A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY
THEORY
CHARLES REZK
Abstract.We describe a category, the objects of which may be viewed as
models for homotopy theories. We show that for such models, "functors be-
tween two homotopy theories form a homotopy theory", or more precisely t*
*hat
the category of such models has a well-behaved internal hom-object.
1.Introduction
Quillen introduced the notion of a closed model category [12], which is a cat*
*e-
gory together with a distinguished subcategory of "weak equivalences", along wi*
*th
additional structure which allows one to do homotopy theory. Examples of closed
model categories include the category of topological spaces with the usual noti*
*on
of weak equivalence, and the category of bounded-below chain complexes, with
quasi-isomorphisms as the weak equivalences. A model category has an associ-
ated homotopy category. More strikingly, a model category has "higher homotopy"
structure. For instance, Quillen observed that one can define homotopy groups a*
*nd
Toda brackets in a closed model category. Dwyer and Kan later showed [4] that f*
*or
any two objects in a model category one can define a function complex.
Quillen's motivation for developing the machinery of closed model categories
was to give criteria which would imply that two models give rise to "equivalent"
homotopy theories, in an appropriate sense; his criterion is now referred to as*
* a
"Quillen equivalence" of closed model categories. For example, the categories *
*of
topological spaces and simplicial sets, which both admit closed model category
structures, should be viewed as alternate models for the same homotopy theory,
since any "homotopy-theoretic" result in one model translates into a similar re*
*sult
for the other. This is similar to the distinction one makes between the notion *
*of a
"space" and a "homotopy type". (In Quillen's case, the problem at hand was that
of finding algebraic models for rational homotopy theory [13].)
Thus it is convenient to distinguish between a "model" for a homotopy theory *
*and
the homotopy theory itself. A "model" could be a closed model category, though
one might want to consider other kinds of models. This notion of an abstract
homotopy theory, as opposed to a model for a homotopy theory, was clarified by
Dwyer and Kan [4]. Their work consists of several parts. First, in their theory*
*, the
minimal data needed to specify a homotopy theory is merely a category equipped
with a distinguished subcategory of "weak equivalences". Second, they show that
any such data naturally gives rise to a simplicial localization, which is a cat*
*egory
enriched over simplicial sets. If the initial data came from a model category, *
*then
____________
Date: November 3, 1998.
1991 Mathematics Subject Classification. Primary 55U35; Secondary 18G30.
Key words and phrases. homotopy theory, simplicial spaces, localization.
1
2 CHARLES REZK
one can recover its homotopy category and higher composition structure from the
simplicial localization.
Furthermore, Dwyer and Kan define a notion of equivalence of simplicial local*
*iza-
tions, which provides an answer to the question posed by Quillen on the equival*
*ence
of homotopy theories. In fact, the category of simplicial localizations togethe*
*r with
this notion of equivalence gives rise to a "homotopy theory of homotopy theory".
A brief discussion of this point of view may be found in [8, x11.6].
On the other hand, one can approach abstract homotopy theory from the study
of diagrams in a homotopy theory. For instance, a category of functors from a
fixed domain category which takes values in a closed model category is itself (*
*under
mild hypotheses) a closed model category. In particular, the domain category may
itself be a closed model category, (or a subcategory of a closed model category*
*).
Thus, just as functors from one category to another form a category, one expects
that functors from one homotopy theory to another should form a new homotopy
theory. Such functor categories are of significant practical interest; applica*
*tions
include models for spectra, simplicial sheaf theory, and the "Goodwillie calcul*
*us"
of functors.
In this paper we study a particular model for a homotopy theory, called a com-
plete Segal space, to be described in more detail below. The advantage of this *
*model
is that a complete Segal space is itself an object in a certain Quillen closed *
*model
category, and that the category of complete Segal spaces has internal hom-objec*
*ts.
Our main results are the following:
(0) A complete Segal space has invariants such as a "homotopy category" and
"function complexes", together with additional "higher homotopy" structure
(x5).
(1) There exists a simplicial closed model category in which the fibrant objec*
*ts
are precisely the complete Segal spaces (7.2). (I.e., there is a "homotopy
theory of homotopy theories".)
(2) This category is cartesian closed, and the cartesian closure is compatible*
* with
the model category structure. In particular, if X is any object and W is a
complete Segal space, then the internal hom-object W X is also a complete
Segal space (7.3). (I.e., the functors between two homotopy theories form
another homotopy theory.)
In fact, the category in question is just the category of simplicial spaces s*
*upplied
with an appropriate closed model category structure. The definition of a comple*
*te
Segal space is a modification of Graeme Segal's notion of a -space, which is a
particular kind of simplicial space which serves as a model for loop spaces. T*
*he
definition of "complete Segal space", given in Section 6, is a special case of *
*that of
a "Segal space", which is defined in Section 4.
1.1. Natural examples. Complete Segal spaces arise naturally in situations where
one can do homotopy theory. Any category gives rise to a complete Segal space
by means of a classifying diagram construction, to be described below. A Quillen
closed model category can give rise to a complete Segal space by means of a cla*
*ssi-
fication diagram construction, which is a generalization of the classifying dia*
*gram.
More generally, a pair (C; W ) consisting of a category C and a subcategory W
gives rise to a complete Segal space by means of a localization of the classifi*
*cation
diagram.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 3
Given a closed model category M and a small category C, it is often the case
that the category M C of functors from C to M is again a closed model category.
In this case, one can ask whether the classification diagram of M C is equivale*
*nt
to the complete Segal space obtained as the internal hom-object of maps from the
classifying diagram of C to the classification diagram of M . A consequence (8.*
*11)
of a result of Dwyer and Kan tells us that this equivalence holds at least when
M is the category of simplicial sets, or more generally a category of diagrams*
* of
simplicial sets; it probably holds true for a general closed model category, bu*
*t we
do not prove that here.
1.2. Classifying diagrams and classification diagrams. We give a brief de-
scription of the classifying diagram and classification diagram constructions h*
*ere,
in order to motivate the definition of a complete Segal space. These constructi*
*ons
are discussed in detail in Section 3.
To any category C one may associate its classifying space BC; this is a space
obtained by taking a vertex for each object of C, attaching a 1-simplex for each
morphism of C, attaching a 2-simplex for each commutative triangle in C, and
so forth. It is well-known that if the category C is in fact a groupoid, then i*
*t is
characterized up to equivalence of categories by its classifying space; for a g*
*roupoid
C the classifying space BC has the homotopy type of a disjoint union of spaces
K(ssX ; 1), where X ranges over the representatives of isomorphism classes of o*
*bjects
in C and each ssX is the group of automorphisms of the object X in C.
Unfortunately, a general category cannot be recovered from its classifying sp*
*ace.
Let isoC denote the subcategory of C consisting of all objects and all isomorph*
*isms
between them; thus isoC is just the maximal subgroupoid of the category C. From
the homotopy type of the classifying space B(isoC) of this groupoid one can rec*
*over
some information about the category C, namely the set of isomorphism classes of
objects in C and the group of automorphisms of any object. For this reason one
may view B(isoC) as a kind of "moduli space" for the category C.
Although a category C is not determined by its classification space, it turns*
* out
(3.7)that it is determined, up to equivalence, by a simplicial diagram of spaces
[n] 7! B iso(C[n]) which we call the classifying diagram of C; here [n] denotes
the category consisting of a sequence of (n + 1) objects and n composable arrow*
*s,
and C[n]denotes the category of functors from [n] ! C. The classifying diagram
of a category is in fact a complete Segal space.
The homotopy theoretic analogue of B(isoC) is Dwyer and Kan's notion of the
classification space of a model category. Given a closed model category M , let
weM M denote the subcategory consisting of all objects and all weak equiva-
lences between them. The classification space of M is denoted class(M ), and is
defined to be B(we M ), the classifying space of the category of weak equivalen*
*ces
of M . The classification space of a model category is in many ways analogous to
the space B(isoC) considered above. For example, class(M ) has the homotopy
type of a disjoint union of spaces B(hautX), where X ranges over appropriate re*
*p-
resentatives of weak equivalence classes of objects in M , and hautX denotes the
simplicial monoid of self-homotopy equivalences of X (8.6). Classification spa*
*ces
arise naturally in the study of realization problems, e.g., the problem of real*
*izing
a diagram in the homotopy category of spaces by an actual diagram of spaces; see
[6], [5].
4 CHARLES REZK
Given a closed model category M , form a simplicial space [n] 7! class(M [n]),
called the classification diagram of M . We show (8.3)that the classification
diagram of a closed model category is essentially a complete Segal space.
1.3. Organization of the paper. In Section 2 we set up notation for simplicial
spaces and discuss the Reedy model category structure for simplicial spaces. In
Section 3 we define the classification diagram construction, which produces a s*
*im-
plicial space from category theoretic data. In Section 4 we define the notion o*
*f a
Segal space, and in Section 5 we discuss in elementary terms how one can view a
Segal space as a model for a homotopy theory. In Section 6 we define the notion*
* of
a complete Segal space. In Section 7 we present our main theorems. In Section 8
we show how the classification diagram of a simplicial closed model category gi*
*ves
rise to a complete Segal space.
In Sections 9 through 14 we give proofs for the more technical results from e*
*arlier
sections.
1.4. Acknowledgements. I would like to thank Dan Kan for his encouragement
and hospitality, and Bill Dwyer for his beautiful talk at the 1993 Cech confere*
*nce,
where I first heard about the homotopy theory of homotopy theory. I would also
like to thank Phil Hirschhorn, Mark Johnson, and Brooke Shipley for their helpf*
*ul
comments on the manuscript.
2. Simplicial spaces
In this section we establish notation for spaces and simplicial spaces, and d*
*iscuss
the Reedy model category structure for simplicial spaces.
