotopy theories.
One of them in particular, Rezk's complete Segal space model category st*
*ruc
ture on the category of simplicial spaces, is much more convenient from *
*the
perspective of making calculations and therefore obtaining information a*
*bout
a given homotopy theory.
1.Introduction
Classical homotopy theory considers topological spaces, up to weak homotopy
equivalence. Eventually, the structure of the category of topological spaces ma*
*king
it possible to talk about its "homotopy theory" was axiomatized; it is known as*
* a
model category structure. In particular, given a model category structure on an
arbitrary category, we can talk about its homotopy category. More generally, we
can think about the "homotopy theory" given by that category with its particular
class of weak equivalences, where the homotopy theory encompasses the homotopy
category as well as higherorder information. One might ask what specifically *
*is
meant by a homotopy theory.
One answer to this question uses simplicial categories, which in this paper we
will always take to mean categories enriched over simplicial sets. Given a mod*
*el
category M, taking its simplicial localization with respect to its subcategory *
*of weak
equivalences yields a simplicial category LM [8, 4.1]. The simplicial localiz*
*ation
encodes the known homotopytheoretic information of the model category, so one
point of view is that this simplicial category is the homotopy theory associate*
*d to
the model category structure. Settheoretic issues aside, we can also construct*
* the
simplicial localization for any category with a subcategory of weak equivalence*
*s, so
therefore we can speak of an associated homotopy theory even in this more gener*
*al
situation.
____________
Date: April 15, 2005.
2000 Mathematics Subject Classification. Primary: 55U35; Secondary 18G30, 18*
*E35.
1
2 J.E. BERGNER
Given two homotopy theories, one can ask whether they are equivalent to one
another in some natural sense. There is a notion of weak equivalence between
two simplicial categories which is a simplicial analogue of an equivalence betw*
*een
categories. These weak equivalences are known as DKequivalences, where the
"DK" refers to the fact that they were first defined by Dwyer and Kan in [7]. In
fact, there is a model category structure SC on the category of all (small) sim*
*plicial
categories in which the weak equivalences are these DKequivalences [2, 1.1]. T*
*he
associated homotopy theory of simplicial categories is what we will refer to as*
* the
homotopy theory of homotopy theories.
In [16], Rezk takes steps toward finding a model other than that of simplicial
categories for the homotopy theory of homotopy theories. He defines complete Se*
*gal
spaces, which are simplicial spaces satisfying some nice properties (Definition*
*s 3.4
and 3.6 below) and constructs a functor which assigns a complete Segal space to*
* any
simplicial category. He considers a model category structure CSS on the categor*
*y of
all simplicial spaces in which the weak equivalences are levelwise weak equival*
*ences
of simplicial sets and then localizes it in such a way that the local objects a*
*re the
complete Segal spaces (Theorem 3.8).
However, Rezk does not construct a functor from the category of complete Segal
spaces to the category of simplicial categories, nor does he discuss the model *
*cat
egory SC. In this paper, we complete his work by showing that SC and CSS have
equivalent homotopy theories. This result is helpful in that the weak equivalen*
*ces
between complete Segal spaces are easy to identify (see Proposition 3.11 below),
unlike the weak equivalences between simplicial categories, and therefore making
any kind of calculations would be much easier in CSS. Using terminology of Dugg*
*er
[6], this model category CSS is a presentation for the homotopy theory of homot*
*opy
theories, since it is a localization of a category of diagrams of spaces.
In order to prove this result, we make use of an intermediate category. Consi*
*der
the full subcategory SeCat of the category of simplicial spaces whose objects a*
*re
simplicial spaces with a discrete simplicial set in degree zero. We will prove*
* the
existence of two model category structures on SeCat, each with the same class of
weak equivalences. The first of these structures, which we denote SeCatc, has as
cofibrations the maps which are levelwise cofibrations of simplicial sets. (An *
*alter
nate proof of the existence of this model category structure is given by Hirsch*
*owitz
and Simpson [12, 2.3]. They actually prove the existence of such a model catego*
*ry
structure for Segal ncategories, whereas we consider only the case where n = 1*
*.)
The second model category structure, which we denote SeCatf, has as fibrations
maps which are essentially levelwise fibrations of simplicial sets. We use thes*
*e model
category structures to produce a chain of Quillen equivalences
SC o SeCatf AE SeCatc AE CSS.
(In each case, the topmost arrow is the left adjoint of the adjoint pair.) Not*
*ice
that we can obtain a single Quillen equivalence SeCatf AE CSS via composition.
Since Quillen equivalent model categories have DKequivalent simplicial localiz*
*a
tions (Proposition 2.8), all three of these categories with their respective we*
*ak
equivalences give models for the homotopy theory of homotopy theories.
1.1. Organization of the Paper. We begin in section 2 by recalling standard
information about model category structures and simplicial objects. In section *
*3,
we state the definitions of simplicial categories, complete Segal spaces, and S*
*egal
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 3
categories, and we give some basic results about each. In section 4, we set up
some constructions on Segal precategories that we will need in order to prove o*
*ur
model category structures. In section 5, we prove the existence of a model cate*
*gory
structure SeCatc on Segal precategories which we then in section 6 show is Quil*
*len
equivalent to Rezk's complete Segal space model category structure CSS. In sect*
*ion
7, we prove the existence of the model category structure SeCatf on Segal preca*
*te
gories and prove that it is Quillen equivalent to SeCatc. We then show in secti*
*on 8
that SeCatf is Quillen equivalent to the model category structure SC on simplic*
*ial
categories. Section 9 contains the proofs of some technical lemmas.
Acknowledgments. This paper is a version of my Ph.D. thesis at the University
of Notre Dame [5]. I would like to thank my advisor Bill Dwyer for his help and
encouragement on this paper. I would also like to thank Charles Rezk and Bertra*
*nd
To"en for helpful comments.
2.Background on Model Categories and Simplicial Objects
2.1. Model Categories. Recall that a model category structure on a category C
is a choice of three distinguished classes of morphisms: fibrations (i), cofibr*
*ations
(,!), and weak equivalences (!"). A (co)fibration which is also a weak equivale*
*nce
is an acyclic (co)fibration. With this choice of three classes of morphisms, C*
* is
required to satisfy five axioms MC1MC5 which can be found in [9, 3.3]:
In all the model categories we will use, the factorizations given by axiom MC5
can be chosen to be functorial [13, 1.1.1]. An object X in a model category is
fibrant if the unique map X ! * to the terminal object is a fibration. Dually, *
*X is
cofibrant if the unique map from the initial object OE ! X is a cofibration. Gi*
*ven
any object X, the functorial factorization of the map X ! * as the composite of
an acyclic cofibration followed by a fibration
"~
X O____//Xf___////_*
gives us the object Xf, the fibrant replacement of X. Dually, we can define its
cofibrant replacement Xc using the functorial factorization
" ~
OEO____//Xc___////_X.
All the model category structures that we will work with will be cofibrantly
generated. In a cofibrantly generated model category, there are two sets of spe*
*cified
morphisms, the generating cofibrations and the generating acyclic cofibrations,*
* such
that a map is an acyclic fibration if and only if it has the right lifting prop*
*erty with
respect to the generating cofibrations, and a map is a fibration if and only if*
* it has
the right lifting property with respect to the generating acyclic cofibrations *
*[11,
11.1.2]. To prove that a particular category with a choice of weak equivalences*
* has
a cofibrantly generated model category structure, we need the following definit*
*ion.
Definition 2.2. [11, 10.5.2] Let C be a category and I a set of maps in C. Then
an Iinjective is a map which was the right lifting property with respect to ev*
*ery
map in I. An Icofibration is a map with the left lifting property with respect*
* to
every Iinjective.
We are now able to state the theorem that we will use in this paper to prove *
*the
existence of specific model category structures.
4 J.E. BERGNER
Theorem 2.3. [11, 11.3.1] Let M be a category with a specified class of weak
equivalences which satisfies model category axioms MC1 and MC2. Suppose further
that the class of weak equivalences is closed under retracts. Let I and J be se*
*ts of
maps in M which satisfy the following conditions:
(1) Both I and J permit the small object argument [11, 10.5.15].
(2) Every Jcofibration is an Icofibration and a weak equivalence.
(3) Every Iinjective is a Jinjective and a weak equivalence.
(4) One of the following conditions holds:
(i)A map that is an Icofibration and a weak equivalence is a Jcofibra*
*tion,
or
(ii)A map that is both a Jinjective and a weak equivalence is an I
injective.
Then there is a cofibrantly generated model category structure on M in which I *
*is
a set of generating cofibrations and J is a set of generating acyclic cofibrati*
*ons.
We now define our notion of "equivalence" between two model categories. Recall
that for categories C and D a pair of functors
F : C____//Do:oR_
is an adjoint pair if for each object X of C and object Y of D there is an isom*
*orphism
' : Hom D(F X, Y ) ! Hom C(X, RY ) which is natural in X and Y [14, IV.1].
Definition 2.4. [13, 1.3.1] If C and D are model categories, then the adjoint p*
*air
F : C____//Do:oR_
is a Quillen pair if one of the following equivalent statements is true:
(1) F preserves cofibrations and acyclic cofibrations.
(2) R preserves fibrations and acyclic fibrations.
Definition 2.5. [13, 1.3.12] A Quillen pair is a Quillen equivalence if for all*
* cofi
brant X in C and fibrant Y in D, a map f : F X ! Y is a weak equivalence in D
if and only if the map 'f : X ! RY is a weak equivalence in C.
We will use the following proposition to prove that a Quillen pair is a Quill*
*en
equivalence. Recall that a functor F : C ! D reflects a property if, for any
morphism f of C, whenever F f has the property, then so does f.
Proposition 2.6. [13, 1.3.16] Suppose that
F : C____//Do:oR_
is a Quillen pair. Then the following statements are equivalent:
(1) This Quillen pair is a Quillen equivalence.
(2) F reflects weak equivalences between cofibrant objects and, for every fi*
*brant
Y in D, the map F ((RY )c) ! Y is a weak equivalence.
(3) R reflects weak equivalences between fibrant objects and, for every cofi*
*brant
X in C, the map X ! R((F X)f) is a weak equivalence.
The existence of a Quillen equivalence between two model categories is actual*
*ly
a stronger condition than we need, but it is a convenient way to show that two
homotopy theories are the same. We will use the following notion of equivalence*
* of
simplicial categories.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 5
Definition 2.7. [7, 2.4] A functor f : C ! D between two simplicial categories *
*is
a DKequivalence if it satisfies the following two conditions:
(1) for any objects x and y of C, the induced map Hom C(x, y) ! Hom D(fx, fy)
is a weak equivalence of simplicial sets, and
(2) the induced map of categories of components ss0f : ss0C ! ss0D is an
equivalence of categories.
Recall that the category of components ss0C of a simplicial category C is the
category with the same objects as C and such that
Hom ss0C(x, y) = ss0Hom C(x, y).
Now, the following result tells us that model categories which are Quillen eq*
*uiv
alent of model categories actually have equivalent homotopy theories.
Proposition 2.8. [7, 5.4] Suppose that C and D are Quillen equivalent model
categories. Then the simplicial localizations LC and LD are DKequivalent.
2.9. Simplicial Objects. Recall that a simplicial set is a functor op! Sets,
where the cosimplicial category has as objects the finite ordered sets [n] =
{1, . .,.n} and as morphisms the orderpreserving maps, and op is its opposite
category. In particular, for n 0, we have [n], the nsimplex, `[n], the boun*
*dary
of [n], and, for n > 0 and 0 k n, V [n, k], which is `[n] with the kth fa*
*ce
removed [10, I.1]. For any simplicial set X, we denote by Xn the image of [n]. *
*There
are face maps di: Xn ! Xn1 for 0 i n and degeneracy maps si: Xn ! Xn+1
for 0 i n, satisfying certain compatibility conditions [10, I.1]. We will d*
*enote
by X the topological space given by geometric realization of the simplicial s*
*et X
[10, I.2].
There is a model category structure on simplicial sets in which the weak equi*
*v
alences are the maps which become weak homotopy equivalences of topological
spaces after geometric realization [10, I.11.3]. We will denote this model cate*
*gory
structure SSets. Note in particular that it is cofibrantly generated. The gener*
*ating
cofibrations are the maps ` [m] ! [m] for all m 0, and the generating acyclic
cofibrations are the maps V [m, k] ! [m] for all m 1 and 0 k m [13, 3.2.*
*1].
This model category structure is Quillen equivalent to the standard model categ*
*ory
structure on topological spaces [13, 3.6.7]. In light of this fact, we will som*
*etimes
refer to simplicial sets as "spaces."
More generally, a simplicial object in a category C is a functor op! C [13, *
*3.1].
In particular, a simplicial space (or bisimplicial set) is a functor op! SSets*
* [10,
IV.1]. Given a simplicial set X, we will also use X to denote the constant simp*
*licial
space with the simplicial set X in each degree. By Xt we will denote the simpli*
*cial
space such that (Xt)n is the constant simplicial set Xn, or the simplicial set *
*which
has the set Xn in each degree.
Notice, however, that our definition of "simplicial category" in this paper is
inconsistent with this terminology. There is a more general notion of simplici*
*al
category by which is meant a simplicial object in the category of (small) categ*
*ories.
Such a simplicial category is a functor op ! Cat where Cat is the category with
objects the small categories and morphisms the functors between them. Our def
inition of simplicial category coincides with this one when the extra condition*
* is
imposed that the face and degeneracy maps be the identity map on objects [7, 2.*
*1].
6 J.E. BERGNER
We will also require the following additional structure on some of our model
category structures. A simplicial model category is a model category which is
also a simplicial category satisfying two additional axioms [11, 9.1.6]. (Again*
*, the
terminology is potentially confusing because a simplicial model category is not*
* a
simplicial object in the category of model categories.) The important part of t*
*his
structure that we will use is the fact that, given objects X and Y of a simplic*
*ial
model category, it makes sense to talk about the function complex, or simplicia*
*l set
Map (X, Y ).
Given a model category M, or more generally a category with weak equivalences,
a homotopy function complex Map h(X, Y ) is a simplicial set which is the morph*
*ism
space between X and Y in the simplicial localization LM [7, x4]. If M is a sim*
*plicial
model category, X is cofibrant in M, and Y is fibrant in M, then Map h(X, Y ) is
weakly equivalent to Map (X, Y ).
2.10. Localized Model Category Structures. Several of the model category
structures that we will use will be obtained by localizing a given model catego*
*ry
structure with respect to a map or a set of maps. Suppose that S = {f : A ! B}
is a set of maps with respect to which we would like to localize a model catego*
*ry
(or category with weak equivalences) M. We define an Slocal object W to be an
object of M such that for any f : A ! B in S, the induced map on homotopy
function complexes
f* : Map h(B, W ) ! Map h(A, W )
is a weak equivalence of simplicial sets. (If M is a model category, a local ob*
*ject
is usually required to be fibrant.) A map g : X ! Y in M is then defined to be
an Slocal equivalence if for every local object W , the induced map on homotopy
function complexes
g* : Map h(Y, W ) ! Map h(X, W )
is a weak equivalence of simplicial sets.
The following theorem holds for model categories M which are left proper and
cellular. We will not define these conditions here, but refer the reader to [11*
*, 13.1.1,
12.1.1] for more details. We do note, in particular, that a cellular model cate*
*gory
is cofibrantly generated. All the model categories that we localize in this pap*
*er can
be shown to satisfy both these conditions.
Theorem 2.11. [11, 4.1.1] Let M be a left proper cellular model category. There
is a model category structure LSM on the underlying category of M such that:
(1) The weak equivalences are the Slocal equivalences.
(2) The cofibrations are precisely the cofibrations of M.
(3) The fibrations are the maps which have the right lifting property with r*
*espect
to the maps which are both cofibrations and Slocal equivalences.
(4) The fibrant objects are the Slocal objects which are fibrant in M.
(5) If M is a simplicial model category, then its simplicial structure induc*
*es a
simplicial structure on LSM.
In particular, given an object X of M, we can talk about its functorial fibra*
*nt
replacement LX in LSM. The object LX is an Slocal object which is fibrant in
M, and we will refer to it is the localization of X in LSM.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 7
2.13. Model Category Structures for Diagrams of Spaces. Suppose that
D is a small category and consider the category of functors D ! SSets, denoted
SSetsD . This category is also called the category of Ddiagrams of spaces. We
would like to consider model category structures on SSetsD .
A natural choice for the weak equivalences in SSetsD is the class of levelwi*
*se
weak equivalences of simplicial sets. Namely, given two Ddiagrams X and Y , we
define a map f : X ! Y to be a weak equivalence if and only if for each object d
of D, the map X(d) ! Y (d) is a weak equivalence of simplicial sets.
There is a model category structure SSetsDfon the category of Ddiagrams with
these weak equivalences and in which the fibrations are given by levelwise fibr*
*ations
of simplicial sets. The cofibrations in SSetsDfare then the maps of simplicial *
*spaces
which have the left lifting property with respect to the maps which are levelwi*
*se
acyclic fibrations. This model structure is often called the projective model c*
*ategory
structure on Ddiagrams of spaces [10, IX, 1.4]. Dually, there is a model categ*
*ory
structure SSetsDc in which the cofibrations are given by levelwise cofibrations*
* of
simplicial sets, and this model structure is often called the injective model c*
*ategory
structure [10, VIII, 2.4].
The small categoryoDpwhich we will use in this paper is op, so that the diag*
*ram
category SSets is just the category of simplicial spaces.
Consider the Reedy model category structure on simplicial spaces [15]. In this
structure, the weak equivalences are again the levelwise weak equivalences of s*
*im
plicial sets. The Reedy model category structure is cofibrantly generated, whe*
*re
the generating cofibrations are the maps
`[m] x [n]t[ [m] x `[n]t! [m] x [n]t
for all n, m 0. The generating acyclic cofibrations are the maps
V [m, k] x [n]t[ [m] x `[n]t! [m] x [n]t
for all n 0, m 1, and 0 k m [16, 2.4].
It turns out that the Reedy model category structure on simplicial spaces is
exactly the same as the injective model category structure on this same categor*
*y,
as given by the following result.
Proposition 2.14. [11, 15.8.7, 15.8.8] A map f : X ! Y of simplicial spaces is
a cofibration in the Reedy model category structure if and only if it is a mono*
*mor
phism. In particular, every simplicial space is Reedy cofibrant.
In light of thisoresult,pwe willodenoteptheoReedypmodel structure on simplici*
*al
spaces by SSetsc . Both SSetsc and SSetsf are simplicial model categories.
In each case, given two simplicial spaces X and Y , we can define Map (X, Y ) by
Map (X, Y )n = Hom (X x [n], Y )
where the set on the righthand side consists of maps of simplicial spaces.
To establish some notation we will need later in the paper, we recall the def*
*inition
of fibration in the Reedy model category structure. If X is a simplicial space,*
* let
sknX denote its nskeleton, generated by the spaces in degrees less than or equ*
*al
to n, and let cosknXodenotepthe ncoskeleton of X [15, x1]. A map X ! Y is a
fibration in SSetsc if
o X0 ! Y0 is a fibration of simplicial sets, and
8 J.E. BERGNER
o for all n 1, the map Xn ! Pn is a fibration, where Pn is defined to be
the pullback in the following diagram:
Pn _____________//_Yn
 
