A CHARACTERIZATION OF FIBRANT SEGAL CATEGORIES
JULIA E. BERGNER
Abstract.In this note we prove that Reedy fibrant Segal categories are f*
*i-
brant objects in the model category structure SeCatc. Combining this res*
*ult
with a previous one, we thus have that the fibrant objects are precisely*
* the
Reedy fibrant Segal categories. We also show that the analogous result h*
*olds
for Segal categories which are fibrant in the projective model structure*
* on
simplicial spaces, considered as objects in the model structure SeCatf.
1.Introduction
Segal categories are simplicial spaces which are meant to look like simplicial
categories, but their morphisms are only associative up to (higher) homotopy. T*
*hey
were first described by Dwyer, Kan and Smith, who called them special op-spac*
*es
[3]. They have since been generalized to the notion of Segal n-categories, vari*
*ants
of which have been studied as models for weak n-categories by Hirschowitz and
Simpson [7], [13] and Tamsamani [14]. Like simplicial categories, they can be
considered as models for homotopy theories [2, 8.6].
In any model category structure, it is useful to have a precise description o*
*f the
fibrant and cofibrant objects, since these are used to define the homotopy cate*
*gory.
Furthermore, various constructions, such as that of mapping spaces, are only ho*
*mo-
topy invariant when the objects involved are both fibrant and cofibrant. In the*
* first
of the model structures which we will consider, SeCatc, all objects are cofibra*
*nt,
but a description of the fibrant objects has not been so clear. In this note, w*
*e give
a complete characterization of them.
We begin by giving some precise definitions. Let denote the cosimplicial
category, or category whose objects are finite ordered sets [n] = (0 ! 1 ! . .!*
*.n)
for n 0 and whose morphisms are order-preserving maps between them. Then
op is the opposite of this category and is called the simplicial category. Re*
*call
that a simplicial set X is a functor op ! Sets. We will denote the category
of simplicial sets by SSets. (In the course of this paper we will sometimes ref*
*er to
simplicial sets as spaces, due to their homotopy-theoretic similarity with topo*
*logical
spaces [8, 3.6.7].) We denote by [n] the n-simplex for each n 0, by `[n] i*
*ts
boundary, and by V [n, k] the boundary with the kth face removed. More details
about simplicial sets can be found in [5, I]. A simplicial set X is discrete if*
* all
elements of Xn are degenerate for n > 0. We denote by |X| the topological space
given by geometric realization of the simplicial set X [5, I.2].
More generally, recall that a simplicial object in a category C is a functor *
* op!
C. In particular, a functor op! SSets is a simplicial space or bisimplicial se*
*t [5,
IV]. Given a simplicial set X, we denote by Xt the simplicial space such that (*
*Xt)0
____________
Date: August 24, 2006.
2000 Mathematics Subject Classification. Primary: 55U35; Secondary 18G30.
1
2 J.E. BERGNER
is the discrete simplicial set X0. Also, given a simplicial space W , we denote*
* by
sknW the n-skeleton of W , or the simplicial space generated by the simplices o*
*f W
of degree less than or equal to n [11, x1]. In this paper, our primary concern *
*is the
case where the simplicial set in degree zero is discrete (or constant).
Definition 1.1. A Segal precategory is a simplicial space X such that X0 is a
discrete simplicial set.
Now note that for any simplicial space X there is a Segal map
'k : Xk ! X1_xX0_._.x.X0X1_-z_______"
k
for each k 2, which we define as follows. Let ffi: [1] ! [n] be the map in *
*such
that ffi(0) = i and ffi(1) = i + 1, defined for each 0 i n - 1. We can then*
* define
the dual maps ffi: [n] ! [1] in op. Given a simplicial space X, for each k 2*
* the
Segal map is defined to be the map
'k : Xk ! X1_xX0_._.x.X0X1_-z_______"
k
induced by the maps
X(ffi) : Xk ! X1.
Definition 1.2. [7, x2] A Segal category X is a Segal precategory such that the
Segal map 'k is a weak equivalence of simplicial sets for each k 2.
For a Segal (pre)category X, we will frequently refer to the discrete simplic*
*ial
set X0 as the set of "objects" of X.