2.1. Spaces. By space we always mean "simplicial set" unless otherwise indicate*
*d;
the category of spaces is denoted by S. Particular examples of spaces which we
shall need are [n], the standard n-simplex, _[n], the boundary of the standard
n-simplex, and k[n], the boundary of the standard n-simplex with the k-th face
removed. If X and Y are spaces we write Map S(X; Y ) for the space of maps from
X to Y ; the n-simplices of Map S(X; Y ) correspond to maps X x [n] ! Y .
We will sometimes speak of a "point" in a space, by which is meant a 0-simple*
*x,
or of a "path" in a space, by which is meant a 1-simplex.
2.2. The simplicial indexing category. For n 0 let [n] denote the category
consisting of n+1 objects and a sequence of n composable arrows: {0 ! 1 ! : :!:
n}. Let denote the full subcategory of the category of categories consisting*
* of
the objects [n]. We write : [n] ! [n] for the identity map in this category.
As is customary, we let di:[n] ! [n + 1] for i = 0; : :;:n denote the injecti*
*ve
functor which omits the ith object, and we let si:[n] ! [n - 1] for i = 0; : :;*
*:n - 1
denote the surjective functor which maps the ith and (i + 1)st objects to the s*
*ame
object. Additionally, we introduce the following notation: let ffi:[m] ! [n] f*
*or
i = 0; : :;:n - m denote the functor defined on on objects by
ffi(k) = k + i:
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 5
2.3. Simplicial spaces. Let sS denote the category of simplicial spaces. An
object in this category is a functor X : op! S, sending [n] 7! Xn. We write
di:Xn ! Xn-1, si:Xn ! Xn+1 and ffi:Xn ! Xm for the maps corresponding
respectively to the morphisms di:[n+1] ! [n], si:[n-1] ! [n], and ffi:[m] ! [n]
in .
The category sS is enriched over spaces. We denote the mapping space by
Map sS(X; Y ) 2 S. It is convenient to identify S with the full subcategory of*
* sS
consisting of constant simplicial objects (i.e., those K 2 sS such that Kn = K0
for all n), whence for a space K and simplicial spaces X and Y ,
Map sS(X x K; Y ) Map S(K; MapsS(X; Y )):
In particular, the n-simplices of Map sS(X; Y ) correspond precisely to the set*
* of
maps X x [n] ! Y of simplicial spaces.
Let F (k) 2 sS denote the simplicial space defined by
[n] 7! ([n]; [k]);
where the set ([n]; [k]) is regarded as a discrete space. The F (k)'s represe*
*nt the
k-th space functor, i.e.,
Map sS(F (k); X) Xk:
We write di:F (n) ! F (n + 1), si:F (n) ! F (n - 1), and ffi:F (m) ! F (n) for
the maps of simplicial spaces corresponding to the maps di, si, and ffi in .
The category of simplicial spaces is cartesian closed; for X; Y 2 sS there is*
* an
internal hom-object Y X 2 sS characterized by the natural isomorphism
sS(X x Y; Z) sS(X; ZY ):
In particular, (Y X)0 Map sS(X; Y ), and
(Y X)k Map sS(X x F (k); Y ):
Furthermore, if K 2 S is regarded as a constant simplicial space, then (XK )n
Map sS(K; Xn).
Finally, we note the existence of a diagonal functor diag:sS ! S, defined so
that the n-simplices of diagX are the n-simplices of Xn.
2.4. Reedy model category. In this paper we will consider several distinct clos*
*ed
model category structures on sS. If the model category structure is not named in
a discussion, assume that the Reedy model category structure is intended.
The Reedy model category structure [14], [7, 2.4-6] on sS has as its weak
equivalences maps which are degree-wise weak equivalences. A fibration (resp.
trivial fibration) in sS is a map X ! Y such that each k 0 the induced map
Map sS(F (k); Y ) ! Map sS(F (k); X) xMapsS(F_(k);X)Map(F_(k); Y )
is a fibration (resp. trivial fibration) of simplicial sets, where F_(k) denote*
*s the
largest subobject of F (k) which does not contain : [k] ! [k] 2 ([k]; [k]). *
*It
follows that the cofibrations are exactly the inclusions.
With the above definitions, all objects are cofibrant, and the fibrant object*
*s are
precisely those X for which each map `k: Map sS(F (k); X) ! Map sS(F_(k); X) is
a fibration of spaces. We note here the fact that discrete simplicial spaces (*
*i.e.,
simplicial spaces X such that each Xn is a discrete space) are Reedy fibrant.
The Reedy model category structure is cofibrantly generated [3]; i.e., there *
*ex-
ist sets of generating cofibrations and generating trivial cofibrations which
6 CHARLES REZK
have small domains, and trivial fibrations (resp. fibrations) are characterized*
* as
having the right lifting property with respect to the generating cofibrations (*
*resp.
generating trivial cofibrations). The generating cofibrations are the maps
a
F_(k) x [`] F (k) x _[`] ! F (k) x [`]; k; ` 0;
_F(k)x _[`]
and the generating trivial cofibrations are the maps
a
_F(k) x [`] F (k) x t[`] ! F (k) x [`]; k 0; ` t 0:
_F(k)xt[`]
2.5. Compatibility with cartesian closure. Given a model category structure
on sS, we say that it is compatible with the cartesian closure if for any
cofibrations i: A ! B and j :C ! D and any fibration k :X ! Y , either (and
hence both) of the following two assertions hold:
(1) The induced map AxDqAxC BxC ! BxD is a cofibration, and additionally
is a weak equivalence if either i or j is.
(2) The induced map Y B ! Y AxXA XB is a fibration, and additionally is a
weak equivalence if either i or k is.
(A closed symmetric monoidal category together with a Quillen closed model cat-
egory structure which satisfies the above properties is sometimes also called a
"Quillen ring".) Assuming (as will always be the case for us) that a weak equiv-
alence or a fibration X ! Y in our model category structure induces a weak
equivalence or a fibration X0 ! Y0 on the degree 0 spaces, then it follows that*
* such
a model category structure makes sS into a simplicial model category in the sen*
*se
of [12], since Map sS(X; Y ) (Y X)0 for any simplicial spaces X and Y .
The Reedy model category structure on sS is compatible with the cartesian
closure; to prove (1) in this case, it suffices to recall that cofibrations are*
* exactly
inclusions, and that weak equivalences are degree-wise.
2.6. Proper model categories. A closed model category is said to be proper if
1. the pushout of a weak equivalence along a cofibration is a weak equivalenc*
*e,
and
2. the pullback of a weak equivalence along a fibration is a weak equivalence.
The Reedy model category structure is proper, because cofibrations and fibratio*
*ns
are in particular cofibrations and fibrations in each degree, and S is proper.
3.Nerve constructions and classification diagrams
In this section we discuss a construction called the classification diagram, *
*which
produces a simplicial space from a pair of categories. A special case of this c*
*onstruc-
tion of particular interest is the classifying diagram of a category, which pro*
*duces a
full embedding N :Cat! sS of the category of small categories into the category*
* of
simplicial spaces, which has the property that N takes equivalences of categori*
*es,
and only equivalences of categories, to weak equivalences of simplicial spaces.*
* An-
other special case of this construction is the application of the classificatio*
*n diagram
to model categories, which will be considered in Section 8.
In what follows we write DC for the category of functors from C to D.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 7
3.1. The nerve of a category. Given a category C, let nerveC denote the nerve
of C; that is, nerveC is a simplicial set whose n-simplices consist of the set *
*of
functors [n] ! C. (The classifying space BC of a category is a topological space
which is the geometric realization of the nerve.) The following is well-known.
Proposition 3.2.The nerve of [n] is [n]. For categories C and D there are
natural isomorphisms
nerve(C x D) nerveC x nerveD and nerve(DC ) nerve(D)nerve(C):
The functor nerve:Cat! Sis a full embedding of categories. Furthermore, if C is
a groupoid then nerve(C) is a Kan complex.
Although the nerve functor is a full embedding of categories, it is awkward f*
*rom
our point of view, since non-equivalent categories may give rise to weakly equi*
*valent
nerves.
3.3. The classification diagram of a pair of categories. Consider a pair
(C; W ) consisting of a category C together with a subcategory W such that obW =
obC; we refer to a morphism of C as a weak equivalence if it is contained in W .
More generally, given a natural transformation ff: f ! g of functors f; g :D ! *
*C,
we say that ff is a weak equivalence if ffd 2 W for each d 2 obD, and write we(*
*CD )
for the category consisting of all functors from D to C and all weak equivalenc*
*es
between them; thus we(C) = W .
For any such pair (C; W ) of categories we define a simplicial space N(C; W ),
called the classification diagram of (C; W ), by setting
N(C; W )m = nervewe(C[m]):
If we view the category [m] x [n] as an m-by-n grid of objects with rows of m
composable horizontal arrows and columns of n composable vertical arrows, then
the set of n-simplices of the mth space of N(C; W ) corresponds to the set of f*
*unctors
[m] x [n] ! C in which the vertical arrows are sent into W C.
We consider several special cases of this construction.
3.4. Discrete nerve construction. A special case of the classification diagram *
*is
the discrete nerve. If we let C0 C denote the subcategory of C consisting of a*
*ll
its objects and only identity maps between them, then let discnerveC = N(C; C0).
Note that nerveC = diag(discnerveC), and that discnerve([n]) = F (n).
It is not hard to see that the functor discnerve:Cat! sS embeds the category
of small categories as a full subcategory of simplicial spaces. The discrete n*
*erve
functor is awkward from our point of view, since equivalent categories can have
non-weakly equivalent discrete nerves.
3.5. The classifying diagram of a category. We give a construction which
embeds the category of categories inside the category of simplicial spaces and *
*which
carries equivalences of categories (and only equivalences) to weak equivalences*
* of
simplicial spaces.
Given a category C, define a simplicial space NC = N(C; isoC), where isoC C
denotes the maximal subgroupoid. Thus, the mth space of NC is (NC)m =
nerveiso(C[m]). We call NC the classifying diagram of C.