 
fflffl fflffl
(coskn1X)n_____//(coskn1Y )n
Notice in particular that this pullback diagram is actually a homotopy pullba*
*ck
diagram, as follows. If f : X ! Y is a Reedy fibration, then it has the right l*
*ifting
property with respect to all Reedy acyclic cofibrations. In particular, there *
*is a
dotted arrow lift in the following diagram, where m 1, 0 k m, and n 0:
V [m, k] x `[n]t___//X99r
 r r 
 r r 
fflfflr fflffl
[m] x `[n]t______//Y.
Since the functors sknand cosknare adjoint [15, x1], we have that
(coskn1X)n ' Map ( [n], cosknX) ' Map (skn [n], X) ' Map ( `[n], X).
Therefore, we have a dotted arrow lift in each diagram
V [m, k]____//(coskn1X)n77
 pp p 
 p p 
fflfflp fflffl
[m] _____//_(coskn1Y )n.
In particular, the righthand vertical arrow is a fibration of simplicial sets.*
* Thus,
the simplicial set Pn is a homotopy pullback and therefore homotopy invariant.op
We will also make use of the projective model category structure SSetsf on
simplicial spaces. This model category is also cofibrantly generated; the gener*
*ating
cofibrations are the maps
`[m] x [n]t! [m] x [n]t
for all m, n 0 [10, IV.3.1].
In the next section, we will localize the Reedy (or injective) and projective*
* model
category structures on simplicial spaces with respect to a map to obtain model
category structures in which the fibrant objects are Segal spaces (Definition 3*
*.4).
We will further localize them to obtain model category structures in which the
fibrant objects are complete Segal spaces (Definition 3.6).
3. Some Definitions and Model Category Structures
In this section, we define and discuss in turn the three main structures that*
* we
will use in the course of this paper: simplicial categories, complete Segal spa*
*ces,
and Segal categories.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 9
3.1. Simplicial Categories. Simplicial categories, most simply stated, are cate
gories enriched over simplicial sets, or categories with a simplicial set of mo*
*rphisms
between any two objects. So, given any objects x and y in a simplicial category*
* C,
there is a simplicial set Hom C(x, y).
Fix an object set O and consider the category of simplicial categories with o*
*bject
set O such that all morphisms are the identity on the objects. Dwyer and Kan de*
*fine
a model category structure SCO in which the weak equivalences are the functors
f : C ! D of simplicial categories such that given any objects x and y of C, the
induced map
Hom C(x, y) ! Hom D(x, y)
is a weak equivalence of simplicial sets [8, x7]. The fibrations are the funct*
*ors
f : C ! D for which these same induced maps are fibrations, and the cofibrations
are the functors which have the left lifting property with respect to the acycl*
*ic
fibrations.
It is more useful, however, to consider the category of all small simplicial *
*cat
egories with no restriction on the objects. Before describing the model catego*
*ry
structure on this category, we need a few definitions. Recall from Definition *
*2.7
above that if C is a simplicial category, then we denote by ss0C the category of
components of C.
If C is a simplicial category and x and y are objects of C, a morphism e 2
Hom C(x, y)0 is a homotopy equivalence if the image of e in ss0C is an isomorph*
*ism.
Theorem 3.2. [2, 1.1] There is a model category structure on the category SC of
small simplicial categories defined by the following three classes of morphisms:
(1) The weak equivalences are the maps f : C ! D satisfying the following two
conditions:
o (W1) For any objects x and y in C, the map
Hom C(x, y) ! Hom D(fx, fy)
is a weak equivalence of simplicial sets.
o (W2) The induced functor ss0f : ss0C ! ss0D on the categories of
components is an equivalence of categories.
(2) The fibrations are the maps f : C ! D satisfying the following two condi
tions:
o (F1) For any objects x and y in C, the map
Hom C(x, y) ! Hom D(fx, fy)
is a fibration of simplicial sets.
o (F2) For any object x1 in C, y in D, and homotopy equivalence e :
fx1 ! y in D, there is an object x2 in C and homotopy equivalence
d : x1 ! x2 in C such that fd = e.
(3) The cofibrations are the maps which have the left lifting property with *
*respect
to the maps which are fibrations and weak equivalences.
Notice that the weak equivalences are precisely the DKequivalences that we
defined above (Definition 2.7).
The proof of this theorem actually shows that this model category structure
is cofibrantly generated. Define the functor U : SSets ! SC such that for any
simplicial set K, the simplicial category UK has two objects, x and y, and only
10 J.E. BERGNER
nonidentity morphisms the simplicial set K = Hom (x, y). Using this functor, we
define the generating cofibrations to be the maps of simplicial categories
o (C1) U `[n] ! U [n] for n 0, and
o (C2) OE ! {x}, where OE is the simplicial category with no objects and
{x} denotes the simplicial category with one object x and no nonidentity
morphisms.
The generating acyclic cofibrations are defined similarly [2, x1].
3.3. Segal Spaces and Complete Segal Spaces. Complete Segal spaces, defined
by Rezk in [16], are more difficult to describe, but ultimately they are actual*
*ly easier
to work with than simplicial categories. The name "Segal" refers to the similar*
*ity
between Segal spaces and Segal's spaces [17].
We begin by defining Segal spaces. In [16, 4.1], Rezk defines for each 0 i *
* k1
a map ffi: [1] ! [k] in such that 0 7! i and 1 7! i+1. Then for each k he def*
*ines
the simplicial space
k1[
G(k)t= ffi [1]t [k]t.
i=0
He shows that, for any simplicial space X, there is an weak equivalence of si*
*m
plicial sets
Map hSSets op(G(k)t, X) ! X1 xhX0. .x.hX0X1,
________z_______"
k
where the right hand side is the homotopy limit of the diagram
X1 _d0_//_X0od1oX1___d0_//._.d.0//_X0d1ooX1_
with k copies of X1.
Now, given any k, define the map 'k : G(k)t ! [k]t to be the inclusion map.
Then for any simplicial space W there is a map
'k = Map hSSets(op'k, W ) : Map hSSets(op [k]t, W ) ! Map hSSets(opG(k)t, W ).
More simply written, this map is
'k : Wk ! W1 xhW0. .x.hW0W1
________z_______"
k
and is often called a Segal map.
Definition 3.4. [16, 4.1] A Reedy fibrant simplicial space W is a Segal space i*
*f for
each k 2 the map 'k is a weak equivalence of simplicial sets. In other words,*
* the
Segal maps
'k : Wk ! W1 xhW0. .x.hW0W1
________z_______"
k
are weak equivalences for all k 2.
Notice that if W is a Segal space, or more generally if W is Reedy fibrant, we
can use ordinary function complexes and a limit in the definition of the Segal *
*maps
[16, x4].
Rezk defines the coproduct of all these inclusion maps
a
' = ('k : G(k)t! [k]t).
k 0
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 11
Using this map ', we have the following result.
Theorem 3.5. [16, 7.1] There is a model category structure on simplicial spaces
which can be obtained by localizing the Reedy model category structure with res*
*pect
to the map '. This model category structure has the following properties :
(1) The weak equivalences are the maps f for which Map hSSets op(f, W ) is a
weak equivalence of simplicial sets for any Segal space W .
(2) The cofibrations are the monomorphisms.
(3) The fibrant objects are the Reedy fibrant 'local objects, which are pre*
*cisely
the Segal spaces.
We will refer to this model category structure on simplicial spaces as the Se*
*gal
space model category structure and denote it SeSpc.
The properties of Segal spaces enable us to speak of them much in the same
way that we speak of categories. Heuristically, a simple example of a Segal spa*
*ce
is the nerve of a category C, regarded as a simplicial space nerve(C)t. (We ne*
*ed
to take a Reedy fibrant replacement of this nerve to be an actual Segal space.)*
* In
particular, we can define "objects" and "maps" of a Segal space. We will summar*
*ize
the particular details here that we will need; a full description is given by R*
*ezk [16,
x5].
Given a Segal space W , define its set of objects, denoted ob(W ), to be the *
*set
of 0simplices of the space W0, namely, the set W0,0. Given any two objects x, *
*y in
ob(W ), define the mapping space map W(x, y) to be the homotopy fiber of the map
(d1, d0) : W1 ! W0 x W0 over (x, y). (Note that since W is Reedy fibrant, this
map is a fibration, and therefore in this case we can just take the fiber.) Giv*
*en a 0
simplex x of W0, we denote by idxthe image of the degeneracy map s0 : W0 ! W1.
We say that two 0simplices of map W(x, y), say f and g, are homotopic, denoted
f ~ g, if they lie in the same component of the simplicial set map W(x, y).
Given f 2 map W (x, y)0 and g 2 map W (y, z)0, there is a composite g O f 2
mapW (x, z)0, and this notion of composition is associative up to homotopy. We
define the homotopy category Ho (W ) of W to have as objects the set ob(W )
and as morphisms between any two objects x and y, the set map Ho(W)(x, y) =
ss0map W (x, y).
A map g in map W(x, y)0 is a homotopy equivalence if there exist maps f, h 2
mapW (y, x)0 such that gOf ~ idyand hOg ~ idx. Any map in the same component
as a homotopy equivalence is itself a homotopy equivalence [16, 5.8]. Therefore*
* we
can define the space Whoequivto be the subspace of W1 given by the components
whose zerosimplices are homotopy equivalences.
We then note that the degeneracy map s0 : W0 ! W1 factors through Whoequiv
since for any object x the map s0(x) = idxis a homotopy equivalence. Therefore,
we make the following definition:
Definition 3.6. [16, x6] A complete Segal space is a Segal space W for which the
map s0 : W0 ! Whoequivis a weak equivalence of simplicial sets.
We now consider an alternate way of defining a complete Segal space which is
less intuitive but will enable us to further localize the Segal space model cat*
*egory
structure in such a way that the complete Segal spaces are the new fibrant obje*
*cts.
Consider the category I[1] which consists of two objects x and y and exactly two
nonidentity maps which are inverse to one another, x ! y and y ! x. Denote
by E the nerve of this category, considered as a constant simplicial space. The*
*re
12 J.E. BERGNER
are two maps [0]t ! E given by the inclusions of [0]t to the objects x and y,
respectively. Let _ : [0]t ! E be the map which takes [0]t to the object x. (*
*It
does not actually matter which one of the two maps we have chosen, as long as
it is fixed.) This map then induces, for any Segal space W , a map on homotopy
function complexes
_* : Map hSSets(opE, W ) ! Map hSSets(op [0]t, W ) = W0.
Proposition 3.7. [16, 6.4] For any Segal space W , the map of homotopy func
tion complexes _* : Map hSSets(opE, W ) ! Map hSSets(op [0], W ) = W0 is a weak
equivalence of simplicial sets if and only if W is a complete Segal space.
Given this proposition, we can further localize the category of simplicial sp*
*aces
with respect to this map.
Theorem 3.8. [16, 7.2] Taking the localization of the Reedy model category stru*
*c
ture on simplicial spaces with respect to the maps ' and _ above results in a m*
*odel
category structure which satisfies the following properties:
(1) The weak equivalences are the maps f such that Map hSSets op(f, W ) is a
weak equivalence of simplicial sets for any complete Segal space W .
(2) The cofibrations are the monomorphisms.
(3) The fibrant objects are the complete Segal spaces.