In [2, 5.1, 7.1], we show that there exist two different model category struc*
*tures
on the category of all Segal precategories in which the fibrant objects are Seg*
*al
categories. (We will discuss model category structures in more detail in the ne*
*xt
section.) For the first of these structures, SeCatc, we show in [2, 5.13] that*
* the
fibrant objects are Segal categories which are fibrant in the Reedy model categ*
*ory
structure on the category of all simplicial spaces. Here, we complete the resul*
*t and
show that the converse holds as well, namely, that all Reedy fibrant Segal cate*
*gories
are fibrant in SeCatc. Similarly, in the second model structure, SeCatf, we show
that the fibrant objects are precisely the Segal categories which are fibrant i*
*n the
projective model category structure on simplicial spaces.
It should be noted that these model category structures on Segal precategorie*
*s fit
into a chain of Quillen equivalences between various model structures. The two *
*are
Quillen equivalent to one another, as well as to a model structure on the categ*
*ory
of simplicial categories and to Rezk's complete Segal space model structure on
simplicial spaces, which Joyal and Tierney prove to be Quillen equivalent to Jo*
*yal's
model structure on quasi-categories [9], [10]. In doing so, they actually obtai*
*n an
alternate proof of Theorem 3.2. The author's interest in comparing these model
structures arose from finding models for the homotopy theory of homotopy theori*
*es,
a project begun by Rezk [12].
Acknowledgments. I would like to thank Bertrand To"en for pointing out an error
in a previous proof of the main result, and the referee for suggestions which l*
*ed to
an improved proof of Proposition 3.1.
A CHARACTERIZATION OF FIBRANT SEGAL CATEGORIES 3
2. Review of model category structures
In this section, we give a brief review of model category structures. In part*
*icu-
lar, we discuss the Reedy model category structure on the category of all simpl*
*icial
spaces and the model category structure SeCatc on the category of Segal precate-
gories.
Recall that a model category structure on a category C is a choice of three
distinguished classes of morphisms: fibrations, cofibrations, and weak equivale*
*nces.
A (co)fibration which is also a weak equivalence is an acyclic (co)fibration. W*
*ith
this choice of three classes of morphisms, C is required to satisfy five axioms*
* MC1-
MC5 which can be found in [4, 3.3]. An object X in a model category is fibrant *
*if
the unique map X ! * to the terminal object is a fibration. Dually, X is cofibr*
*ant
if the unique map from the initial object OE ! X is a cofibration.
There is a model category structure on the category of simplicial sets in whi*
*ch
the weak equivalences are the maps f : X ! Y such that the induced map |f| :
|X| ! |Y | of topological spaces is a weak homotopy equivalence [5, I.11.3]. We*
* can
then use this model category structure to define model structures on the catego*
*ry
of simplicial spaces.
A natural choice for the weak equivalences in the category of all simplicial *
*spaces
is the class of levelwise weak equivalences of simplicial sets. Namely, given *
*two
simplicial spaces X and Y , we define a map f : X ! Y to be a weak equivalence *
*if
and only if for each n 0, the map fn : Xn ! Yn is a weak equivalence of simpl*
*icial
sets.
In the Reedy model category structure on simplicial spaces [11], the weak equ*
*iv-
alences are the levelwise weak equivalencesoofpsimplicial sets. We will denote *
*the
Reedy model structure by SSetsc . While it is defined somewhat differently, the
Reedy model category structure on simplicial spaces is exactly the same as the
injective model category structure on this same category, in which the cofibrat*
*ions
are defined to be levelwise cofibrations of simplicial sets [6, 15.8.7].
In sectiono4,pwe will also make use of the projective model category structure
SSetsf on simplicial spaces, in which the fibrations are given by levelwise f*
*ibra-
tions of simplicial sets [5, IX 1.4].
In [12, 7.1],oRezkpdefines a model category structure which is obtained by lo*
*cal-
izing SSetsc with respect to a set of maps [6, 4.1.1]. Its fibrant objects ar*
*e called
Segal spaces, and they satisfy two properties: they are fibrant in the Reedy mo*
*del
structure, and each Segal map 'k is a weak equivalence for k 2. In particular,
there is a fibrant replacement functor taking any simplicial space to a Segal s*
*pace.