Let I[n] denote the category having n + 1 distinct objects, and such that the*
*re
exists a unique isomorphism between any two objects. We suppose further that
there is a chosen inclusion [n] ! I[n]. Then the set of n-simplices of the mth *
*space
8 CHARLES REZK
of NC corresponds to the set of functors [m] x I[n] ! C. Note that there is a
natural isomorphism
(3.6) (NC)m (NC)1 x(NC)0. .x.(NC)0(NC)1
(where the right-hand side is an m-fold fiber-product), and that the natural map
(d1; d0): (NC)1 ! (NC)0 x (NC)0 is a simplicial covering space, with fiber over
any vertex (x; y) 2 (NC)20isomorphic to the set hom C(x; y).
Note that if the category C is a groupoid, then the natural map nerveC ! NC
from nerveC viewed as a constant simplicial space is a weak equivalence; this
follows from the fact that for C a groupoid, iso(C[m]) and C[m] are equivalent
categories. It is therefore natural to regard the classifying diagram construct*
*ion as
a generalization of the notion of a classifying space of a groupoid.
The following theorem says that N :Cat! sS is a full embedding of categories
which preserves internal hom-objects, and furthermore takes a functor to a weak
equivalence if and only if it is an equivalence of categories.
Theorem 3.7. Let C and D be categories. There are natural isomorphisms
N(C x D) NC x ND and N(DC ) (ND)NC
of simplicial spaces. The functor N :Cat ! sS is a full embedding of categorie*
*s.
Furthermore, a functor f :C ! D is an equivalence of categories if and only if *
*Nf
is a weak equivalence of simplicial spaces.
Proof.That N preserves products is clear.
To show that N(DC ) ! (ND)NC is an isomorphism, we must show that for
each m; n 0 this map induces a one-to-one correspondence between functors
[m] x I[n] ! DC and maps F (m) x [n] ! (ND)NC . By (3.8)it will suffice to
show this for the case m = n = 0; that is, to show that functors C ! D are in o*
*ne-
to-one correspondence with maps NC ! ND, or in other words, that N :Cat! sS
is a full embedding of categories.
To show that N is a full embedding, we note that any map NC ! ND is
determined by how it acts on the 0th and 1st spaces of NC. Thus the result foll*
*ows
from a straightforward argument using (3.2)and the fact that (d1; d0) is a simp*
*licial
covering map such that d1s0 = 1 = d0s0 for both NC and ND.
It is immediate that naturally isomorphic functors induce simplicially homo-
topic maps of simplicial spaces since N(CI[1]) (NC)[1] by (3.8), and thus an
equivalence of categories induces a weak equivalence of simplicial spaces. To p*
*rove
the converse, note that (3.9)will show that (ND)NC N(DC ) is Reedy fibrant,
and in particular Map S(NC; ND) (NDNC )0 is a Kan complex. Therefore, if
Nf :NC ! ND is a weak equivalence of simplicial spaces it must be a simpli-
cial homotopy equivalence. Furthermore, the homotopy inverse is a 0-simplex of
N(CD )0 and the simplicial homotopies are 1-simplices of N(DC )0 and N(CD )0; by
what we have already shown these correspond precisely to a functor g :D ! C and_
natural isomorphisms fg ~ 1D and gf ~ 1C , as desired. |__|
Lemma 3.8. Let C be a category. Then there are natural isomorphisms
N([m] x C) F (m) x NC and N(CI[n]) (NC)[n]
of simplicial spaces.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 9
Proof.The first isomorphism follows from the fact that N preserves products and
that N([m]) F (m). The second isomorphism may be derived from the fact that
iso(DI[n]) (isoD)I[n] (isoD)[n]for any category D, and thus in particular_when
D = C[m]. |__|
Lemma 3.9. If C is a category, then NC is a Reedy fibrant simplicial space.
Proof.We must show that
`n :(NC)n Map sS(F (n); NC) ! Map sS(F_(n); NC)
is a fibration for each n 0. We have the following cases:
n = 0: (NC)0 = nerve(isoC) is a Kan complex by (3.2).
n = 1: `1: (NC)1 ! (NC)0x (NC)0 is a simplicial covering space with discrete
fiber, and thus is a fibration.
n = 2: `2 is isomorphic to an inclusion of path-components, and so is a fibra*
*tion.
n 3: `n is an isomorphism, and thus a fibration.
|___|
3.10. Classification diagrams of functor categories. The following generalizes
one of the statements of (3.7), and we note it for future reference.
Proposition 3.11.Let C and D be categories, and W D a subcategory such
that isoD W . Then there are natural isomorphisms
N(DC ; we(DC )) N(D; W )NC N(D; W )discnerveC:
Proof.We must show that for each m; n 0 the natural maps N(DC ; we(DC )) !
N(D; W )NC ! N(D; W )discnerveCinduce one-to-one correspondences amongst the
sets of
1. functors [m] x [n] ! DC which carry "vertical" maps into we(DC ),
2. maps F (m) x [n] ! N(D; W )NC of simplicial spaces, and
3. maps F (m) x [n] ! N(D; W )discnerveCof simplicial spaces.
By (3.8)and (3.12)it will suffice to show this in the case m = n = 0, in which_*
*case
the result becomes a straightforward computation. |__|
Lemma 3.12. Let C a category, and W a subcategory with obW = obC. Then
there is a natural isomorphism
N(Cg[n]; we(Cg[n])) (NC)[n];
where gC[n] C[n]denotes the full subcategory whose objects are those functors
[n] ! C which factor through W C, and we(Cg[n]) = we(C[n]) \ gC[n].
Proof.For any pair (D; W ) of category D and subcategory W , we have that
we(Dg[n]) = W [n], and that nerve(W [n]) = (nerveW )[n]. We obtain the result_
by substituting C[m]for D. |__|
4. Segal spaces
In this section we define the notion of a Segal space. This is a modification*
* of
the notion of a -space as defined by Graeme Segal; a -space is a simplicial spa*
*ce
X such that Xn is weakly equivalent to the n-fold product Xn. It was proposed as
a model for loop spaces, and is closely related to Segal's -space model for inf*
*inite
loop spaces, as in [15]. (To my knowledge, Segal never published anything about
10 CHARLES REZK
-spaces. The first reference in the literature appears to be by Anderson [1]. T*
*he
fact that -spaces model loop spaces was proved by Thomason [16].)
Our definition of a "Segal space" introduces two minor modifications to that *
*of
a -space: we allow the 0th space of the simplicial space to be other than a poi*
*nt,
and we add a fibrancy condition.
4.1. Definition of a Segal space. Let G(k) F (k) denote the smallest subobject
having G(k)0 = F (k)0 and such that G(k)1 contains the elements ffi 2 F (k)1 =
([1]; [k]) defined in (2.2). In other words,
k-1[
G(k) = ffiF (1) F (k);
i=0
where ffiF (1) denotes the image of the inclusion map ffi:F (1) ! F (k). Let
'k: G(k) ! F (k) denote the inclusion map. It is straight-forward to check that
Map sS(G(k); X) X1 xX0 . .x.X0X1;
where the right-hand side denotes the limit of a diagram
(4.2) X1 d0-!X0 -d1-X1 d0-!X0 -d1-. .d.1--X1
with k copies of X1.
We define a Segal space to be a simplicial space W which is Reedy fibrant,
and such that the map 'k = Map sS('k; W ): Map sS(F (k); W ) ! Map sS(G(k); W )
is a weak equivalence. In plain language, this means that W is a Reedy fibrant
simplicial space such that the maps
(4.3) 'k: Wk ! W1 xW0 . .x.W0W1
are weak equivalences for k 2. Because the maps 'k are inclusions and W is
Reedy fibrant, the maps 'k acting on a Segal space are trivial fibrations. Note*
* also
that the maps d0; d1: W1 ! W0 are fibrations as well, so that the fiber-product*
* of
(4.2)is in fact a homotopy fiber-product.
4.4. Examples. Recall that every discrete simplicial space is Reedy fibrant. A
discrete simplicial space W is a Segal space if and only if the maps in (4.3)are
isomorphisms. Thus, W is a discrete Segal space if and only if it is isomorphic*
* to
the discrete nerve of some small category. In particular, the objects F (k) are*
* Segal
spaces.
If C is a category, then its classifying diagram NC (as defined in (3.5)) is *
*a Segal
space, by (3.6)and (3.9).
5. Homotopy theory in a Segal space
In this section we describe how to obtain certain invariants of a Segal space,
including its set of objects, the mapping spaces between such objects, homotopy
equivalences between such objects, and the homotopy category of the Segal space.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 11
5.1. "Objects" and "mapping spaces". Fix a Segal space W . We define the
set of objects of a Segal space W to be the set of 0-simplices of W0, and we de*
*note
the set of objects by obW .
Given two objects x; y 2 obW we define the mapping space map W(x; y) be-
tween them to be the fiber of the map (d1; d0): W1 ! W0 x W0 over the point
(x; y) 2 W0 x W0. Note that since W is Reedy fibrant the map (d1; d0) is a fibr*
*a-
tion, and thus the homotopy type of map W(x; y) depends only on the equivalence
classes of x and y in ss0W0. We will sometimes write map (x; y) when W is clear
from the context.
Given a vertex x 2 W0 we have that d0s0x = d1s0x = x. Thus for each object
x 2 obW the point s0x 2 W1 defines a point in map W(x; x), called the identity
map of x, and denoted idx.
Given (n + 1) objects x0; : :;:xn in obW we write map W(x0; x1; : :;:xn) for *
*the
fiber of the map (ff0; : :;:ffn): Wn ! W0 n+1 over (x0; : :;:xn) 2 W0 n+1. The
commutative triangle
Map sS(F (n);OW_)_____~'k______//MapsS(G(n); W )
OOO oooo
OOOO oooo
OO''O wwooo
W0n+1
induces trivial fibrations
'k: map (x0; x1; : :;:xn) ! map(xn-1; xn) x . .x.map(x0; x1)
between the fibers of the slanted maps.