We will refer to this model category structure on simplicial spaces as the co*
*mplete
Segal space model category structure, denoted CSS. It turns out that when the
objects involved are Segal spaces, the weak equivalences in this model category
structure can be described more explicitly.
Definition 3.9. A map f : U ! V of Segal spaces is a DKequivalence if
(1) for any pair of objects x, y 2 U0, the induced map mapU (x, y) ! mapV (f*
*x, fy)
is a weak equivalence of simplicial sets, and
(2) the induced map Ho(f) : Ho(U) ! Ho(V ) is an equivalence of categories.
We then have the following result by Rezk:
Theorem 3.10. [16, 7.7] Let f : U ! V be a map of Segal spaces. Then f is a
DKequivalence if and only if it becomes a weak equivalence in CSS.
Note that these weak equivalences have been given the same name as the ones
in SC. While this may at first seem strange, the two definitions are very simil*
*ar,
in fact rely on the same generalization of the idea of equivalence of categorie*
*s to a
simplicial setting.
However, what is especially nice about the complete Segal space model category
structure is the simple characterization of the weak equivalences between the f*
*ibrant
objects.
Proposition 3.11. [16, 7.6] A map f : U ! V between complete Segal spaces is a
DKequivalence if and only if it is a levelwise weak equivalence.
This proposition is actually a special case of a more general result. In any
localized model category structure, a map is a local equivalence between fibrant
objects if and only if it is a weak equivalence in the original model category *
*structure
[11, 3.2.18].
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 13
It is also possible to localize the projective model category structure SSets*
*fop
on the category of simplicial spaces to obtain analogous model category structu*
*res.
We will denote the localization of the projective model category structure by w*
*ith
respect to the map ' by SeSpf. There is also a localization of the projective
model category structure with respect to the maps ' and _ analogous to the model
category structure CSS, but we will not need this structure here.
3.12. Segal Categories. Lastly, we consider the Segal categories. We begin by
defining the preliminary notion of a Segal precategory.
Definition 3.13. [12, x2] A Segal precategory is a simplicial space X such that*
* the
simplicial set X0 in degree zero is discrete, i.e. a constant simplicial set.
In the case of Segal precategories, it again makes sense to talk about the Se*
*gal
maps
'k : Xk ! X1 xhX0. .x.hX0X1
________z_______"
k
for each k 2. Since X0 is discrete, we can actually take the limit
X1_xX0_._.x.X0X1_z_______"
k
on the righthand side.
Definition 3.14. [12, x2] A Segal category X is a Segal precategory such that e*
*ach
Segal map 'k is a weak equivalence of simplicial sets for k 2.
Note that the definition of a Segal category is similar to that of a Segal sp*
*ace,
with the additional requirement that the degree zero space be discrete. However,
Segal categories are not required to be Reedy fibrant, so they are not necessar*
*ily
Segal spaces. op
Given a fixed set O, we can consider the category SSetsO whose objects are
the Segal precategories with O in degree zero and whoseomorphismspare the ident*
*ity
on this set. There is a model category structure SSetsO,f on this category in w*
*hich
the weak equivalences are levelwise [4]. In other words, f : X ! Y is a weak
equivalence if for each n 0, the map fn : Xn ! Yn is a weak equivalence of
simplicial sets. Furthermore, the fibrations are also levelwise. This model str*
*ucture
can then be localized with respect to a map similar to the map which we used to
obtain the Segal space model category structure.
We first need to determine what this map should be. We begin by considering
the maps of simplicial spaces 'k : G(k)t! [k]t and adapting them to the case at
hand. op
The first problem is that [k]t is not going to be in SSetsO,f for all values*
* of
k. Instead, we need to define a separate ksimplex for any ktuple x0, . .,.xk*
* of
objects in O, denoted [k]tx0,...,xk, so that the objects are preserved. Note t*
*hat this
object [k]tx0,...,xkalso needs to have all elements of O as 0simplices, so we*
* add
any of these elements that have not already been included in the xi's, plus the*
*ir
degeneracies in higher degrees.
Then we can define
k1[
G(k)tx0,...,xk= ffi [1]txi,xi+1.
i=0
14 J.E. BERGNER
Now, we need to take coproducts not only over all values of k, but also over all
ktuples of vertices. Hence, the resulting map 'O looks like
a a
'O = ( (G(k)tx0,...,xk! [k]tx0,...,xk)).
k 0 (x0,...,xk)2Ok
Setting x_= (x0, . .,.xk), we can write the component maps as G(k)tx_! [k]tx_.*
* We
can then localize SSetsOop,fwith respect to the map 'O to obtain a model catego*
*ry
which we denote LSSetsOop,f.
There are also analogous model category structures SSetsOop,cand LSS etsOop,c
on the category of Segal precategories with a fixed set O in degree zero with t*
*he
same weak equivalences but where the cofibrations, rather than the fibrations, *
*are
defined levelwise, and then we can localize with respect to the same map [4], [*
*18,
A.1.1].
However, we would like a model category structure on the category of all Segal
precategories, not just on these more restrictive subcategories. In the course*
* of
this paper, we will prove the existence of two model category structures on Seg*
*al
precategories. Unlike in the fixed object set case, we cannot actually obtain *
*the
model category structure via localization of a model category structure with le*
*v
elwise weak equivalences since it is not possible to put a model structure on t*
*he
category of Segal precategories in which the weak equivalences are levelwise an*
*d in
which the cofibrations are monomorphisms.
To see that there is no such model structure, suppose that one did exist and
consider the map f : [0]tq [0]t ! [0]t. By model category axiom MC5, f
could be factored as the composite of a cofibration [0]tq [0]t ! X followed
by an acyclic fibration X ! [0]t. However, since the weak equivalences would
be levelwise weak equivalences, X0 would have to consist of one point. However,
the only map ( [0]tq [0]t)0 ! X0 is not an inclusion. Thus, there is no such
factorization of the map f, and therefore there can be no model category struct*
*ure
satisfying the two given properties.
3.15. Relationship Between Simplicial Categories and Segal Categories
in Fixed Object Set Cases. Recall from above that there is a model category
structure SCO on the category whose objects are the simplicial categories with a
fixed set O of objects and whose morphisms are the functors which areothepident*
*ity
on the objects and that there is a model category structure LSS etsO,f on the
category whose objects are the Segal precategories with the set O in degree zero
and whose morphisms are the identity on degree zero.
Theorem 3.16. [4] There is an adjoint pair
FO : LSSetsOop,f__//SCO:oROo_
which is a Quillen equivalence.
The proof of this theorem uses a generalization of a result by Badzioch [1, 6*
*.5]
which relates strict and homotopy algebras over an algebraic theory. This gener*
*al
ization uses the notion of multisorted algebraic theory [3].
A key step in this proof is a explicit description of the localization of the*
* objects
[n]tx_. Up to homotopy, this localization is the same as the localization of *
*the
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 15
objects G(n)tx_and is obtained by taking the colimit of stages of a filtration
G(n)tx_= 1G(n)tx_ 2G(n)tx_ . . .
Let ei denote the nondegenerate simplex xi1! xi in G(n)tx_and let wj denote
a word in the ei's which can be obtained via "composition" of these 1simplices.
The kth stage of the filtration is given by
( k(G(n)tx_))m = {(w1 . ..wm )l(w1. .w.m) k}
where l(w1. .w.n) denotes the length of the word w1. .w.n.opThe colimit of this
filtration is weakly equivalent to LcG(n)tx_in LSSetsO,f.
We show in [4] that for each i 1 the map
iG(n)tx_! i+1G(n)tx_
is a DKequivalence, and that the unique map from G(n)tx_to the colimit of this
directed system is also a DKequivalence.
In the current paper, we use some of the ideas of the proof from the fixed ob*
*ject
set case,obutpwe no longer use multisorted theories as we pass from SCO to SC *
*and
SSetsO to SeCat.
4. Methods of Obtaining Segal Precategories from Simplicial Spaces
In the course of proving these two model category structures SeCatc and SeCat*
*f,
we will need sets of generating cofibrations and generating acyclic cofibration*
*s which
are similar to those of the Reedy and projective model category structures on
simplicial spaces. However, we will need to modify these maps so that they are
actually maps between Segal precategories. We have two canonical ways of turning
any simplicial space to a Segal precategory. The purpose of this section is to *
*define
these two methods and prove a result which we will need to prove the existence *
*of
each of these two model structures.
The first method we will call reduction, and we use it to define the generati*
*ng
cofibrations in SeCatc. Consider the forgetful functor from the category of Se*
*gal
precategories to the category of simplicial spaces. This map has a left adjoin*
*t,
which we will call the reduction map. Given a simplicial space X, we will denote
its reduction by (X)r. The degree n space of (X)r is obtained from Xn by collap*
*sing
the subspace sn0X0 of Xn to the discrete space ss0(sn0X0), where sn0is the iter*
*ated
degeneracy map.
Recall that the cofibrations in the Reedy model category structure on simpli
cial spaces are monomorphisms (Proposition 2.14) and that the Reedy generating
cofibrations are of the form
`[m] x [n]t[ [m] x `[n]t! [m] x [n]t
for all n, m 0. In general, these maps are not in SeCat because the objects
involved are not Segal precategories. Therefore, we apply this reduction functo*
*r to
these maps.
Thus, we consider the maps
( `[m] x [n]t[ [m] x `[n]t)r ! ( [m] x [n]t)r.
However, we still need to make some modifications to assure that all these maps*
* are
actually monomorphisms. In particular, we need to check the case where n = 0.
If n = m = 0, and if OE denotes the empty simplicial space, we obtain the map
16 J.E. BERGNER
OE ! [0]t, which is a monomorphism. However, when n = 0 and m = 1, we get
the map [0]tq [0]t ! [0]t, which is not a monomorphism. When n = 0 and
m 2, we obtain the map [0]t ! [0]t. This map is an isomorphism, and thus
there is no reason to include it in the generating set. Therefore, we define th*
*e set
Ic = {( `[m] x [n]t[ [m] x `[n]t)r ! ( [m] x [n]t)r}
for all m 0 when n 1 and for n = m = 0. This set Ic will be a set of genera*
*ting
cofibrations of SeCatc.
This reduction process works well in almost all situations, but weohavepprobl*
*ems
when we try to reduce some of the generating cofibrations in SSetsf , namely t*
*he
maps
`[1] x [n]t! [1] x [n]t
for any n 0. The object [1] x [n]t reduces to a Segal precategory with n + 1
points in degree zero, but the object `[1] x [n]t reduces to a Segal precateg*
*ory
with 2(n + 1) points in degree zero. In other words, the reduced map in this ca*
*se
is no longer a monomorphism.
Consider the set [n]0 and denote by [n]t0the doubly constant simplicial spa*
*ce
defined by it. For m 1 and n 0, define Pm,n to be the pushout of the diagram
`[m] x [n]t0___//`[m] x [n]t
 