In [2, x5], we construct a similar functor Lc which takes a Segal precategory X*
* to
a Segal space LcX which is also a Segal category such that X0 = (LcX)0.
Theorem 2.1. [2, 5.1] There is a model category structure SeCatc on the categor*
*y of
Segal precategories with the following weak equivalences, fibrations, and cofib*
*rations:
o Weak equivalences are the maps f : X ! Y such that the induced map
LcX ! LcY is a weak equivalence in Rezk's model structure on simplicial
spaces.
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.
4 J.E. BERGNER
Corollary 2.2. [2, 5.13] Fibrant objects in SeCatc are Reedy fibrant Segal cate-
gories.
We would like to prove the converse of this corollary. We begin by proving it
in a more restricted setting, namely in the model category of Segal precategori*
*es
which have a fixed set O in degree zero.
3. Reedy fibrant Segal categories
We begin this section by characterizing the fibrant objects in the fixed obje*
*ct set
case, and then we proceed to the more general case. We begin by briefly describ*
*ing
the model structure in the more restricted situation.
We show in [1, 3.9] that there is a model structure SSpO,con the category of *
*Segal
precategories which have a fixed set O in degree zero in which the weak equival*
*ences
and cofibrations are each defined levelwise. This model category can be localiz*
*ed
with respect to a set of maps to obtain a model structure LSSpO,c, in which the
fibrant objects are Segal categories.
Thus, we will first prove that the fibrant objects of LSS pO,care precisely t*
*he
Reedy fibrant Segal categories with the set O in degree zero. Note that any Ree*
*dy
fibrant Segal precategory is a fibrant object in the appropriate SSpO,c, and he*
*nce
that any Reedy fibrant Segal category is fibrant in the appropriate LSSpO,c. The
following proposition shows that the converse statement holds as well.
Proposition 3.1. A fibrant object in SSpO,cis fibrant in the Reedy model struct*
*ure
on simplicial spaces. In particular, a fibrant object of LSSpO,cis fibrant in t*
*he Reedy
structure.
Proof.Let W be a fibrant object in SSpO,c. We need to show that the map W !
[0]thas the right lifting property with respect to all levelwise acyclic cofib*
*rations,
not just the ones in SSpO,c. We first consider the case where A ! B is an acycl*
*ic
cofibration where A and B are Segal precategories, say in SSpO0,cfor some set
O06= O.
Using the 0-skeleta sk0(A) and sk0(W ), we can define a simplicial space A0 as
the pushout
sk0(A)_____//sk0(W )
| |
| |
|fflffl fflffl|
A ________//_A0.
Note in particular that the induced map A00! W0 is an isomorphism. Now, we
can define B0 as a pushout
A _____////_A0
| |
| |
fflffl|fflffl|
B _____//B0
A CHARACTERIZATION OF FIBRANT SEGAL CATEGORIES 5
and thus a diagram
A _____//A0_______//W
| | |
| | |
fflffl| fflffl| fflffl|
B ____//_QQQB0//_qO [0]t
QQ
QQQQ |
QQQQ |
QQ((Qfflffl|
[0]t
Because it is defined as a pushout along an acyclic cofibration in the Reedy st*
*ruc-
ture, the map A0 ! B0 is also an acyclic cofibration, and A00~=B00~=qO [0]t.
Therefore there exists a lift B0! W , from which there exists a lift B ! W .
Now, suppose that A ! B is an acyclic cofibration between simplicial spaces
which are not necessarily Segal precategories. Since W and [0]t are Segal prec*
*at-
egories, we can factor our diagram as
A _____//eA____//W
| | |
| | |
fflffl|fflffl| fflffl|
B _____//eB___//_ [0]t
where eAand eBare obtained from A and B, respectively, by collapsing the space
in degree zero to its components. Then, we obtain a lifting from the previous
argument.
Since fibrant objects in LSSpO,care fibrant in SSpO,c, the second statement of
the proposition follows as well.
Using this result, we turn to the converse to Corollary 2.2.
Theorem 3.2. Any Reedy fibrant Segal category is fibrant in SeCatc.