As an example, if C is a category and either W = discnerveC or W = NC, then
obW obC and map W(x; y) homC (x; y).
5.2. "Homotopies" and "compositions" of "maps". Let W be a Segal space,
and suppose x; y 2 obW . Given points f; g 2 map (x; y), we say that f and g are
homotopic if they lie in the same component of map (x; y). We write f ~ g if f
and g are homotopic.
A Segal space is not a category, so we cannot compose maps in the usual way.
Nonetheless, given f 2 map (x; y) and g 2 map (y; z), we define a composition to
be a lift of (g; f) 2 map(y; z) x map(x; y) along '2 to a point k 2 map(x; y; z*
*). The
result of the composition k is the point d1(k) 2 map (x; z). Since '2 is a triv*
*ial
fibration by definition the results of any two compositions of f and g are homo*
*topic.
Sometimes we write g O f 2 map(x; z) to represent the result of some composite *
*of
f and g.
Proposition 5.3.Given points f 2 map(w; x), g 2 map(x; y), and h 2 map(y; x),
we have that (h O g) O f ~ h O (g O f) and that f O idw~ f ~ idxOf.
Proof.We prove the proposition by producing particular choices of compositions
which give equal (not just homotopic) results.
12 CHARLES REZK
To construct h O (g O f) consider the diagram
map(w; x; y; z)____d1______//_map(w; y;_z)d1_//_map(w; z)
d0d0xd3~|| '2|~|
fflffl| 1xd1 fflffl|
map (y; z) x map(w; x;_y)___//_map(y; z) x map(w; y)
1x'2~||
fflffl|
map (y; z) x map(x; y) x map(w; x)
Note that the composite of the vertical maps in the left-hand column is '3. Any
choice of k 2 map (w; x; y; z) such that '3(k) = (h; g; f) determines compositi*
*ons
d3k 2 map(w; x; y) and d1k 2 map(w; y; z) with results g O f and h O (g O f) re*
*spec-
tively. By considering an analogous diagram we see that such a k also determines
compositions d0k 2 map (x; y; z) and d2k 2 map (w; x; z) with results h O g and
(h O g) O f respectively, and that for this choice of compositions there is an *
*equality
h O (g O f) = (h O g) O f of results, as desired.
To show that f O idw ~ f for f 2 map (w; x), let k = s0(f) 2 map (w; w; x).
Then '2(k) = (f; idw) and d1(k) = f, showing that f O idw= f. The proof that_
idzOf ~ f is similar. |__|
5.4. Homotopy category and homotopy equivalences. In view of (5.3)we de-
fine the homotopy category of a Segal space W , denoted by Ho W , to be the cat-
egory having as objects obW , and having as maps homHo W(x; y) = ss0map W (x; y*
*).
For any f 2 mapW (x; y) we can write [f] 2 homHo W(x; y) for its associated equ*
*iv-
alence class.
Remark 5.5.Recall (3.5)in which we defined an embedding N :Cat! sS via the
classifying diagram construction. This N admits a left adjoint L: sS ! Cat, and*
* it
is possible to show that L(W ) = Ho W whenever W is a Segal space.
A homotopy equivalence g 2 map (x; y) is a point for which [g] admits an
inverse on each side in Ho W . That is, there exist points f; h 2 map(y; x) suc*
*h that
g O f ~ idyand h O g ~ idx. Note that this implies by (5.3)that h ~ h O g O f ~*
* f.
Furthermore, for each x 2 obW the map idx2 map(x; x) is a homotopy equivalence
by (5.3).
We give another characterization of homotopy equivalences in a Segal space. L*
*et
Z(3) = discnerve(0 ! 2 1 ! 3) F (3) be the discrete nerve of a "zig-zag" cat*
*e-
gory; it follows that there is a fibration W3 = Map sS(F (3); W ) ! Map sS(Z(3)*
*; W ),
and an isomorphism
MapsS(Z(3); W ) lim(W1 d1-!W0 -d1-W1 d0-!W0 -d0-W1) W1 x W1 x W1:
W0 W0
(We can thus write simplices of Map sS(Z(3); W ) as certain ordered triples of *
*sim-
plices of W1.) Then a point g 2 map (x; y) W1 is a homotopy equivalence if
and only if the element (idx; g; idy) 2 Map sS(Z(3); W ) admits a lift to an el*
*ement
H 2 W3; note that if g 2 map(x; y) then s0d1g = idxand s0d0g = idy.
5.6. The space of homotopy equivalences. Clearly, any point in map (x; y)
which is homotopic to a homotopy equivalence is itself a homotopy equivalence.
More generally, we have the following.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 13
Lemma 5.7. If g 2 W1 is a vertex which can be connected by a path to a homotopy
equivalence g02 W1, then g is itself a homotopy equivalence.
Proof.Let G: [1] ! W1 denote the path connecting g and g0. Then it suffices to
note that a dotted arrow exists in
[0] __________H___________//W344___________
_______
| _____________|
| ______________ |
fflffl|_______________ fflffl|
[1] ___(s0d1G;G;s0d0G)//_MapsS(Z(3); W )
where H is a lift of (s0d1g0; g0; s0d0g0) = (idx0; g0; idy0) to W3, since the_r*
*ight-hand
vertical map is a fibration. |__|
Thus, we define the space of homotopy equivalences of W to be the sub-
space Whoequiv W1 consisting of exactly those components whose points are ho-
motopy equivalences. Note that the map s0: W0 ! W1 necessarily factors through
Whoequiv, since s0x = idxis a homotopy equivalence for any vertex x 2 W0.
6.Complete Segal spaces
A complete Segal space is defined to be a Segal space W for which the map
s0: W0 ! Whoequivis a weak equivalence, where Whoequivis the space of homotopy
equivalences defined in (5.6).
Proposition 6.1.If C is a small category, then the classifying diagram NC of
(3.5)is a complete Segal space.
Proof.This follows from (3.6)and (3.9), together with the fact that (NC)hoequiv
is isomorphic to nerveiso(CI[1]) and the fact that the natural inclusion isoC !_
iso(CI[1]) is an equivalence of categories. |_*
*_|
Let E denote the Segal space which is the discrete nerve of the category I[1]
which consists of exactly two objects x and y, and two non-identity maps x ! y
and y ! x which are inverses of each other.
There is an inclusion i: F (1) ! E associated to the arrow x ! y, inducing a
map Map sS(E; W ) ! Map sS(F (1); W ) W1. The following is crucial.
Theorem 6.2. If W is a Segal space, then the map Map sS(E; W ) ! W1 factors
through Whoequiv W1, and induces a weak equivalence Map sS(E; W ) ! Whoequiv.
The proof of (6.2)is technical, and we defer it to Section 11.
Suppose that W is a Segal space, and let x; y 2 W0 be objects, and consider t*
*he
diagram
W0 ____________________________
| __________________________________*
*_______________________
| _________________________________*
*__
fflffl|_______________________________*
*_________
(6.3) hoequiv(x; y)___//Whoequi_________________________________*
*____________________v
________________________________
| | __________________________________
| (d1;d0)| _________________________
fflffl| fflf______________________fl|
{(x; y)}______//W0 x W0
14 CHARLES REZK
Here hoequiv(x; y) map (x; y) denotes the subspace of map (x; y) consisting of
homotopy equivalences. This square is a pullback square, and also is a homotopy
pullback since (d1; d0) is a fibration. We have the following result.
Proposition 6.4.Let W be a Segal space. The following are equivalent.
(1) W is a complete Segal space.
(2) The map W0 ! Map sS(E; W ) induced by E ! F (0) is a weak equivalence.
(3) Either of the maps Map sS(E; W ) ! W0 induced by a map F (0) ! E is a
weak equivalence.
(4) For each pair x; y 2 obW , the space hoequiv(x; y) is naturally weakly equ*
*iva-
lent to the space of paths in W0 from x to y.
Proof.
(1) ) (2):This follows from (6.2).
(2) , (3):Straightforward.
(2) ) (4):Part (2) and (6.2)imply that W0 ! Whoequivis a weak equivalence,
whence the result follows from the fact that the space of paths in W0 with
endpoints x and y is equivalent to the homotopy fiber of the map in (6.3).
(4) ) (1):Immediate from the diagram (6.3).
|___|
Corollary 6.5.Let obW=~ denote the set of homotopy equivalence classes of ob-
jects in Ho W . If W is a complete Segal space, then ss0W0 obW=~ .
Proof.This is immediate from (6.4, (4)). |___|
Corollary 6.6.Let W be a complete Segal space. Then Ho W is a groupoid if and
only if W is Reedy weakly equivalent to a constant simplicial space.
Proof.The category Ho W is a groupoid if and only if Whoequiv= W1, if and only
if s0: W0 ! W1 is a weak equivalence (since W is complete). A simplicial space W
is weakly equivalent to a constant simplicial space if and only if s0: W0_! W1 *
*is a
weak equivalence. |__|
7.Closed model category structures
Our main results deal with the existence of certain closed model category str*
*uc-
tures on sS related to Segal spaces and complete Segal spaces.
Theorem 7.1. There exists a simplicial closed model category structure on the *
*cat-
egory sS of simplicial spaces, called the Segal space model category structure,
with the following properties.
1. The cofibrations are precisely the monomorphisms.
2. The fibrant objects are precisely the Segal spaces.
3. The weak equivalences are precisely the maps f such that Map sS(f; W ) is a
weak equivalence of spaces for every Segal space W .
4. A Reedy weak equivalence between any two objects is a weak equivalence in
the Segal space model category structure, and if both objects are themselv*
*es
Segal spaces then the converse holds.
Moreover, this model category structure is compatible with the cartesian closed*
* struc-
ture on sSet in the sense of Section 2.