 
fflffl fflffl
[n]t0__________//_Pm,n.
If m = 0, then we define Pm,0 to be the empty simplicial space. For all m 0 a*
*nd
n 1, define Qm,n to be the pushout of the diagram
[m] x [n]t0____// [m] x [n]t
 
 
fflffl fflffl
[n]t0__________//_Qm,n.
For each m and n, the map ` [m] x [n]t induces a map im,n : Pm,n ! Qm,n. We
then define the set If = {im,n : Pm,n ! Qm,nm, n 0}. Note that when m 2
this construction gives exactly the same objects as those given by reduction, n*
*amely
that Pm,n is precisely ( `[m]x [n]t)r and likewise Qm,n is precisely ( [m]x [n]*
*t)r.
If Hom denotes morphism set and X is an arbitrary simplicial space, notice th*
*at
we can use the pushout diagrams defining the objects Pm,n and Qm,n to see that
a
Hom (Pm,n, X) ~= Hom ( `[m], Xn(v0, . .,.vn))
v0,...,vn
and a
Hom (Qm,n, X) ~= Hom ( [m], Xn(v0, . .,.vn)).
v0,...,vn
We now state and prove a lemma using the maps in If. Given a Segal precategory
X, we denote by Xn(v0, . .v.n) the fiber of the map Xn ! Xn+10over (v0, . .,.vn*
*) 2
Xn+10, where this map is given by iterated face maps of X. More specifically,
Xn+10= (cosk0X)n and the map Xn ! Xn+10is given by the map X ! cosk0X.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 17
Lemma 4.1. Suppose a map f : X ! Y has the right lifting property with respect
to the maps in If. Then the map X0 ! Y0 is surjective and each map
Xn(v0, . .,.vn) ! Yn(fv0, . .,.fvn)
is an acyclic fibration of simplicial sets for each n 1 and (v0, . .,.vn) 2 X*
*n+10.
Proof.The surjectivity of X0 ! Y0 follows from the fact that f has the right li*
*fting
property with respect to the map P0,0! Q0,0.
In order to prove the remaining statement, it suffices to show that there is a
dotted arrow lift in any diagram of the form
(4.2) `[m]______//Xn(v0,7.7.,.vn)
p
 p p 
 p p 
fflfflp fflffl
[m] _____//Yn(fv0, . .,.fvn)
for m, n 0.
By our hypothesis, there is a dotted arrow lift in diagrams of the form
(4.3) Pm,n_____//X==

  
  
fflffl fflffl
Qm,n ____//_Y
for all m, n 0. The existence of the lift in diagram 4.3 is equivalent to the
surjectivity of the map Hom (Qm,n, X) ! P in the following diagram, where P
denotes the pullback and Hom denotes morphism set:
Hom (Qm,n, X)__________//_P_________//_Hom(Pm,n, X)
 
 
fflffl fflffl
Hom (Qm,n, Y )____//Hom(Pm,n, Y ).
Now, as noted above we have that
a
Hom (Qm,n, X) ~= Hom ( [m], Xn(v0, . .,.vn))
v0,...,vn
and analogous weak equivalences for the other objects of the diagram.
Using these weak equivalences and being particularly careful in the cases whe*
*re
m = 1 and m = 0, one can show that for each m, n 0 the dottedarrow lift in
diagram 4.2 exists and therefore that each map
Xn(v0, . .,.vn) ! Yn(fv0, . .,.fvn)
is an acyclic fibration of simplicial sets for each n 1.
5. A Segal Category Model Category Structure on Segal
Precategories
In this section, we prove the existence of the model category structure SeCat*
*c.
We would like to define a functorial "localization" functor Lc on SeCat such *
*that
given any Segal precategory X, its localization LcX is a Segal space which is a*
* Segal
category weakly equivalent to X in SeSpc. (It is unclear whether Lc is actually*
* a
fibrant replacement functor in SeCatc. We will discuss this issue at the end of*
* the
18 J.E. BERGNER
section.) We begin by considering a functorial fibrant replacement functor in S*
*eSpc
and then modifying it so that it takes values in SeCat.
A choice of generating acyclic cofibrations for SeSpc is the set of maps
V [m, k] x [n]t[ [m] x G(n)t! [m] x [n]t
for n 0, m 1, and 0 k m [11, x4.2]. Therefore, one can use the small ob*
*ject
argument to construct a functorial fibrant replacement functor by taking a coli*
*mit
of pushouts, each of which is along the coproduct of all these maps [11, x4.3].
If we apply this functor to a Segal precategory, the maps with n = 0 will be
problematic because taking pushouts along them will not result in a space which
is discrete in degree zero. We claim that we can obtain a functorial localizat*
*ion
functor Lc on the category SeCat by taking a colimit of iterated pushouts along*
* the
maps
V [m, k] x [n]t[ [m] x G(n)t! [m] x [n]t
for n, m 1 and 0 k m.
To see that this restricted set of maps is sufficient, consider a Segal preca*
*tegory
X and the Segal category LcX we obtain from taking such a colimit. Then for any
0 k m, consider the diagram
V [m, k]___//Maph(G(0)t,7LcX)7
n
 n n 
 nn 
fflfflnn fflffl
[m]______//Maph( [0]t, LcX).
Since [0]t is isomorphic to G(0)t, and since LcX is discrete in degree zero, t*
*he
righthand vertical map is an isomorphism of discrete simplicial sets. Therefor*
*e, a
dotted arrow lift exists in this diagram. It follows that the map LcX ! [0]t h*
*as
the right lifting property with respect to the maps
V [m, k] x [n]t[ [m] x G(n)t! [m] x [n]t
for all n 0, m 1, and 0 k m. Therefore, LcX is fibrant in SeSpc, namely,
a Segal space.
Since LcX is a Segal space, it makes sense to talk about the mapping space
mapLcX (x, y) and the homotopy category Ho (LcX). Given these facts, we will
show that there exists a model category structure SeCatc on Segal precategories
with the following three distinguished classes of morphisms:
o Weak equivalences are the maps f : X ! Y such that the induced map
LcX ! LcY is a DKequivalence of Segal spaces. (Again, we will call such
maps DKequivalences.)
o Cofibrations are the monomorphisms. (In particular, every Segal precate
gory is cofibrant.)
o Fibrations are the maps with the right lifting property with respect to *
*the
maps which are both cofibrations and weak equivalences.
Theorem 5.1. There is a cofibrantly generated model category structure SeCatc on
the category of Segal precategories with the above weak equivalences, fibration*
*s, and
cofibrations.
We first need to sets Ic and Jc which will be our candidates for generating
cofibrations and generating acyclic cofibrations, respectively.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 19
We take as generating cofibrations the set
Ic = {( `[m] x [n]t[ [m] x `[n]t)r ! ( [m] x [n]t)r}
for all m 0 when n 1 and for n = m = 0. Notice that since taking a pushout
along such a map amounts to attaching an msimplex to the space in degree n, any
cofibration can be written as a directed colimit of pushouts along the maps of *
*Ic.
We then define the set Jc = {i : A ! B} to be a set of representatives of
isomorphism classes of maps in SeCat satisfying two conditions:
(1) For all n 0, the spaces An and Bn have countably many simplices.
(2) The map i : A ! B is a monomorphism and a weak equivalence.
Given these proposed generating acyclic cofibrations, we need to show that any
acyclic cofibration in SeCatc is a directed colimit of pushouts along such maps*
*. To
prove this result, we require several lemmas. The proofs of the first three omi*
*t here;
proofs can be found in the author's thesis [5].
Lemma 5.2. Let A ! B be a CWinclusion. The following statements are equiv
alent:
(1) A ! B is a weak equivalence of topological spaces.
(2) For all n 1, any map of pairs (Dn, Sn1) ! (B, A) extends over the map
of cones (CDn, CSn1).
(3) For all n 1, any map (Dn, Sn1) ! (B, A) is homotopic to a constant
map.
Lemma 5.3. Let f : X ! Y be a an inclusion of simplicial sets which is a weak
equivalence, and let W and Z be simplicial sets such that we have a diagram of
inclusions
W _____//Z
 