Proof.Let W be a Reedy fibrant Segal category and suppose that f : A ! B is a
generating acyclic cofibration in SeCatc. We need to show that the map W ! [0]t
has the right lifting property with respect to the map f. We know that it has t*
*he
right lifting property with respect to any such f which preserves a fixed objec*
*t set
O, by Proposition 3.1. Therefore, we assume that f is a monomorphism but is not
surjective.
Choose b 2 B0 which is not in the image of f : A ! B. Since f is a weak
equivalence, we know that b is equivalent in LcB to f(a) for some a 2 (LcA)0 = *
*A0.
Define (LcB)a to be the full Segal subcategory of LcB whose objects are in the
essential image of a. Let Ba be the sub-simplicial space of B whose image is in
(LcB)a. Note that (Ba)0 = ((LcB)a)0.
Now, define Aa to be the Segal precategory which has as 0-space the union of
(Ba)0 and A0 and for which
(Aa)n(a0, . .,.an) = An(a, . .,.a)
6 J.E. BERGNER
for all ai2 (Aa)0. Letting a also denote the doubly constant simplicial space g*
*iven
by a, we can then define A1 to be a pushout given by
a_____//_Aa
| |
| |
fflffl| fflffl|
A ____//_A1.
Notice that the map A ! A1 has a section and that we can factor f as the compos*
*ite
A ! A1 ! B.
We now repeat this process by choosing a b0 which is not in the image of the
map A1 ! B and a corresponding a0, and continue to do so, perhaps infinitely
many times, and take a colimit to obtain a Segal precategory bAsuch that the map
f factors as A ! bA! B and there is a section bA! A. Furthermore, notice that
bA0= B0 and that the map bA! B is object-preserving. Notice also that it is an
acyclic cofibration. Therefore, the dotted-arrow lift exists in the following d*
*iagram:
bA_____//W??
| "" |
| " |
fflffl|fflffl|"
B _____//_*
which implies, using the section Ab! A, that there is a dotted arrow lift in the
diagram
A ____//_W>>.
| " " |
| " |
fflffl|"fflffl|
B _____//_*
Note. One might wonder why, in the general case, the model category structure
SeCatc is not defined as a localization of a model structure with object-preser*
*ving
weak equivalences, as it is in the fixed object set situation. We prove in [2, *
*3.12]
that it is impossible, simply because the more basic model structure cannot exi*
*st.
Therefore, we cannot use the tools of localized model structures. Much of the
difficulty with working with SeCatc arises from this fact.
4.Projective fibrant Segal categories
In this section, we show that the analogous statement holds in our other model
category structure, SeCatf. In this structure, the weak equivalences are the sa*
*me
as those of SeCatc, but the fibrations and cofibrations are defined differently*
*. They
are technical to describe, so we will refer the interested reader to [2, x4, 7.*
*1] for
a complete description. The fibrations should be thought of heuristically as th*
*ose
we would obtain via a localization of a model structure given by levelwise weak
equivalences and fibrations, although, as mentioned at the end of the previous
section, such a construction is impossible in the general case.
As before, we begin by considering the special case where we have a fixed set*
* O
in degree zero. Analogously to the situation in the previous section, there exi*
*sts a
model category structure SSpO,f on simplicial spaces with a fixed set O in degr*
*ee
zero in which the weak equivalences and fibrations are levelwise. Localizing t*
*his
A CHARACTERIZATION OF FIBRANT SEGAL CATEGORIES 7
model structure with respect to a set of maps results in a model structure LSSp*
*O,f
in which the fibrant objects are Segal categories with O in degree zero [1, 3.8*
*].
We can prove the following result using the same techniques that we used to
prove Proposition 3.1.
Proposition 4.1. The fibrant objects in LSSpO,fare precisely the Segal categori*
*es
which have the set O in degree zero and are fibrant in the projective model str*
*ucture
on simplicial spaces.
As in the Reedy case, there is a corresponding functor Lf (where "Segal space*
*s"
are now obtained as a localization in the projective, rather than the Reedy, mo*
*del
structure), and in [2, x7] we show that the weak equivalences are actually inde*
*pen-
dent of whether we define them in terms of Lc or Lf.
Theorem 4.2. Segal categories which are fibrant in the projective model category
structure on simplicial spaces are fibrant in SeCatf.
Proof.The argument given for SeCatc still holds in this case.
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Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506
E-mail address: bergnerj@member.ams.org