We will prove (7.1)in Section 10.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 15
Theorem 7.2. There exists a simplicial closed model category structure on the
category sS of simplicial spaces, called the complete Segal space model category
structure, with the following properties.
1. The cofibrations are precisely the monomorphisms.
2. The fibrant objects are precisely the complete Segal spaces.
3. The weak equivalences are precisely the maps f such that Map sS(f; W ) is a
weak equivalence of spaces for every complete Segal space W .
4. A Reedy weak equivalence between any two objects is a weak equivalence in
the complete Segal space model category structure, and if both objects are
themselves complete Segal spaces then the converse holds.
Moreover, this model category structure is compatible with the cartesian closed*
* struc-
ture on sSet in the sense of Section 2.
We will prove (7.2)in Section 12.
These theorems have the following important corollary.
Corollary 7.3.If W is a complete Segal space (resp. a Segal space) and X is any
simplicial space, then W X is a complete Segal space (resp. a Segal space).
Proof.This is a direct consequence of the compatibility of these model category
structures with the cartesian closure, since any object is cofibrant in either_*
*of these
model category structures. |__|
7.4. Dwyer-Kan equivalences. We would like to understand the relationship
between the model category structures for Segal spaces and for complete Segal
spaces.
We say a map f :U ! V of Segal spaces is a Dwyer-Kan equivalence if
1. the induced map Ho f :Ho U ! Ho V on homotopy categories is an equiva-
lence of categories, and
2. for each pair of objects x; x0 2 U the induced function on mapping spaces
mapU (x; x0) ! mapV (fx; fx0) is a weak equivalence.
For a Segal space W let obW=~ denote the set of objects of W modulo the equiv-
alence relation of homotopy equivalence (or equivalently, the set of isomorphism
classes in Ho W ). If we define a condition 1' by
1'.the induced map obU=~ ! obV=~ on equivalence classes of objects is a
bijection,
then it is not hard to see that conditions 1' and 2 together are equivalent to *
*condi-
tions 1 and 2.
Lemma 7.5. If U -f!V g-!W are maps of Segal spaces such that f and g are
Dwyer-Kan equivalences, then gf is a Dwyer-Kan equivalence.
If U -f!V -g!W -h!X are maps of Segal spaces such that gf and hg are Dwyer-
Kan equivalences, then each of the maps f, g, and h is a Dwyer-Kan equivalence.
Proof.Straightforward. |___|
Proposition 7.6.A map f :U ! V of complete Segal spaces is a Dwyer-Kan
equivalence if and only if it is a Reedy weak equivalence.
Proof.It is clear that a Reedy weak equivalence between any two Segal spaces is*
* a
Dwyer-Kan equivalence.
16 CHARLES REZK
Conversely, suppose f :U ! V is a Dwyer-Kan equivalence between complete
Segal spaces. Then ss0U0 obU=~ and ss0V0 obV=~ by (6.5), so that ss0U0 !
ss0V0 is a bijection. In the commutative diagram
(d1;d0)
U0 __s0_//U1____//U0 x U0
| | |
| | |
fflffl|sfflffl|0(d|fflffl1;d0)
V0 _____//V1____//V0 x V0
the right-hand square is a homotopy pullback (since the induced maps of fibers *
*are of
the form mapU (x; y) ! mapV (fx; fy), which is assumed to be a weak equivalence*
*),
and the large rectangle is a homotopy pullback (since by (6.4)the induced maps
of fibers are of the form hoequivU(x; y) ! hoequivV(fx; fy), which is also a we*
*ak
equivalence). We conclude that U0 ! V0 is a weak equivalence, and therefore that
U1 ! V1 is a weak equivalence. Since both U and V are Segal spaces, it follows_
that the map f :U ! V is a Reedy weak equivalence as desired. |__|
Theorem 7.7. Let f :U ! V be a map between Segal spaces. Then f is a Dwyer-
Kan equivalence if and only if it becomes a weak equivalence in the complete Se*
*gal
space model category structure.
We will prove (7.7)in Section 14.
Remark 7.8.Note that if C is a category, then the natural inclusion discnerve(C*
*) !
N(C) of simplicial spaces is a Dwyer-Kan equivalence; thus by (7.7)this map is a
weak equivalence in the complete Segal space model category structure.
Corollary 7.9.The homotopy category of complete Segal spaces may be obtained
by formally inverting the Dwyer-Kan equivalences in the homotopy category of Se*
*gal
spaces.
8. Complete Segal spaces from model categories
In this section we show that complete Segal spaces arise naturally from closed
model categories.
Recall from Section 3 that given a category C and a subcategory W we can
construct a simplicial space N(C; W ). If the category C = M is a closed model
category with weak equivalences W , we will usually write N(M ) for N(M ; W ),
assuming that W is clear from the context. Let Nf(M ) denote a functorial Reedy
fibrant replacement of N(M ).
Given such a pair (C; W ) we can always construct a complete Segal space by
taking the fibrant replacement of N(C; W ) in the complete Segal space model ca*
*t-
egory structure. Because this fibrant replacement is a localization functor, it*
* seems
to be difficult to compute anything about it. Our purpose in this section is to
show that if we start with an appropriate closed model category M , then we obt*
*ain
a complete Segal space by taking a Reedy fibrant replacement of N(M ), which is
easy to understand since Reedy fibrant replacement does not change the homotopy
type of the spaces which make up N(M ).
8.1. Universes. Because the usual examples of closed model categories are not
small categories, their classification diagrams are not simplicial simplicial s*
*ets. We
may elude this difficulty by positing, after Grothendieck, the existence of a u*
*niverse
U (a model for set theory) in which M is defined. Then N(M ; W ) is an honest
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 17
simplicial space (though not modeled in the universe U, but rather in some high*
*er
universe U0).
Alternately, we note that there is no difficulty if the model category M is a*
* small
category, and that such exist in practise. As an example, choose an uncountable
cardinal fl, and let Sfldenote a skeleton of the category of all simplicial set*
*s which
have fewer than fl simplices. Then Sflis a small category, and is in fact a sim*
*plicial
closed model category. (Of course, the category Sflis not suitable for all purp*
*oses;
for example, it is not cartesian closed.)
8.2. The classification space of a closed model category. If M is a simplicial
model category, and X and Y objects in M , we write mapM (X; Y ) for the functi*
*on
complex from X to Y .
Theorem 8.3. Let M be simplicial closed model category, and let W M denote
the subcategory of weak equivalences. Then V = Nf(M ; W ) is a complete Segal
space. Furthermore, there is an equivalence of categories Ho V Ho M and there
are weak equivalences of spaces map V(X; Y ) mapM (X; Y ).
We prove (8.3)below.
Remark 8.4.Note that any category C can be made into a closed model category
in which the weak equivalences are precisely the isomorphisms (and all maps are
fibrations and cofibrations). In this case N(C) = N(C; isoC) coincides with the
classifying diagram construction described in (3.5), and we have noted (6.1)that
this is already a complete Segal space.
8.5. Results about classification spaces. Recall that the classification space
classM of a model category is defined to be nervewe(M ). Given a closed model
category M and an object X 2 M , write scX for the component of class(M )
containing X.
Proposition 8.6 (Dwyer-Kan [5, 2.3, 2.4]).Given a simplicial closed model cate-
gory M , and an object X 2 M which is both fibrant and cofibrant, let hautX
mapM (X; X) be its simplicial monoid of weak equivalences. Then the classifying
complex W hautX is weakly equivalent to scX; in fact, W hautX and scX can
be connected by a finite string of weak equivalences which is natural with resp*
*ect to
simplicial functors f :M ! N between closed model categories which preserve weak
equivalences and are such that fX 2 N is both fibrant and cofibrant.
Remark 8.7.We can interpret (8.6)as saying that for any two fibrant-and-cofibra*
*nt
objects X; Y 2 M , the space of paths from X to Y in class(M ) is naturally wea*
*kly
equivalent to the space hoequivM(X; Y ) mapM (X; Y ) of homotopy equivalences
from X to Y . Compare with (6.4, 4).
Let M be a simplicial closed model category. Then M [n]also admits a simplici*
*al
closed model category structure, in which a map f :X ! Y in M [n]is
1. a weak equivalence if fi: Xi ! Y i is a weak equivalence in M for each
0 i n,
2. a fibration if fi: Xi ! Y i is a fibration in M for each 0 i n, and
3. a cofibration if the induced maps Xi qX(i-1)Y (i - 1) ! Y i are cofibratio*
*ns
in M for each 0 i n, and we let X(-1) = Y (-1) denote the initial object
in M .
18 CHARLES REZK
Furthermore, a map ffi :[m] ! [n] induces a functor ffi*: M [n]! M [m]which is
simplicial and which preserves fibrations, cofibrations, and weak equivalences.
If Y is a fibrant-and-cofibrant object in M [n], with restriction Y 02 M [n-*
*1]
formed from the first n objects and (n - 1) maps in [n], then the homotopy fiber
of the map
W hautM [n]Y ! W hautM [n-1]Y 0x W hautM Y (n)
is weakly equivalent the union of those components of map M(Y (n - 1); Y (n)) c*
*on-
taining conjugates of the given map Yn-1: Y (n - 1) ! Y (n); by conjugate we
mean maps of the form j O Yn-1 O i where i and j are self-homotopy equivalences
of Y (n - 1) and Y (n) respectively. Here W denotes the classifying complex as*
* in
[11, p. 87]. Applying this fibration iteratively shows that the homotopy fiber *
*of the
map
W hautM[n]Y ! W hautM Y (0) x . .x.W hautM Y (n)
is naturally weakly equivalent to the union of those components of
map M (Y (0); Y (1)) x . .x.mapM (Y (n - 1); Y (n))
containing "conjugates" of the given sequence of maps Yi:Y (i) ! Y (i + 1).