 
fflfflfflffl
X _____//Y
Let u : (Dn, Sn1) ! (Z, W ) be a relative map of CWpairs. Then the inclus*
*ion
i : (Z, W ) ! (Y , X) can be factored as a composite
(Z, W ) ! (K, L) ! (Y , X)
where K is a subspace of Y obtained from Z by attaching a finite number of non
degenerate simplices, L is a subspace of X, and the composite map of relative C*
*W
complexes
(Dn, Sn1) ! (Z, W ) ! (K, L)
is homotopic rel Sn1 to a map Dn ! L.
Lemma 5.4. Let (Y, X) be a CWpair such that X and Y have only countably many
cells. Then for a fixed n 0, there are only countably many homotopy classes of
maps (Dn, Sn1) ! (Y, X).
If A ! B is a monomorphism of Segal precategories, then taking the localizati*
*on
via the small object argument gives us that LcA ! LcB is a monomorphism of Segal
categories. In particular, if A B is an inclusion, then we can regard LcA L*
*cB
as an inclusion as well.
20 J.E. BERGNER
Lemma 5.5. Let A and B be Segal precategories such that A B. Let oe be a
simplex in LcB which is not in LcA. Then there exists a Segal precategory A0such
that A0is obtained from A by attaching a finite number of nondegenerate simplic*
*es
and oe is in LA0.
Proof.By our description of our localization functor at the beginning of the se*
*ction,
LcB is obtained from B by taking a colimit of pushouts, each of which is along *
*the
map a a
V [m, k] x [n]t[ [m] x G(n)t! [m] x [n]t
m,k,n m,k,n
for n, m 1 and 0 k m. The Segal category LcB is the colimit of a filtrati*
*on
B 1B 2B . . .
where each iis given by a colimit of iterated pushouts along this map. Since o*
*e is
a single simplex, it is small and therefore oe is in nB for some n.
Therefore, oe is obtained by attaching [m] x [n]t along a finite number of
nondegenerate simplices of n1B. We can then apply the preceding argument
to each of these simplices and inductively obtain a finite number of nondegener*
*ate
simplices of B which form a subSegal precategory which we will call C. We then
define A0= A [ C.
We then state one more lemma, which is a generalization of a lemma given by
Hirschhorn [11, 2.3.6].
Lemma 5.6. Let the map g : A ! B be an inclusion of Segal precategories, each
of which has countably many simplices. If X is a Segal precategory with count
ably many simplices, then its localization LX with respect to the map g has only
countably many simplices.
We are now able to state and prove our result about generating cofibrations.
Proposition 5.7. Any acyclic cofibration j : C ! D in SeCatc can be written as
a directed colimit of pushouts along the maps in Jc.
Proof.Note that by definition j : C ! D is a monomorphism of Segal precategorie*
*s.
We assume that it is an inclusion. Let U be a subsimplicial space of D such that
U has countably many simplices in each degree. Apply the localization functor Lc
to obtain a diagram of Segal categories
Lc(U \ C)_____//LcU
 
 
fflffl fflffl
LcC _______//_LcD.
Since U has only countably many simplices, this localization process adds at mo*
*st
a countable number of simplices to the original simplicial space by Lemma 5.6.
We would like to find a Segal precategory W such that U W D and such
that the map W \ C ! W is in the set Jc.
First consider the map
Ho(Lc(U \ C)) ! Ho(LcU)
which we want to be an equivalence of categories. If it is not an equivalence, *
*then
there exists z 2 (LcU)0 which is not equivalent to some z02 (Lc(U \C))0. Howeve*
*r,
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 21
there is such a z0 when we consider z as an element of (LcD)0, since j : C ! D *
*is
a DKequivalence. If this z0is not in (U \ C)0, then we add it. Repeat this pro*
*cess
for all such z.
Now for each such z, consider the four mapping spaces in LcU involving the
objects z and z0: map LcU(z, z), map LcU(z, z0), map LcU(z0, z), and map LcU(z0*
*, z0).
We want the sets of components of these four spaces to be isomorphic to one ano*
*ther
in Ho (LcU). We can attach a countable number of simplices via an analogous
argument to the one in the proof of Lemma 5.5 such that these sets of components
are isomorphic. We then repeat the same argument to assure that ss0map LcU(x, z)
is isomorphic to ss0map LcU(x, z0) for each x 2 U0 and analogously for the sets*
* of
components of the mapping spaces out of each such x.
By repeating this process for each such z, we obtain a Segal precategory Y
with a countable number of simplices such that Ho (Lc(Y \ C)) ! Ho (LcY ) is
an equivalence of categories. However, we do not necessarily have that for each
x, y 2 Lc(Y \ C),
map Lc(Y \C)(x, y) ! mapLcY (x, y)
is a weak equivalence of simplicial sets. Therefore we consider all maps
(Dn, Sn1) ! (map LcY(x, y), map Lc(Y \C)(x, y)) ! (map LcD(x, y), map L*
*cC(x, y))
for each x, y 2 (Y \ C)0 and n 0. Identify all x, y, and n such that the map
(Dn, Sn1) ! (map LcY(x, y), map Lc(Y \C)(x, y))
is not homotopic to a constant map.
However each composite map
(Dn, Sn1) ! (map LcY(x, y), map Lc(Y \C)(x, y)) ! (map LcD(x, y), map L*
*cC(x, y))
is homotopic to a constant map by Lemma 5.2 since
map LcC(x, y) ! map LcD(x, y)
is a weak equivalence.
For each such x, y, and n, it follows from Lemma 5.3 that there exists some p*
*air
of simplicial sets
(map LcY(x, y), mapLc(Y \C)(x, y)) (K, L) (map LcD(x, y), mapLcC(x, y))
such that the composite map
(Dn, Sn1) ! (map LcY(x, y), map Lc(Y \C)(x, y)) ! (K, L)
is homotopic to a constant map, and the pair (K, L) is obtained from the pair
(map LcY(x, y), mapLc(Y \C)(x, y)) by attaching a finite number of nondegenerate
simplices. We apply Lemma 5.5 to each of these new simplices obtained by consid
ering each nontrivial homotopy class to obtain some Segal precategory Y 0with a
countable number of number of simplices such that each composite map
(Dn, Sn1) ! (map LcY(x, y), map Lc(Y \C)(x, y)) ! (map LcY 0(x, y), map*
* Lc(Y 0\C)(x, y))
is homotopic to a constant map.
However, the process of adding simplices may have created more maps
(Dn, Sn1) ! (map LcY 0(x, y), map Lc(Y 0\C)(x, y))
22 J.E. BERGNER
that are not homotopic to a constant map. Therefore we repeat this argument,
perhaps countably many times, until, taking a colimit over all of them, we obta*
*in
a Segal precategory W such that each map
(Dn, Sn1) ! (map LcW(x, y), map Lc(W\C)(x, y))
is homotopic to a constant map. Since each of these steps added only countably
many simplices to the original Segal precategory U, and since by Lemma 5.2
mapLc(W\C)(x, y) ! mapLcW (x, y)
is a weak equivalence for all x, y 2 (Lc(W \ C))0, the map W \ C ! W is in the
set Jc.
Now, take some eUobtained from W by adding a countable number of simplices,
consider the inclusion map eU\C ! eU, and repeat the entire process. To show th*
*at
we can repeat this argument, taking a (possibly transfinite) colimit, and event*
*ually
obtain the map j : C ! D, it suffices to show that the localization functor Lc
commutes with arbitrary directed colimits of inclusions. However, this fact fol*
*lows
from [11, 2.2.18].
Now, we have two definitions of acyclic fibration that we need to show coinci*
*de:
the fibrations which are weak equivalences, and the maps with the right lifting
property with respect to the maps in Ic.
Proposition 5.8. The maps with the right lifting property with respect to the m*
*aps
in Ic are fibrations and weak equivalences.
Before giving a proof of this proposition, we begin by looking at the maps in
Ic and determining what an Icinjective looks like. Recall the definition of t*
*he
coskeleton of a simplicial space from the paragraph following Proposition 2.14.*
* If
f : X ! Y has the right lifting property with respect to the maps in Ic, then f*
*or
each n 1, the map Xn ! Pn is an acyclic fibration of simplicial sets, where P*
*n is
the pullback in the diagram
Pn _____________//_Yn
 
 
fflffl fflffl
(coskn1X)n_____//(coskn1Y )n.
In the case that n = 0, the restrictions on m and n give us that the map X0 ! Y0
is a surjection rather than the isomorphism we get in the Reedy case. Notice th*
*at
by the same argument given for the Reedy model category structure (in the secti*
*on
following Proposition 2.14 above), the simplicial sets Pn can be characterized *
*up to
weak equivalence as homotopy pullbacks and are therefore homotopy invariant.
This characterization of the maps with the right lifting property with respec*
*t to
Ic will enable us to prove Proposition 5.8. Before proceeding to the proof, how*
*ever,
we state a lemma, whose proof we defer to section 9.
Lemma 5.9. Suppose that f : X ! Y is a map of Segal precategories which is an
Icinjective. Then f is a DKequivalence.
Proof of Proposition 5.8.Suppose that f : X ! Y is an Icinjective, or a map
which has the right lifting property with respect to the maps in Ic. Note that
f then has the right lifting property with respect to all cofibrations. Since,*
* in
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 23
particular, it has the right lifting property with respect to the acyclic cofib*
*rations,
it is a fibration by definition. It remains to show that f is a weak equivalenc*
*e.
However, this fact follows from Lemma 5.9, proving the proposition.
We now state the converse, which we will prove in section 9.
Proposition 5.10. The maps in SeCatc which are both fibrations and weak equiv
alences are Icinjectives.
Now we prove a lemma which we need to check the last condition for our model
category structure.
Lemma 5.11. A pushout along a map of Jc is also an acyclic cofibration in SeCat*
*c.
Proof.Let j : A ! B be a map in Jc. Notice that j is an acyclic cofibration in *
*the
model category CSS. Since CSS is a model category, we know that a pushout along
an acyclic cofibration is again an acyclic cofibration [9, 3.14(ii)]. If all th*
*e objects
involved are Segal precategories, then the pushout will again be a Segal precat*
*egory
and therefore the pushout map will be an acyclic cofibration in SeCatc.
Proposition 5.12. If a map of Segal precategories is a Jccofibration, then it *
*is
an Iccofibration and a weak equivalence.
Proof.By definition and Proposition 5.7, a Jccofibration is a map with the left
lifting property with respect to the maps with the right lifting property with *
*respect
to the acyclic cofibrations. However, by the definition of fibration, these map*
*s are
the ones with the left lifting property with respect to the fibrations.
Similarly, using Propositions 5.8 and 5.10, an Iccofibration is a map with t*
*he
left lifting property with respect to the acyclic fibrations. Thus, we need to *
*show
that a map with the left lifting property with respect to the fibrations has th*
*e left
lifting property with respect to the acyclic fibrations and is a weak equivalen*
*ce.
Since the acyclic fibrations are fibrations, it remains to show that the maps w*
*ith
the left lifting property with respect to the fibrations are weak equivalences.
Let f : A ! B be a map with the left lifting property with respect to all
fibrations. By Lemma 5.11 above, we know that a pushout along maps of Jc is an
acyclic cofibration. Therefore, we can use the small object argument [11, 10.5.*
*15]
to factor the map f : A ! B as the composite of an acyclic cofibration A ! A0
and a fibration A0! B. Then there exists a dotted arrow lift in the diagram
A __'__//A0>>
 "" 
 " 
fflfflfflfflid"
B _____//B
showing that the map A ! B is a retract of the map A ! A0and therefore a weak
equivalence.
Proof of Theorem 5.1.Axiom MC1 follows since limits and colimits of Segal pre
categories (computed as simplicial spaces) still have discrete zero space and a*
*re
therefore Segal precategories. MC2 and MC3 (for weak equivalences) work as usua*
*l,
for example see [9, 8.10].
It remains to show that the four conditions of Theorem 2.3 are satisfied. The
set Ic permits the small object argument because the generating cofibrations in*
* the
Reedy model category structure do. We can show that the objects A which appear
24 J.E. BERGNER
as the sources of the maps in Jc are small using an analogous argument to the o*
*ne
for simplicial sets [11, 10.4.4], so the set Jc permits the small object argume*
*nt.
Thus, condition 1 is satisfied.
Condition 2 is precisely the statement of Proposition 5.12. Condition 3 and
condition 4(ii) are precisely the statements of Propositions 5.8 and 5.10.
Note that the reduced Reedy acyclic cofibrations
(V [m, k] x [n]t[ [m] x `[n]t)r ! ( [m] x [n]t)r
are acyclic cofibrations in SeCatc for m 0 when n 1 and for n = m = 0.
Corollary 5.13. The fibrant objects in SeCatc are Reedy fibrant Segal categorie*
*s.
Proof.Suppose that X is fibrant in SeCatc. Then, since the reduced Reedy cofi
brations are acyclic cofibrations in SeCatc and since X has discrete zero space*
*, it
follows that X is Reedy fibrant.
Then, since the maps
( [m] x G(n)t)r ! ( [m] x [n]t)r
for all m, n 0 are acyclic cofibrations in SeCatc, it follows that X is a Seg*
*al
category.
It would also be nice to know that the converse statement is also true: that *
*every
Reedy fibrant Segal category is fibrant in Secatc. While it seems plausible tha*
*t this
fact is also true, we do not prove it here.
6.A Quillen Equivalence Between SeCatc and CSS
In this section, we will show that there is a Quillen equivalence between the*
* model
category structure SeCatcon Segal precategories and the Segal space complete Se*
*gal
space model category structure CSSon simplicial spaces. We first need to show t*
*hat
we have an adjoint pair of maps between the two categories.
Let I : SeCatc ! CSS be the inclusion functor of Segal precategories into the
category of all simplicial spaces. We will show that there is a right adjoint f*
*unctor
R : CSS ! SeCatc which "discretizes" the degree zero space.
Let W be a simplicial space. Define simplicial spaces U = cosk0(W0) and V =
cosk0(W0,0). There exist maps W ! U V . Then we take the pullback RW in
the diagram
RW _____//V
 