Proof of (8.3).Let U = N(M ), so that Un = nervewe(M [n]) and Un ! Vn is a
weak equivalence of spaces. For each n 0 there is a map ssn :Un ! Un+10which
"remembers" only objects. The remarks above together with (8.6)show that for
each (n + 1)-tuple of objects (X0; : :;:Xn) in M the homotopy fiber of ssn over
the point corresponding to (X0; : :;:Xn) is in a natural way weakly equivalent *
*to a
product
map M(X0n-1; X0n) x . .x.mapM (X00; X01);
where X0iis a fibrant-and-cofibrant object of M which is weakly equivalent to *
*Xi.
Note that it is an immediate consequence of the above that V is a Segal space.
Since ss0U0 is just the set of weak homotopy types in M , and since Ho M (X; Y )
ss0map M (X0; Y 0) where X0and Y 0are fibrant-and-cofibrant replacements of X a*
*nd
Y respectively, we see that Ho M Ho V .
Let Uhoequiv U1 denote the subspace of U1 which corresponds to the subspace
Vhoequiv V1. By the equivalence of homotopy categories above, we see that
Uhoequivconsists of precisely the components of U1 whose points go to isomorphi*
*sms
in Ho M . Since M is a closed model category, this means that the 0-simplices *
*of
Uhoequivare precisely the objects of M [1]which are weak equivalences, so Uhoeq*
*uiv=
nervewe((we M )[1]). There is an adjoint functor pair F :M [1]o M :G in which
the right adjoint takes G(X) = idX, and the left adjoint takes F (X ! Y ) = X; *
*this
pair restricts to an adjoint pair we((we M )[1]) o we M and thus induces a weak_
equivalence Uhoequiv U0 of the nerves. Thus V is a complete Segal space. |_*
*_|
8.8. Categories of diagrams. Let M be a closed model category, and let I denote
a small indexing category; recall that the weak equivalences in the category M *
*Iof
functors are the object-wise weak equivalences. Consider
(8.9) f :N(M I) N(M )discnerveI! Nf(M )discnerveI;
where the isomorphism on the left-hand side is that described in (3.11), and the
map on the right-hand side is that induced by the Reedy fibrant replacement of
N(M ). If f can be shown to be a weak equivalence, then this means we can
compute the homotopy type of the classification diagram associated to I-diagrams
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 19
in M knowing only the homotopy type of the classification diagram of M itself.
In particular, knowing N(M ) determines the homotopy category Ho(M I) of the
category of I-diagrams in M for every small category I.
A result of Dwyer and Kan shows that this holds at least for certain cases of*
* M .
Theorem 8.10. The map f of (8.9)is a Reedy weak equivalence when M = SJ,
where S denotes the category of simplicial sets and J is a small indexing categ*
*ory.
Taken together with (3.11)we obtain the following corollary.
Corollary 8.11.There is a natural weak equivalence N(M I) -~!Nf(M )N(I) of
complete Segal spaces if M = SJ and I and J are small categories.
We prove (8.10)below.
Remark 8.12.It seems that the theorem of Dwyer and Kan, and hence the state-
ments of (8.10)and (8.11)should hold for any "reasonable" model category M ,
where the class of "reasonable" closed model categories includes at least the "*
*cofi-
brantly generated" simplicial closed model categories. We hope that future work
will provide a generalization of these theorems to arbitrary closed model categ*
*ories.
Let opI denote the category of simplices of I. This is a category in which *
*the
objects are functors f :[m] ! I, and the morphisms (f :[m] ! I) ! (g :[n] ! I)
consist of functors ffi :[n] ! [m] making f O ffi = g. The actual theorem of Dw*
*yer
and Kan [5], [6] is the following:
Theorem 8.13 (Dwyer-Kan). Let I be a small category. The natural map
class(SI) lim([k]!I)2 opIclass(S[k]) ! holim([k]!I)2 opIclass(S[k])f
is a weak equivalence, where Xf denotes the fibrant replacement of a space X, a*
*nd
holimis the homotopy inverse limit construction of [2].
Proof.That this map is a weak equivalence from each component of class(SI) to
the corresponding component of the homotpy limit follows from [5, 3.4(iii)]. Th*
*at
the map is surjective on path components is a consequence of Proposition_3.4_and
Theorem 3.7 of [6]. |__|
To derive (8.10)from (8.13)we use the following lemma.
Lemma 8.14. Let I be a small category and let W be a Reedy fibrant simplicial
space. Then the natural map
Map sS(discnerveI; W ) lim([k]!I)2 opIWk ! holim([k]!I)2 opIWk
is a weak equivalence.
Proof.Let A be an object in s(sS) (i.e., a simplicial object in sS) defined by
a
A(m) = F (k0) 2 sS:
[k0]!:::![km ]2I
There is an augmentation map A(0) ! discnerveI, and the induced map diag0A !
discnerveI is a Reedy weak equivalence in sS, where diag0:s(sS) ! sS denotes
the prolongation of the diagonal functor, in this case defined by (diag0A)n
diag([m] ! A(m)n). The result follows from isomorphisms
Map sS(diag0A; W ) Tot(Map sS(A(- ); W )) holim[k]!I2 opIWk;
20 CHARLES REZK
and the fact that Map sS(discnerveI; W ) ! Map sS(diag0A; W ) is a weak equiva-_
lence since W is Reedy fibrant. |__|
Proof of (8.10).Using (8.14)we can reinterpret (8.13)as stating that there is a
weak equivalence
class(SI) ~-!Map sS(discnerveI; Nf(S)):
Substituting [m] x I for I in the above for all m 0 leads to a Reedy weak
equivalence
N(SI) ~-!Nf(S)discnerveI;
which is the special case of (8.9)with M = S. To obtain the case of M = SJ, n*
*ote
that by what we have just shown the maps in
N(SIxJ) ~-!Nf(S)discnerve(IxJ) Nf(S)discnerveIxdiscnerveJ~-Nf(SJ)discnerveI
must be Reedy weak equivalences. |___|
9.Localization model category
In this section we state the properties of localization model category struct*
*ures
which we will need in order to prove (7.1)and (7.2).
Given an inclusion f :A ! B 2 sS, we can construct a localization model
category structure on sS. More precisely,
Proposition 9.1.Given a inclusion f :A ! B 2 sS, there exists a cofibrantly
generated, simplicial model category structure on sS with the following propert*
*ies:
(1) the cofibrations are exactly the inclusions,
(2) the fibrant objects (called f-local objects) are exactly the Reedy fibrant*
* W 2
sS such that
MapsS(B; W ) ! Map sS(A; W )
is a weak equivalence of spaces,
(3) the weak equivalences (called f-local weak equivalences) are exactly the
maps g :X ! Y such that for every f-local object W , the induced map
MapsS(Y; W ) ! Map sS(X; W )
is a weak equivalence, and
(4) a Reedy weak equivalence between two objects is an f-local weak equivalenc*
*e,
and if both objects are f-local then the converse holds.
Proof.The proposition is just a statement of the theory of localization applied
to the category of simplicial spaces. Although localization is now considered a
standard technique, it seems that no treatment at the level of generality which*
* we
require has yet appeared in print. Two such will appear soon: the book of Goerss
and Jardine [9, Ch. 9] and the book of Hirschhorn [10]. In the terminology of
Goerss and Jardine, we construct f-local simplicial model category structure on
the pointwise cofibration model category structure of op-diagrams in S.
We give a brief sketch of the proof here. Since the desired classes of cofibr*
*ations
and f-local weak equivalences have been characterized, the class of f-local fib*
*ra-
tions must be determined by these choices. To construct the localization model
category structure, we must find a cofibration j :A ! B which is also an f-local
weak equivalence with the property that a map is an f-local fibration if and on*
*ly
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 21
if it has the right lifting property with respect to j. The proof of the model *
*cate-
gory structure follows using the "small object argument" to prove the factoriza*
*tion
axiom. (That such a small object argument works here makes use of the fact that
simplicial spaces is a left proper model category.)
` It is still necessary to choose a j. Given an uncountable cardinal fl, take *
*j =
ffiff, where iff:Aff! Bffranges over isomophism classes of maps which are
cofibrations, f-local weak equivalences, and such that Bffhas fewer than fl sim*
*plices
in each degree. That a sufficiently large fl produces a map j with the desired*
*__
properties follows from the "Bousfield-Smith cardinality argument". |*
*__|
Such a localization model category structure need not be compatible (in the
sense of (2.5)) with the cartesian closure of sS. However, there is a simple cr*
*iterion
for this to happen.
Proposition 9.2.Suppose that for each f-local object W , the simplicial space
W F(1)is also f-local. Then the f-local model category structure on sS is com-
patible with the enrichment over itself.
Proof.The proof proceeds in several stages.kSuppose that W is an f-local object.
Then it follows by hypothesis that W (F(1))is f-local for all k. Next one obser*
*ves
by elementary computation that F (k) is a retract of (F (1))k; thus it follows *
*that
W F(k)is a retract of W (F(1))kand hence is also f-local.
Since the f-local model category is a simplicial model category, we see that *
*for
any K 2 Swe have that (W F(n))K = W F(n)xKis f-local (recall that we regard K
as a constant simplicial space). Since any simplicial space X is a homotopy col*
*imit
(in the Reedy model category structure) of a diagram of simplicial spaces of the
form F (k) x K where K is a space, it follows that W X is a homotopy limit (aga*
*in
in the Reedy model category structure, assuming W is Reedy fibrant) of a diagram
of simplicial spaces of the form W F(k)xK. Since a homotopy limit of f-local ob*
*jects
is f-local, we see that W X is f-local for arbitrary X.
Now, to show that the f-local model category is compatible with the enrichmen*
*t,
it suffices to show that for a cofibration i: X ! Y and an f-local trivial cofi*
*bration
j :U ! V , the induced map
a
U x Y V x X ! V x Y
UxX
is an f-local equivalence. Equivalently, we must show that for every f-local ob*
*ject
W the square
Map sS(V x Y; W )____//MapsS(V x X; W )
| |
| |
fflffl| fflffl|
Map sS(U x Y; W )____//MapsS(U x X; W )
is a homotopy pull-back of spaces. But this diagram is isomorphic to
Map sS(V; W Y)____//MapsS(V; W X)
| |
| |
fflffl| fflffl|
Map sS(U; W Y)____//MapsS(U; W X)
22 CHARLES REZK
and since W X and W Yare f-local, the columns are weak equivalences, whence_the_
square is in fact a homotopy pull-back. |__|
10. Segal space model category structure
In this section we prove (7.1).
The Segal space closed model category structure on sS is defined using `
(9.1)to be the localization of simplicial spaces with respect to the map ' = *
*i0 'i,
where 'n :G(n) ! F (n) is the map defined in (4.1). Parts (1)-(4) of (7.1)follow
immediately from (9.1). The only thing left to prove is the compatibility of t*
*his
model category structure with the cartesian closure.
To prove this, we need the notion of a cover of F (n). Let ffi:[k] ! [n] for *
*i =
0; : :;:n-k denote the maps defined by ffi(j) = i+j; we also write ffi:F (k) ! *
*F (n)
for the corresponding map of simplicial spaces. We say that a subobject G F (n)
is a cover of F (n) if
1. G and F (n) have the same 0-space, i.e., G0 = F (n)0, and
2. G has the form [
G = ffi F (k )
where k 1 and i = 0; : :;:k - 1.
In particular, F (n) covers itself, and G(n) F (n) is the smallest cover of F *
*(n).
Lemma 10.1. Let G F (n) be a cover. Then the inclusion maps G(n) -i!G -j!
F (n) are weak equivalences in the Segal space model category structure.
Proof.By weak equivalence, we shall mean weak equivalence in the Segal space
model category structure. The composite map ji is a weak equivalence by con-
struction, so it suffices to show that i is also a weak equivalence. Given any
ffi1F (k1); ffi2F (k2) F (n), we see that the intersection ffi1F (k1) \ ffi2F *
*(k2) is
either empty, or is equal to ffi3F (k3) for some i3 and k3. Thus G can be writt*
*en
as a colimit over a partially ordered set of subcomplexes of the form ffiF (k).*
* Since
G(n) \ ffiF (k) = ffiG(k), we see that G(n) is obtained as a colimit over the s*
*ame
indexing category of subobjects of the form ffiG(k). Since by hypothesis the map
Map sS(ffiF (k); W ) ! Map sS(ffiG(k); W ) is a weak equivalence for any Segal *
*space
W , we conclude that Map sS(G; W ) ! Map sS(G(n); W ) is also a weak equivalence
for any Segal space W , and hence i is a weak equivalence in the Segal space_mo*
*del
category, as desired. |__|
Remark 10.2.The class of subobjects which are weakly equivalent to F (n) is not
exhausted by the coverings. For example, one can show that for 0 < i < n the
subobject _F(n) \ diF (n - 1) (the "boundary" of F (n) with a "face" removed wh*
*ich
is neither the first nor the last face) is weakly equivalent to F (n) in the Se*
*gal space
model category structure, but is not a cover.
To finish the proof of (7.1), we note that by (9.2)it suffices to show that f*
*or
a Segal space W , the simplicial space W F(1)is also a Segal space; i.e., that *
*the
induced maps 'k: (W F(1))k Map sS(F (k); W F(1)) ! Map sS(G(k); W F(1)) are
weak equivalences. This follows immediately from (10.3)below.
Lemma 10.3. The inclusion F (1) x G(n) ! F (1) x F (n) is a weak equivalence in
the Segal space model category structure.
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 23
Proof.Let fli:[n + 1] ! [1] x [n] denote the map defined by
(
fli(j) = (0; j) if j i,
(1; j - 1)if j > i.
Likewise, let ffii:[n] ! [1] x [n] denote the map defined by
(
ffii(j) = (0; j)if j i,
(1; j)if j > i.
Then one can write F (1) x F (n) as a colimit of the diagram
(10.4) fl0F (n + 1) ffi0F (n) ! fl1F (n + 1) ffi1F (n) ! : :!:flnF (n + 1)
of subobjects. (This is analogous to the decomposition of the simplicial set [1*
*] x
[n] into a union of (n + 1) copies of [n + 1], attached along faces.) A straigh*
*tfor-
ward computation shows that the maps fliF (n + 1) \ (F (1) x G(n)) ! fliF (n + *
*1)
and ffiiF (n) \ (F (1) x G(n)) ! ffiiF (n) are covers, and hence by (10.1)are w*
*eak
equivalences. Thus the result follows by comparing diagram (10.4)with the diagr*
*am_
obtained by intersecting each object of (10.4)with F (1) x G(n). |*
*__|
11. Equivalences in Segal spaces
In this section we give a proof of (6.2). We use the Reedy model category
structure in what follows.
We make use of an explicit filtration of E = discnerve(I[1]). Note that the
category I[1] has two objects, which we call x and y, and exactly four morphism*
*s:
x ! x, x ! y, y ! x, y ! y. Thus the morphisms are in one-to-one correspondence
with the "words" xx, xy, yx, yy. In general the points of Ek are in one-to-one
correspondence with words of length k +1 in the letters {x; y}. The non-degener*
*ate
points correspond to the words which alternate the letters x and y; there are e*
*xactly
two such non-degenerate points in Ek for each k.
We define a filtration
F (1) E(1) E(2) E(3) . . .E
of E where E(k)is the smallestSsubobject containing the word xyxyx . .o.f length
(k + 1). Note that E = k E(k), and so Map sS(E; W ) limMap sS(E(n); W ). We
will prove (6.2)by actually proving the following stronger result.
Proposition 11.1.If W is a Segal space and n 3, the map Map sS(E(n); W ) !
W1 factors through the subspace Whoequiv W1, and induces a weak equivalence
Map sS(E(n); W ) ! Whoequiv.
We prove (11.1)in (11.7).
Proof of (6.2)from (11.1).Since E is the colimit of the E(n) along a sequence
of cofibrations, it follows by (11.1)that Map sS(E; W ) is the inverse limit of*
* the
Map sS(E(n); W ) along a tower of trivial fibrations. The proposition now_foll*
*ows
easily. |__|
24 CHARLES REZK
11.2. Morphisms induced by compositions. Let W be a Segal space. Given
g 2 map(y; z), consider the zig-zag
map (x; y) {g}x1----!map(y; z) x map(x; y) -'2-~map(x; y; z) d1-!map(x; z);
this induces a morphism g*: map (x; y) ! map (x; z) in the homotopy category of
spaces. Likewise, given f 2 map(x; y), consider the zig-zag
map (y; z) 1x{f}----!map(y; z) x map(x; y) -'2-~map(x; y; z) d1-!map(x; z);
this induces a morphism f* :map (y; z) ! map (x; z) in the homotopy category of
spaces. Note that if f 2 map(x; y) and g 2 map(y; z), then g*([f]) = f*(g) = [g*
*Of].
We have the following.
Proposition 11.3.
(1) Given f 2 map(x; y) and g 2 map(y; z), and g O f the result of a compositi*
*on,
then (g O f)* ~ g* O f* and (g O f)* ~ f* O g*.
(2) Given x 2 obW then (idx)* ~ (idx)* ~ idmap(x;x).
Proof.To prove (1), let k 2 map(x; y; z) be a composition of f and g which resu*
*lts
in a composite g O f. To show that (g O f)* ~ g* O f*, it suffices to show that*
* both
sides of the equation are equal (in the homotopy category of spaces) to the zig*
*-zag
map (w; x) {k}x1----!map(x; y; z) x map(w; x) -~ map (w; x; y; z) ! map(w; z):
The proof that (g O f)* ~ f* O g* is similar. __
The proof of (2) is straightforward. |__|
Proposition 11.4.Let f; g 2 map (x; y). Then f ~ g if and only if the maps
f*; g*: map (w; x) ! map (w; y) are homotopic for all w 2 obW , if and only if *
*the
maps f*; g*: map (y; z) ! map(x; z) are homotopic for all z 2 obW .
Proof.The only if direction is straightforward. To prove the if direction, supp*
*ose
that f* and g* are homotopic for all w 2 obW . Then in particular they are
homotopic for w = x. The following commutative diagram demonstrates that
f*(idx) ~ f.
{idx}
map (x; x)oo___________pt
{f}x1|| {f}||
fflffl| 1x{idx} fflffl|
map (x; y)OxOmap(x; x)oo_map(x; y)
| s0llllll |
(d0;d2~)| lll 1|
| uulllld1 fflffl|
map(x; x; y)_______//map(x; y)
Similarly g*(idx) ~ g, whence f ~ g, as desired. |__*
*_|
Corollary 11.5.If f 2 map(x; y) is a homotopy equivalence (in the sense of (5.4*
*))
then f* and f* are weak equivalences of spaces.
It is convenient to write map (x; y)f to denote the component of map (x; y) c*
*on-
taining f. More generally, we write map (x0; : :;:xk)f1;:::;fkfor the componen*
*t of
map(x0; : :;:xk) corresponding to the component of (f1; : :;:fk) in map (x0; x1*
*) x
. .x.map(xk-1; xk). The following lemma will be used in the proof of (11.1).
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 25
Lemma 11.6. Given a Segal space W and f 2 map(x; y) and g 2 map(y; z) such
that f is a homotopy equivalence, the induced map
map(x; y; z)f;g-(d1;d2)---!map(x; z)gOfx map(x; y)f
is a weak equivalence.