 
fflffl fflffl
W ______//U.
Note that RW is a Segal precategory. If W is a complete Segal space, then so
are U and V , and in this case RW is a Segal category, which we can see as foll*
*ows.
The pullback at degree 1 gives
(RW )1_____//W0,0x W0,0
 
 
fflffl fflffl
W1 ________//W0 x W0
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 25
and at degree 2 we get
(RW )1 x(RW)0 (RW )1________//_(W0,0)3
 
 
fflffl fflffl
W2 ' W1 xW0 W1 _______//W0 x W0 x W0.
Looking at these pullbacks, and the analogous ones for higher n, we notice that
RW is in fact a Segal category.
We define the functor R : CSS ! SeCatc which takes a simplicial space W to the
Segal precategory RW given by the description above.
Proposition 6.1. The functor R : CSS ! SeCatc is right adjoint to the inclusion
map I : SeCatc ! CSS.
Proof.We need to show that there is an isomorphism
Hom SeCatc(Y, RW ) ~=Hom CSS(IY, W )
for any Segal precategory Y and simplicial space W .
Suppose that we have a map Y = IY ! W . Since Y is a Segal precategory,
Y0 is equal to Y0,0viewed as a constant simplicial set. Therefore, we can restr*
*ict
this map to a unique map Y ! V , where V is the Segal precategory defined above.
Then, given the universal property of pullbacks, there is a unique map Y ! RW .
Hence, we obtain a map
' : Hom CSS(IY, W ) ! Hom SeCatc(Y, RW ).
This map is surjective because given any map Y ! RW we can compose it with
the map RW ! W to obtain a map Y ! W .
Now for any Segal precategory Y , consider the diagram
Y ____________________________________________________*
*______________________________________D
____________________________________________________*
*________________D
____________________________________________________*
*__________D
___________________________________________D
______________$$___________________________________*
*_____!!
_________________________RW//_V
_________________________
__________________________
_______________
OEOE________________fflfflfflffl
W ______//U
Because this diagram must commute and the image of the map Y0 ! W0 is con
tained in W0,0since Y is a Segal precategory, this map uniquely determines what
the map Y ! V has to be. Therefore, given a map Y ! RW , it could only have
come from one map Y ! W . Thus, ' is injective.
Now, we need to show that this adjoint pair respects the model category struc
tures that we have.
Proposition 6.2. The adjoint pair of functors
I : SeCatc___//CSS:oRo_
is a Quillen pair.
26 J.E. BERGNER
Proof.It suffices to show that the inclusion map I preserves cofibrations and a*
*cyclic
cofibrations. I preserves cofibrations because they are defined to be monomor
phisms in each category. Also in each of the two categories, a map is a weak
equivalence if it is a DKequivalence after localizing to obtain a Segal space,*
* as
given in Theorem 3.10. In each case an acyclic cofibration is an inclusion sati*
*sfying
this property. Therefore, the map I preserves acyclic cofibrations.
Theorem 6.3. The Quillen pair
I : SeCatc___//CSS:oRo_
is a Quillen equivalence.
Proof.We need to show that I reflects weak equivalences between cofibrant objec*
*ts
and that for any fibrant object W (i.e. complete Segal space) in CSS, the map
I((RW )c) = IRW ! W is a weak equivalence in SeCatc.
The fact that I reflects weak equivalences between cofibrant objects follows *
*from
the same argument as from the proof of the Quillen pair. To prove the second pa*
*rt,
it remains to show that the map j : RW ! W in the pullback diagram
RW _____//V
j 
fflffl fflffl
W ______//U
is a DKequivalence. It suffices to show that the map of objects ob(RW ) ! ob(W*
* )
is surjective and that the map map RW (x, y) ! map W (jx, jy) is a weak equiva
lence, where the object set of a Segal space is defined as in section 3.3. Howe*
*ver,
notice by the definition of RW that ob(RW ) = ob(W ). In particular, jx = x and
jy = y. Then notice, using the pullback that defines (RW )1 that map RW (x, y) '
mapW (x, y). Therefore, the map RW ! W is a DKequivalence.
7.Another Segal Category Model Category Structure on Segal
Precategories
The model category structure SeCatc that we defined above is helpful for the
Quillen equivalence with the complete Segal space model category structure, but
there does not appear to be a Quillen equivalence between it and the model cate*
*gory
structure SC on simplicial categories. Therefore, we need another model category
structure SeCatf to obtain such a Quillen equivalence.
In the model category structure SeCatc, we started with the generating cofibr*
*a
tions in the Reedy model category structure and adapted them to be generating
cofibrations of Segal precategories. In this second model category structure, w*
*e will
use modified generating cofibrations from the projective model category structu*
*re
on simplicial spaces so that the objects involved are Segal precategories.
We make the following definitions for a model category structure SeCatf on the
category of Segal precategories.
o The weak equivalences are the same as those of SeCatc.
o The cofibrations are the maps which can be formed by taking iterated
pushouts along the maps of the set If defined in section 4.
o The fibrations are the maps with the right lifting property with respect*
* to
the maps which are cofibrations and weak equivalences.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 27
Notice that to define the weak equivalences in this case we will want to use a
functorial localization in SeSpf rather than SeSpc. We define a localization fu*
*nctor
Lf in the same way that we defined Lc at the beginning of section 5 but making
necessary changes in light of the fact that we are starting from the model stru*
*cture
SeSpf. So, in a sense, the weak equivalences are not defined identically in the
two categories, since they make use of the same localization of different model
category structures on the category of simplicial spaces. However, in each case*
* the
weak equivalences are the same in the unlocalized model category, so we can def*
*ine
homotopy function complexes using only the underlying category and the weak
equivalences. Recall by the definition of local objects that a map LX ! LY is a
local equivalence if and only if the induced map of homotopy function complexes
Map h(Y, Z) ! Map h(X, Z)
is a weak equivalence of simplicial sets for any local Z. In particular, the w*
*eak
equivalences of the localized category depend only on the weak equivalences of *
*the
unlocalized category. Therefore the weak equivalences in SeCatc and SeCatf are
actually the same.
Theorem 7.1. There is a cofibrantly generated model category structure SeCatf on
the category of Segal precategories in which the weak equivalences, fibrations,*
* and
cofibrations are defined as above.
We define the set Jf to be a set of isomorphism classes of maps {i : A ! B}
such that
(1) for all n 0, the spaces An and Bn have countably many simplices, and
(2) i : A ! B is an acyclic cofibration.
We would like to show that If (defined in section 4) is a set of generating c*
*ofi
brations and that Jf is a set of generating acyclic cofibrations for SeCatf.
We begin with the following lemma.
Lemma 7.2. Any acyclic cofibration j : C ! D in SeCatf can be written as a
directed colimit of pushouts along the maps in Jf.
Proof.The argument that we used to prove Proposition 5.7 still holds, applying
the functor Lf rather than Lc.
Proposition 7.3. A map f : X ! Y is an acyclic fibration in SeCatf if and only
if it is an Ifinjective.
Proof.First suppose that f has the right lifting property with respect to the m*
*aps
in If. Then we claim that for each n 0 and (v0, . .,.vn) 2 Xn+10, the map
Xn(v0, . .,.vn) ! Yn(fv0, . .,.fvn) is an acyclic fibration of simplicial sets.*
* This
fact, however, follows from Lemma 4.1. In particular, it is a weak equivalence,*
* and
therefore we can apply the proof of Lemma 5.9 to show that the map X ! Y
is a DKequivalence, completing the proof of the first direction. (The proof do*
*es
not follow precisely in this case, in particular because not all monomorphisms *
*are
cofibrations. However, we can use the fact that weak equivalences are the same *
*in
SeCatc and SeCatf to see that the argument still holds.)
Then, to prove the converse, assume that f is a fibration and a weak equivale*
*nce.
Then we can apply the proof of Proposition 5.10, making the factorizations in
the projective model category structure rather than in the Reedy model category
structure. The argument follows analogously.
28 J.E. BERGNER
Proposition 7.4. A map in SeCatf is a Jfcofibration if and only if it is an If
cofibration and a weak equivalence.
Proof.This proof follows just as the proof of Proposition 5.12, again using the
projective structure rather than the Reedy structure.
Proof of Theorem 7.1.As before, we must check the conditions of Theorem 2.3.
Condition 1 follows just as in the proof of Theorem 5.1. Condition 2 is precise*
*ly
the statement of Proposition 7.4. Condition 3 and condition 4(ii) follow from P*
*ropo
sition 7.3 after applying Lemma 7.2.
We now prove that both our model category structures on the category of Segal
precategories are Quillen equivalent.
Theorem 7.5. The identity functor induces a Quillen equivalence
I : SeCatf____//SeCatco:oJ._
Proof.Since both maps are the identity functor, they form an adjoint pair. We
then show that this adjoint pair is a Quillen pair.
We first make some observations between the two categories. Notice that the
cofibrations of SeCatf form a subclass of the cofibrations of SeCatc since they*
* are
monomorphisms. Similarly, the acyclic cofibrations of SeCatf form a subclass of
the acyclic cofibrations of SeCatc.
In particular, these observations imply that the left adjoint I : SeCatf ! Se*
*Catc
preserves cofibrations and acyclic cofibrations. Hence, we have a Quillen pair.
It remains to show that this Quillen pair is a Quillen equivalence. To do so,
we must show that given any cofibrant X in SeCatf and fibrant Y in SeCatc, a
map f : IX ! Y is a weak equivalence in SeCatf if and only if 'f : X ! JY is
a weak equivalence in SeCatc. However, this follows from the fact that the weak
equivalences are the same in each category.
Note. One might ask at this point why we could not just use the SeCatf model ca*
*t
egory structure and show a Quillen equivalence between it and the model category
structure CSSf where we localize the projective model category structure (rather
than the Reedy) with respect to the maps ' and _. The existence of such a Quill*
*en
equivalence would certainly simplify this paper!
However, if we work with "complete Segal spaces" which are fibrant in the pro
jective model structure rather than in the Reedy structure, then for a fibrant *
*object
W the map W ! U used in defining the right adjoint CSS ! SeCatc is no longer
necessarily a fibration. Therefore, the pullback RW is no longer a homotopy pul*
*l
back and in particular not homotopy invariant. If RW is not homotopy invariant,
then there is no guarantee that the map RW ! W is a DKequivalence, and the ar
gument for a Quillen equivalence fails. Thus, the SeCatc and CSS model structur*
*es
are necessary.
8. A Quillen Equivalence Between SC and SeCatf
We begin, as above, by defining an adjoint pair of functors between the two
categories SC and SeCatf. We have the nerve functor R : SC ! SeCatf. In order
to define a left adjoint to this map, we need some terminology.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 29
Definition 8.1. Let D be a small category and SSetsD the category of functors
D ! SSets. Let S be a set of morphisms in SSetsD . An object Y of SSetsD is
strictly Slocal if for every morphism f : A ! B in S, the induced map on funct*
*ion
complexes
f* : Map (B, Y ) ! Map (A, Y )
is an isomorphism of simplicial sets. A map g : C ! D in SSetsD is a strict Sl*
*ocal
equivalence if for every strictly Slocal object Y in SSetsD , the induced map
g* : Map (D, Y ) ! Map (C, Y )
is an isomorphism of simplicial sets.
Now, we can view Segal precategories as functors op ! SSets. Because we
require the image of [0] to be a discrete simplicial set, the category of Segal*
* pre
categories is a subcategory of the category of all such functors. In this secti*
*on, we
are going to regard simplicial categories as the strictly local objects in Seca*
*tf with
respect to the map ' described in section 3.3.
Although we are actually working in a subcategory, we can still use the follo*
*wing
lemma to obtain a left adjoint functor F to our inclusion map R, since the con
struction will always produce a simplicial space with discrete 0space when app*
*lied
to such a simplicial space.
Lemma 8.2. [3] Consider two categories, the category of all diagrams X : D !
SSets and the category of strictly local diagrams with respect to the set of ma*
*ps
S = {f : A ! B}. The forgetful functor from the category of strictly local diag*
*rams
to the category of all diagrams has a left adjoint.
We define the map F : SeCatf ! SC to be this left adjoint to the inclusion map
of strictly local diagrams into all diagrams R : SC ! SeCatf.
Proposition 8.3. The adjoint pair
F : SeCatf____//SC:oRo_
is a Quillen pair.
Proof.We will prove that this adjoint pair is a Quillen pair by showing that th*
*e left
adjoint F preserves cofibrations and acyclic cofibrations. We begin by consider*
*ing
cofibrations.
Since F is a left adjoint functor, it preserves colimits. Therefore, it suffi*
*ces to
show that F preserves the set If of generating cofibrations in SeCatf. Recall t*
*hat
the elements of this set are the maps Pm,n ! Qm,n as defined in section 4. We
begin by considering the maps Pn,1! Qn,1for any n 0. The strict localization
of such a map is precisely the map of simplicial categories U `[n] ! U [n] (sec*
*tion
3.1) which is a generating cofibration in SC. We can also see that any the stri*
*ct
localization of any Pm,n ! Qm,n can be obtained as the colimit of iterated push*
*outs
along the generating cofibrations of SC. Therefore, F preserves cofibrations.
We now need to show that F preserves acyclicocofibrations.p To do so, first
consider the model category structure LSSetsO,f (defined in section 3.15) on Se*
*gal
precategories with a fixed set O in degree zero and the model category structure
SCO of simplicial categories with a fixed object set O. Recall from section 3.1*
*5 that
there is a Quillen equivalence
FO : LSSetsOop,f__//_SCO:oROo._
30 J.E. BERGNER
In particular, if X is a cofibrant object of LSSetsOop,f, then there is a weak *
*equiv
alence X ! RO ((FO X)f). Notice that FO agrees with F on Segal precategories
with the set O in degree zero, and similarly foroROpand R. op
Suppose, then, that X is an object of LSSetsO,f, Y is an object of LSSetsO0,f,
and X ! Y is an acyclic cofibration in SeCatf. We have a commutative diagram
X __'__//LfX
 
 
fflffl'fflffl
Y _____//LfY
where the upper and lowerohorizontalpmapsoarepweak equivalences not only in
SeCatf, but in LSSetsO,f and LSSetsO0,f, respectively. However, using the fixed
object case Quillen equivalence, the functors FO and FO00(and hence F ) will pr*
*e
serve these weak equivalences, giving us a diagram
F X __'__//F LfX
 
 
fflffl' fflffl
F Y ____//_F LfX.
Using these weak equivalences and our assumption that LfX ! LfY is a DK
equivalence, we obtain a diagram
LfX _'__//_RF LfX
' 
fflffl' fflffl
LfY ____//_RF LfY
in which the upper horizontal arrow is a weak equivalence in LSSetsOop,fand the
lower horizontal arrow is a weak equivalence in LSS etsOop0,f. The commutativi*
*ty
of this diagram implies that the map RF LfX ! RF LfY is a DKequivalence
also. Thus, we have shown that F preserves acyclic cofibrations between cofibra*
*nt
objects.
It remains to show that F preserves all acyclic cofibrations. Suppose that f :
X ! Y is an acyclic cofibration in SeCatf. Apply the cofibrant replacement func*
*tor
to the map X ! Y to obtain an acyclic cofibration X0 ! Y 0, and notice that in
the resulting commutative diagram
X0 _____//Y 0
 
 
fflffl fflffl
X _____//_Y
the vertical arrows are levelwise weak equivalences.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 31
Now consider the following diagram, where the top square is a pushout diagram:
X0__'__//Y 0
 
 
fflffl'fflffl
X _____//Y 00
=  
fflffl'fflffl
X _____//_Y.
Notice that all three of the horizontal arrows are acyclic cofibrations in SeCa*
*tf,
the upper and lower by assumption and the middle one because pushouts preserve
acyclic cofibrations [9, 3.14]. Now we apply the functor F to this diagram to o*
*btain
a diagram
(8.4) F X0__'__//F Y 0
 
 
fflffl' fflffl
F X _____//F Y 00
 
 
fflffl fflffl
F X _____//_F Y.
The top horizontal arrow is an acyclic cofibration since F preserves acyclic co*
*fibra
tions between cofibrant objects. Furthermore, since F is a left adjoint and hen*
*ce
preserves colimits, the middle horizontal arrow is also an acyclic cofibration *
*because
the top square is a pushout square.
Now, recall that, given an object X in a model category C, the category of
objects under X has as objects the morphisms X ! Y in C for any object Y , and
as morphisms the maps Y ! Y 0in C making the appropriate triangular diagram
commute [11, 7.6.1]. There is a model category structure on this under category*
* in
which a morphism is a weak equivalence, fibration, or cofibration if it is in C*
* [11,
7.6.5]. In particular, a object X ! Y is cofibrant in the under category if it *
*is a
cofibration in C.
With this definition in mind, to show that the bottom horizontal arrow of dia
gram 8.4 is an acyclic cofibration, consider the following diagram in the categ*
*ory
of cofibrant objects under X:
X ____//_BY 00
BB 
BBB 
B!!Bfflffl
Y.
Now, let O00denote the set in degree zero of Y 00(and also of Y ) which is not *
*in the
image of the map from X. Now we have the diagram in the category of cofibrant
objects under X q O00with the same set in degree zero
X q O00_____//IY 00
III 
III 
I$$Ifflffl
Y.
32 J.E. BERGNER
However, since we are now working in a fixed object set situation, we know by
Theorem 3.16 that FO00is the left adjoint of a Quillen pair, and therefore the *
*map
FO00Y 00! FO00Y is a weak equivalence in SCO00, and in particular a DKequivale*
*nce
when regarded as a map in SC. It follows that the map F X ! F Y is a weak
equivalence, and F preserves acyclic cofibrations.
Recall that we are regarding a Segal category as a local diagram and a simpli*
*cial
category as a strictly local diagram in SeCatf.
Lemma 8.5. The map X ! F X is a DKequivalence for every cofibrant object X
in SeCatf.
Proof.First consider a free diagram in SeCatf, namely some qiQmi,ni, where each
Qmi,niis defined as in section 4. If Y is a fibrant object in SeCatf, then we h*
*ave
a Y
Map SeCatf( Qmi,ni,'Y ) MapSeCatf(Qmi,ni, Y )
i Yi a
' Map SSets( [mi], Yni(v0, . .,.vn))
i v0,...,vn
Y a
' , Map SSets( [0], Yni(v0, . .,.vn))
i v0,...,vn
a
' Map SeCatf( Q0,ni, Y )
ai
' Map SeCatf( [ni]t, Y )
i
Therefore, it suffices to consider free diagrams qi [ni]t. Such a diagram is a *
*Segal
category. It is also the nerve of a category and thus a strictly local diagram*
*. It
follows that the map a a
[ni]t! F ( [ni]t)
i i
is a DKequivalence.
Now suppose that X is any cofibrant object in SeCatf. Then X can be written
as a directed colimit X ' colim opXj, where each Xj can be written as qi [ni]t.
As before we regard F X as a strictly local object in SeCatf. If Y is a fibrant*
* object
in SeCatf which is strictly local, we have
Map SeCatf(colim opXj, Y')limMap SeCatf(Xj, Y )
' limMap SeCatf(F Xj, Y )
' Map SeCatf(colim opF Xj, Y )
' Map SeCatf(F (colim op(F Xj)), Y )
We can now apply the result that
F (colim(F Xj)) ' F (colimXj).
(This fact is proved in [4] for ordinary localization, but it holds for strict *
*localization
in this case since each Xj is cofibrant and F preserves cofibrant objects.) The*
*refore
we have
Map SeCatf(F (colim op(F Xj)), Y ) ' Map SeCatf(F X, Y ).
It follows that the map X ! F X is a DKequivalence.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 33
We are now able to prove the main result of this section.
Theorem 8.6. The Quillen pair
F : SeCatf____//SC:oRo_
is a Quillen equivalence.
Proof.We will first show that F reflects weak equivalences between cofibrant ob
jects. Let f : X ! Y be a map of cofibrant Segal precategories such that
F f : F X ! F Y is a weak equivalence of simplicial categories. (Since F preser*
*ves
cofibrations, both F X and F Y are again cofibrant.) Then consider the following
diagram:
F X _____//LfF Xoo___LfX
'   
fflffl fflffl fflffl
F Y _____//LfF Yoo__ LfY.
By assumption, the leftmost vertical arrow is a DKequivalence. The horizontal
arrows of the lefthand square are also DKequivalences by definition. Since X *
*and
Y are cofibrant, Lemma 8.5 shows that the horizontal arrows of the righthand
square are DKequivalences. The commutativity of the whole diagram shows that
the map LfF X ! LfF Y is a DKequivalence and then that the map LfX ! LfY
is also. Therefore, F reflects weak equivalences between cofibrant objects.
Now, we will show that given any fibrant simplicial category Y , the map
F ((RY )c) ! Y
is a DKequivalence. Consider a fibrant simplicial category Y and apply the fun*
*ctor
R to obtain a Segal category which is levelwise fibrant and therefore fibrant in
SeCatf. Its cofibrant replacement will be DKequivalent to it in SeCatf. Then, *
*by
the above argument, strictly localizing this object will again yield a DKequiv*
*alent
simplicial category.
9. Proofs of Lemma 5.9 and Proposition 5.10
In this section, we will give a proof of two results stated in section 5. We *
*begin
with a lemma which we will use in the proof of Lemma 5.9.
Lemma 9.1. Suppose that f : X ! Y is a map of Segal precategories with the
right lifting property with respect to the maps in Ic. Then
(1) The map f0 : X0 ! Y0 is surjective, and
(2) The map Xn(v0, . .,.vn) ! Yn(fv0, . .,.fvn) is a weak equivalence of sim
plicial sets for all n 1 and (v0, . .,.vn) 2 Xn+10.
Proof.Since f : X ! Y has the right lifting property with respect to the maps in
Ic, it has the right lifting property with respect to all cofibrations. In part*
*icular, it
has the right lifting property with respect to the maps in the set If. Therefor*
*e we
can apply Lemma 4.1 and the result follows.
Proof of Lemma 5.9.To prove Lemma 5.9, we will consider a given map f : X ! Y
with the right lifting property with respect to the maps in Ic. It follows from
34 J.E. BERGNER
Lemma 9.1 that the map X0 ! Y0 is surjective and such that for all n 1 and
(v0, . .v.n) 2 Xn+10the map
Xn(v0, . .,.vn) ! Yn(fv0, . .,.fvn)
is a weak equivalence of simplicial sets.
We must prove that map LcX(x, y) ! map LcY(fx, fy) is a weak equivalence of
simplicial sets. Once we have proved that fact, combining it with the surjectiv*
*ity
of the map X0 ! Y0 will imply that Ho(LcX) ! Ho(LcY ) is an equivalence of
categories.
We construct a factorization X ! Y ! Y such that ( Y )0 = X0 and the map
( Y )n(v0, . .,.vn) ! Yn(fv0, . .,.fvn)
is an isomorphism of simplicial sets for all (v0, . .,.vn) 2 ( Y )n+10. We begi*
*n by
defining the object Y as the pullback of the diagram
Y ___________//Y
 