Proof.This follows from the diagram
map4(y;4z)g_x_map(x;_y)f_
_______________________________________________*
*___________________
________________|1x|________________________________*
*______________________________________________________________@
(d0;d2)_________________fflffl|_____________________________*
*_____________________________________________________
_______map(y;_z)g_x_map(x;_y)f_x_map(x;_y)f_______________*
*__________________
_____________________33hhhhhOO
________________________________hh~|
________________________________(d0;d2;d2)hhhh(d0;d2)x1|
________________________________hhh|
map(x; y; z)f;g(1;d__//map(x; y; z)f;gx map(x; y)f
_____________2)__________________________________
___________________________________________________|
____________________________________________d1x1|
(d1;d2)________--_____________________________________________*
*______________________________________________________________@
map (x; z)gOfx map(x; y)f
Here the vertical column is a weak equivalence since f is a homotopy equivalence
(restricting to the fiber over f 2 map(x; y)f gives exactly the zig-zag which d*
*efines
f* :map (y; z)g ! map (x; z)gOf). Since (d0; d2) is a weak equivalence,_the_lem*
*ma
follows. |__|
11.7. Proof of (11.1). For k 2 there are push-out diagrams
H(k) ______//F (k)
| |
(11.8) | |oek
fflffl| fflffl|
E(k-1)_____//E(k)
where oek is the map corresponding to the word xyx . .o.f length (k +1), and wh*
*ere
H(k) denotes the largest subobject of F (k) not containing d0.
We next note that H(k) can itself be decomposed. Thus let C(k) F (k) denote
the largest subobject of F (k) not containing d0d0. If we let d1: F (k - 1) ! F*
* (k)
denote the inclusion of the "face" d1, then we have that d1F (k - 1) \ C(k) =
d1H(k - 1), and thus an isomorphism
(11.9) H(k) C(k) [d1H(k-1)d1F (k - 1):
Let X be a simplicial space and W a Segal space. Then each map fl :F (1) ! X
induces a map
fl*: Map sS(X; W ) ! Map sS(F (1); W ) W1
of spaces. We introduce the following notation. Let Map sS(X; W )hoequivdenote *
*the
subspace of Map sS(X; W ) consisting of all simplices x such that fl*(x) 2 Whoe*
*quiv
W1 for all fl :F (1) ! X. Then MapsS(X; W )hoequivis isomorphic to a union of s*
*ome
of the path components of Map sS(X; W ). In particular, Map sS(F (1); W )hoequiv
Whoequivby definition, and MapsS(F (k); W )hoequiv WhoequivxW0 . .x.W0Whoequiv.
26 CHARLES REZK
Lemma 11.10. Let W be a Segal space. Then for k 2 the induced map
Map sS(F (k); W )hoequiv! Map sS(H(k); W )hoequiv
is a weak equivalence.
Proof.The proof is by induction on k. The case k = 2 is immediate from (11.6).
Now suppose the lemma is proved for the map Map sS(F (k - 1); W )hoequiv!
Map sS(H(k - 1); W )hoequiv. From (11.9)we get a commutative square
Map sS(H(k); W )hoequiv______//MapsS(C(k); W )hoequiv
| |
| |
fflffl| fflffl|
Map sS(F (k - 1); W )hoequiv//_MapsS(H(k - 1); W )hoequiv
This square would be a pullback square if we left off the "hoequiv" decorations.
Even with these decorations the square is a pullback (and hence a homotopy pull-
back), as can be seen by recalling that H(k)1 = C(k)1 [ d1F (k - 1)1.
Thus by induction we see that the map
a: Map sS(H(k); W )hoequiv! Map sS(C(k); W )hoequiv
is a weak equivalence. The proof now follows from (11.11)and the fact that the
map
axW01 1
Wk Wk-1 xW0 W1 ----! Map sS(C(k); W ) Map sS(d H(k - 1); W ) xW0 W1
is a weak equivalence after restricting to the "hoequiv" components. *
*|___|
Lemma 11.11. There is a natural weak equivalence
Map sS(C(k); W ) Map (d1H(k - 1); W ) xW0 W1:
Proof.Let d0H(k - 1) C(k) denote the image of H(k - 1) in C(k) induced by
the map d0: F (k - 1) ! F (k). There is a square
ff1F (0)_____//ff0F (1)
| |
| |
fflffl| fflffl|
d0H(k - 1)______//C(k)
of subobjects of C(k); we need to show that the inclusion map d0H(k - 1) [
ff0F (1) ! C(k) of the union of these subobjects is a weak equivalence in the
Segal space model category structure.
Now C(k) can be written as a colimit of the poset of subcomplexes each of whi*
*ch
1. are isomorphic to F (`) for some ` < k, and
2. include 0; 1 2 F (k)0.
Straightforward calculation shows that the intersection of d0H(k-1)[ff0F (1) wi*
*th __
each of the objects F (`) in the above diagram is a cover of F (`). *
* |__|
Proof of (11.1).It is clear that for k 3 every map
MapsS(E(k); W ) ! Map sS(F (1); W ) W1
induced by an inclusion F (1) ! E(k)must factor through Whoequiv W1, since
each point of the mapping space maps to a homotopy equivalence in the sense of
(5.4). Let rk denote the map Map sS(E(k); W ) ! W1 associated to the inclusion
A MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORY 27
F (1) ! E(k)classifying the point xy 2 E(k)1. We have that Map sS(E(k); W ) =
Map sS(E(k); W )hoequivfor k 3, and even when k = 2 we have that
Map sS(E(2); W )hoequiv Map sS(E(2); W ) xW1 Whoequiv:
Then we must show that for each k 2 the fiber of rk over any point in the subs*
*pace
Whoequiv W1 is contractible. The result now follows from (11.10)applied to_the
pushout diagrams (11.8). |__|
12.Complete Segal space closed model category structure
In this section we prove (7.2).
The complete Segal space closed model category structure is defined
using (9.1)to be the localization of the Reedy model category of simplicial spa*
*ces
with respect to the map g obtained as a coproduct of the maps 'iof Section 4 and
the map
x: F (0) ! E
which corresponds to the object x 2 I[1]. Parts (1)-(4) of (7.2)follow immediat*
*ely
from (9.1). The only thing left to prove is the compatibility of this model cat*
*egory
structure with the cartesian closure.
By (9.2)it suffices to show that f W is a complete Segal space, then so is W *
*F(1).
In (7.1)we have already proved that W F(1)is a Segal space; thus it suffices to*
* show
that
Proposition 12.1.If W is a complete Segal space, then the map g :(W F(1))0 !
(W F(1))hoequivis a weak equivalence.
12.2. Homotopy monomorphisms. Say a map f :X ! Y of spaces is a homo-
topy monomorphism if
1. it is injective on ss0, and
2. it is a weak equivalence of each component of X to the corresponding com-
ponent of Y .
Equivalently, f is a homotopy monomorphism if the square
X __1__//X
1 || |f|
fflffl|fflffl|f
X _____//Y
is a homotopy pullback square. Since homotopy limits commute, the homotopy
limit functor applied to a homotopy monomorphism between two diagrams yields
a homotopy monomorphism.
12.3. Proof of (12.1). The map s0: (W F(1))0 ! (W F(1))1 is obtained by taking
limits of the rows in the diagram:
W1 ______W1______W1
s0|| |1| s1||
fflffl|dfflffl|1dfflffl|1
W2 ____//_W1oo__W2
By hypothesis, s0: W0 ! W1 is a homotopy monomorphism. Thus the maps
s0; s1: W1 ! W2 are homotopy monomorphisms, since they are weakly equivalent
28 CHARLES REZK
to W1xW0 s0: W1xW0 W0 ! W1xW0 W1 and s0xW0 W1: W0xW0 W1 ! W1xW0 W1.
It follows that s0: (W F(1))0 ! (W F(1))1 is a homotopy monomorphism.
Thus both s0: (W F(1))0 ! (W F(1))1 and (W F(1))hoequiv! (W F(1))1 are homo-
topy monomorphisms. So to prove the proposition it suffices to show that both t*
*hese
maps hit the same components. As we already know that (W F(1))0 ! (W F(1))1
factors through a map (W F(1))0 ! (W F(1))hoequiv, it suffices to show that thi*
*s last
map is surjective on ss0.
Using the part of the proof already completed and (12.4), it is not hard to s*
*ee that
a point x 2 (W F(1))hoequivlies in a component hit by g :(W F(1))0 ! (W F(1))ho*
*equiv
if and only if the images fx; gx 2 (W F(0))1 W1 are homotopy equivalences
in W , where f; g :W F(1)! W F(0)are the maps induced by the two inclusions
d0; d1: F (0) ! F (1). But if x 2 (W F(1))1 is a homotopy equivalence of W F(1)*
*then
certainly its images under f and g are homotopy equivalences. Thus the result is
proved.
Lemma 12.4. Let W be a Segal space. Then the squares
W0 od1o_W1 W1 _d0_//_W0
s0|| s0|| s1|| |s0|
fflffl|dfflffl|2 fflffl|dfflffl|0
W1 oo___W2 W2 ____//_W1
are homotopy pullback squares.
Proof.Recall that for a Segal space W2 ~-!W1 xW0 W1, so that W2 xW1 W0 ~-!
(W1xW0 W1) xW1 W0 W1 and W0xW1 W2 ~-!W0xW1 (W1xW0 W1) W1. |___|
13.Categorical equivalences
In this section we provide a generalization to Segal spaces of the category t*
*heo-
retic concepts of "natural isomorphism of functors" and "equivalence of categor*
*ies",
and show that for the complete Segal spaces, these concepts correspond precisely
to those of "homotopy between maps" and "(weak) homotopy equivalence".
13.1. Categorical homotopies. We define a categorical homotopy between
maps f; g :U V of Segal spaces to be any one of the following equivalent data:
a map H :U x E ! V , a map H0:U ! V E, or a map H00:E ! V U, making the
appropriate diagram commute:
U EE V>>OO F (0)E
| EEEf f""""| | EE{f}E
Uxi0| EE "" |V i0 i0| EEEE
fflffl|E""EH ""H0 | fflffl|H""00
U xOEO____//_V<