 
fflffl fflffl
cosk0(X0)_____//cosk0(Y0).
Note in particular that ( Y )0 = X0.
Now, notice that for each n 1 and (v0, . .,.vn) 2 ( Y )n+10the map
( Y )n(v0, . .,.vn) ! Yn(fv0, . .,.fvn)
is an isomorphism of simplicial sets. Since each
Xn(v0, . .,.vn) ! Yn(fv0, . .,.fvn)
is a weak equivalence, we can apply model category axiom MC2 to simplicial sets
to see that the map
Xn(v0, . .,.vn) ! ( Y )n(v0, . .,.vn)
is a weak equivalence for each n 1 and (v0, . .v.n) also.
Thus we have shown that if X ! Y has the right lifting property with respect
to the maps in Ic, then each map Xn(v0, . .,.vn) ! ( Y )n(v0, . .,.vn) is a weak
equivalence of simplicial sets for n 1 and (v0, . .,.vn) 2 Xn+10. Since X0 = *
*( Y )0,
the map X ! Y is actually a Reedy weak equivalence and therefore also a DK
equivalence. To prove Lemma 5.9, it remains to show that the map Y ! Y is a
DKequivalence, implying that the map X ! Y is also. We will prove this fact by
induction on the skeleta of Y .
We will denote by sknY the nskeleton of Y , as defined above in the paragraph
below Proposition 2.14. We seek to prove that the map
(sknY ) ! sknY
is a DKequivalence for all n 0.
We first consider the case where n = 0. In this case, sk0( Y ) and sk0Y are
already Segal categories. They can be observed to be DKequivalent as follows.
In the case of sk0Y , given any pair of elements (x, y) 2 (sk0Y )0 x (sk0Y )0, *
*the
mapping space map sk0Y(x, y) is the homotopy fiber of the map
(sk0Y )1 = (sk0Y )0 x (sk0Y )0 ! (sk0Y )0 x (sk0Y )0
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 35
over (x, y). If x = y, this fiber is just the point (x, y), since in this case *
*this map is
the identity. If x 6= y, then the fiber is empty. For (a, b) 2 (sk0 Y )0 x (sk0*
* Y )0,
the fiber of the analogous map over (a, b) is equivalent to (a, b) if a and b m*
*ap to
the same point x in Y0. Otherwise the fiber is empty. The definition of Y and
the map Y ! Y then show that the two are DKequivalent.
We now assume that the map (skn1Y ) ! skn1Y is a DKequivalence and
seek to show that the map
(sknY ) ! sknY
is also for n 2. Notice that sknY is obtained from skn1Y via iterations of
pushouts of diagrams of the form
(9.2) Qm,n oo___Pm,n____//_skn1Y
For simplicity, we will assume that m = 0 and we require only one such pushout *
*to
obtain sknY . Notice that (skn1Y )0 = (sknY )0 = Y0 and that the map
skn1Y ! sknY
is the identity on the discrete space in degree zero. Therefore we use the dist*
*inct
nsimplex [n]ty0,...,ynfor each (y0, . .,.yn) 2 Y0n+1as defined above in secti*
*on 3.12.
Setting y_= (y0, . .y.n), we write this nsimplex as [n]ty_.
We can then apply the map to diagram 9.2 (and its pushout) to obtain the
diagram
(9.3) `[n]ty___// skn1Y
 
 
fflffl fflffl
[n]ty_____// sknY.
We would like to know that we still have a pushout diagram. In other words, we
want to know that the functor preserves pushouts. To see that it does, consid*
*er
the levelwise pullback diagram defining ( Y )n:
( Y )n_______//Yn
 
 
fflffl fflffl
Xn+10_____//Y0n+1.
We can regard the map f : X ! Y as inducing a pullback functor f* from the
category of simplicial sets over Y0n+1to the category of simplicial sets over X*
*n+10.
(Recall that the category of objects over a simplicial set Z has as objects map*
*s of
simplicial sets W ! Z and as morphisms the maps of simplicial sets making the
appropriate triangle commute.) However, this functor between over categories can
be shown to have a right adjoint. Therefore it is a left adjoint and hence pres*
*erves
pushouts.
We know that the maps
`[n]ty_! `[n]ty_
and
(skn1Y ) ! skn1Y
are DKequivalences by our inductive hypothesis, since the nondegenerate simpli*
*ces
in each case are concentrated in degrees less than n. Since the lefthand verti*
*cal
36 J.E. BERGNER
maps of diagrams 9.2 and 9.3 above are cofibrations, the righthand vertical map
in diagram 9.3 is also a cofibration, and therefore it remains only to show tha*
*t the
map [n]ty_! [n]ty_is a DKequivalence in order to show that the pushouts of
the two diagrams are weakly equivalent.
If n = 0, then [0]ty_! [0]ty_is a DKequivalence since everything is alrea*
*dy
local and [0]ty_is just the nerve of some contractible category. In fact, giv*
*en any
n 0 and y_= (y0, . .,.yn), if yi6= yjfor each 0 i, j n, the map [n]ty_!*
* [n]ty_
is a DKequivalence, since [n]ty_is already local.
Now suppose that n = 1 and y_= (y0, y0). Consider g : [1]ty_! [1]ty_and
let k be the number of 0simplices of g1(y0). If Ck denotes the category with k
objects and a single isomorphism between any two objects, then we have that
[1]ty_' [1]ty_x nerve(Ck).
Thus, it suffices to show that
Lc [1]ty_' Lc [1]ty_x Lcnerve(Ck).
To prove this fact, first note that the fibrant objects in SeSpc are closed u*
*nder
internal hom, namely that given a Segal space W and any simplicial space Y , th*
*ere
is a Segal space W Y given by (W Y)k = Map h(Y x [k]t, W ) [16, 7.1]. Therefor*
*e,
given any Segal precategories X and Y and any Segal space W , we can work in the
category SeSpc and make the following calculation.
Map h(LcX x LcY, W )' Map h(LcX, W LcY))
' Map h(X, W Y)
' Map h(X x Y, W )
' Map h(Lc(X x Y ), W )
In other words, the map
Lc(X x Y ) ! LcX x LcY
is a DKequivalence, and in particular the statement above for Lc [n]ty_holds.
Now consider the case where n = 2. Then if y_= (y0, y1, y2), we have that G(2*
*)ty_
can be written as a pushout
(9.4) G(0)ty1_____//G(1)ty0,y1
 
 
fflffl fflffl
G(1)ty1,y2____//G(2)ty_.
Now consider the map g : G(2)ty_! G(2)ty_. We have that g1(G(0)ty1) is the
nerve of some contractible category. Similarly, the map g1(G(1)ty0,y1) ! G(1)t*
*y0,y1
is a DKequivalence, as is the map g1(G(1)ty1,y2) ! G(1)ty1,y2. Since we have*
* a
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 37
pushout diagram
g1(G(0)ty1)_____//g1(G(1)ty0,y1)
 
 
fflffl fflffl
g1(G(1)ty1,y2)______// G(2)ty_
and the lefthand vertical maps of this diagram and of diagram 9.4 are cofibrat*
*ions,
it follows that the map G(2)ty_! G(2)ty_is a DKequivalence. In fact, for any
n 2, G(n)ty_can be obtained by iterating such pushouts. Therefore, we have
shown that the map G(n)ty_! G(n)ty_is a DKequivalence.
To see that [n]ty_! [n]ty_is a DKequivalence for any choice of y_, we need
a variation on this argument. Again using a pushout construction, we will use t*
*he
fact that this map is a DKequivalence when each yi is distinct to show that it
is also a DKequivalence even if yi = yj for some i 6= j. We will describe this
construction for a specific example, but it works in general. Specifically, we *
*show
that [2]ty0,y1,y0! [2]ty0,y1,y0is a DKequivalence.
Define the Segal precategory eY= Y q {ey}, where eyis a 0simplex not in Y0,
and we regard {ey} as a doubly constant simplicial space. Then, using the map
g : Y ! Y and some vertex y0 of Y , we let Z be a Segal precategory isomorphic
to (g1y0) and define eX= X q Z. There is a map eX! eYsuch that Z maps to ey.
We define a functor e and factorization
Xe_____//eeYg__//eY
just as we defined Y above. More generally, we apply e to any Segal precategory
with 0simplices those of eYto obtain a Segal precategory with 0simplices thos*
*e of
eX, just as we have been doing with .
Now consider the objects G(2)ty0,y1,eyand [2]ty0,y1,ey, each with 0simplice*
*s those
of eY. There is a natural map
G(2)ty0,y1,ey! G(2)ty0,y1,y0
where ey7! y0, and an analogous map
[2]ty0,y1,ey! [2]ty0,y1,y0.
We have a pushout diagram
G(2)ty0,y1,ey__//G(2)ty0,y1,y0
 
 
fflffl fflffl
[2]ty0,y1,ey_//_ [2]ty0,y1,y0.
Since the lefthand vertical map is a cofibration, this map is actually a homot*
*opy
pushout diagram.
Now, from above we know that the maps
eG(2)ty0,y1,ey! G(2)ty0,y1,ey
and
G(2)ty0,y1,y0! G(2)ty0,y1,y0
38 J.E. BERGNER
are DKequivalences. We also know that the map
e [2]ty0,y1,ey! [2]ty0,y1,ey
is a DKequivalence since the 0simplices y0, y1, eyare distinct. We can consid*
*er the
pushout diagram
eg1G(2)ty0,y1,ey_//g1G(2)ty0,y1,y0
 
 
fflffl fflffl
eg1 [2]ty0,y1,ey//_ [2]ty0,y1,y0.
which is again a homotopy pushout diagram. It follows that the map
[2]ty0,y1,y0! [2]ty0,y1,y0
is a DKequivalence, completing the proof.
We now proceed with the other remaining proof from section 5.
Proof of Proposition 5.10.Suppose that f : X ! Y is a fibration and a weak
equivalence. First, consider the case where f0 : X0 ! Y0 is an isomorphism.
Without loss of generality,oassumepthat X0 = Y0 and factor the map f : X ! Y
functorially in SSetsc as the composite of a cofibration and an acyclic fibra*
*tion
in such a way that the Y00remains a discrete space:
" '
X O____//Y_0_////_Y.
(We can obtain a Y 0with discrete zero space by taking a factorization in SSets*
*cop
analogous to the one we defined for SeSpc at the beginning of section 5.) Since
the map X ! Y is a DKequivalence and the map Y 0! Y is a Reedy weak
equivalence and therefore a DKequivalence, it follows that the map X ! Y 0is a
DKequivalence. In particular, X ! Y 0is an acyclic cofibration and therefore by
the definition of fibration in SeCatf the dotted arrow lift exists in the follo*
*wing
solidarrow diagram:
X __id_//X>>
 "" 
 " 
fflfflfflffl"
Y 0____//Y.
Thus, f : X ! Y is a retract of Y 0! Y and therefore a Reedy acyclic fibration.
In particular, f has the right lifting property with respect to the maps in Ic,*
* since
they are monomorphisms and therefore Reedy cofibrations.
Now consider the general case, where X0 ! Y0 is surjective but not necessarily
an isomorphism. Then, as in the proof of Proposition 5.8, define the object Y
and consider the composite map X ! Y ! Y . since by the first case X ! Y
has the right lifting property with respect to the maps in Ic, it remains to sh*
*ow
that Y ! Y has the right lifting property with respect to the maps in Ic.
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES 39
Let A ! B be an acyclic cofibration. Then there is a dotted arrow lift in any
solidarrow diagram of the form
(9.5) A _______//_X___=_____//X44ii
i i
'  i ii 
fflffliiifflffli fflffl
B _______// Y_________//_Y
 
 
fflffl fflffl
cosk0X0 ____//_cosk0Y0
We would like to know that this lift B ! X also makes the upper lefthand square
commute.
Suppose that A0 = B0 = X0. In this case, a map B ! Y together with a lifting
X0==
__ 
_ 
_ fflffl
B0 _____//Y0
completely determines a map B ! Y . Therefore, in this fixed object set case,
there is only one possible lifting B ! X in diagram 9.5, and one which makes the
upper lefthand square commute.
The map X !opY is a fibration in the fixed object model category strucop
ture LSSetsO,f where O = X0. However, since the cofibrations in LSSetsO,f are
precisely the monomorphisms, the acyclic fibrations are Reedy acyclic fibration*
*s.
Therefore, the map X ! Y is a Reedy acyclic fibration and thus has the right l*
*ift
ing property with respect to all monomorphisms of simplicial spaces. In particu*
*lar,
it has the right lifting property with respect to the maps in Ic.
Using the construction of Y and the fact that X ! Y is a fibration and a
weak equivalence, we can see that X0 ! Y0 is surjective. In particular, the map
cosk0X0 ! cosk0Y0 has the right lifting property with respect to the maps in Ic.
Using the universal property of pullbacks, we can see that the map Y ! Y also
has the right lifting property with respect to the maps in Ic.
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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
Email address: bergnerj@member.ams.org