A Giraud-type characterization of the simpli-
cial categories associated to closed model cat-
egories as 1-pretopoi
Carlos Simpson
CNRS, UMR 5580, 31062 Toulouse CEDEX
In SGA 4 [1], one of the principal building blocks of the theory of topoi is
Giraud's theorem, which says that the condition of a 1-category A being the
category of sheaves on a site, may be characterized by intrinsic, internal con-
ditions in A. The intrinsic conditions are basically existence of certain limits
and colimits, plus a condition about generation by a small set of objects.
In this paper, we will present a generalization of this theorem to the
situation of simplicial categories (by which we mean simplicially enriched
categories) or equivalently Segal categories ([11] [28], [32]), or complete Seg*
*al
spaces (Rezk [25]). One can easily imagine generalizing the internal condi-
tions of existence of limits or colimits (these become conditions of existence
of homotopy limits or colimits). On the other hand, the condition which we
take as a generalization of the condition of being the category of sheaves on
a site, is the condition of coming from a closed model category [22]. Recall
that Dwyer and Kan associate to any closed model category M its simpli-
cial localization L(M) which is a simplicial category [9]. If M is a simplicial
closed model category in the sense of Quillen, then L(M) is equivalent to the
simplicial category of fibrant and cofibrant objects of M. It is this simplicial
category L(M) which represents the homotopy theory (including information
about all higher-order homotopies) which comes out of M.
We attack the very natural question of characterizing which simplicial
categories A are equivalent to ones of the form L(M) for closed model cate-
gories M. This formulation of the question is closely related to some of the
entries in the "Model Category" section of M. Hovey's recent "problem list"
[18].
The first, easy but fundamental observation is that if M is a closed model
category (admitting all small limits and colimits as it is now customary to
assume), then L(M) admits small homotopy limits and colimits. In par-
ticular, not every simplicial category will be equivalent to one of the form
L(M). Our characterization is that this necessary condition is basically suffi-
1
cient; however, one has to add in an additional set-theoretic hypothesis about
small generation which in practice will always hold. This first easy observa-
tion came from thinking about C. Rezk's terminology of calling his version
of the closed model category of Segal categories, the "homotopy theory of
homotopy theories".
Our answer is, as stated above, analogous to Giraud's theorem. To be
quite precise, the analogy is not complete. In effect, the internal conditions *
*on
A which come out are existence of homotopy colimits, and small generation.
These turn out to imply existence of homotopy limits; however one does
not get any sort of exactness properties allowing one to commute limits and
colimits, and indeed one can find examples of closed model categories M such
that L(M) does not have these exactness properties. Thus, in the statement
of our theorem, we refer to our equivalent conditions as defining a notion
of 1-pretopos, and reserve the name 1-topos for an 1-pretopos satisfying
additional exactness properties.
Another remark is that we are not able to treat all closed model categories,
nor does this seem natural in the context of a Giraud-type theorem. Rather
we speak only of cofibrantly generated closed model categories see [8] [17] [15*
*].
Almost all known closed model categories (here as usual we only consider ones
in which all small limits and colimits exist) are cofibrantly generated.
Hovey also states in [18] that D. Dugger has shown that any cofibrantly
generated closed model category is Quillen-equivalent to a simplicial one;
thus the reader of the present introduction who is unfamiliar with Dwyer-
Kan may assume that we are speaking of simplicial model categories and
may replace L(M) by the simplicial category Mcf of cofibrant and fibrant
objects.
Here is a shortened version of the statement. As a matter of notation, we
speak in the introduction of "simplicial categories"; the notion of equivalence
is that which was explored by Dwyer and Kan [9]. This is just the obvi-
ous notion of "fully faithful and essentially surjective" where "fully faithful"
means inducing weak equivalences of simplicial Hom sets, and "essentially
surjective" means essential surjectivity of the truncated morphism on homo-
topy 1-categories. However, with this definition an equivalence between two
simplicial categories means a string of functors which are equivalences, pos-
sibly going in different directions. See below for a bit more explanation. We
also refer to the body of the paper for the definitions of homotopy colimit,
generation by homotopy colimits, and smallness.
2
Theorem 1 (cf Theorem 14 p. 42) Suppose A is a simplicial category. The
following conditions are equivalent:
(i) There is a cofibrantly generated closed model category M such that A is
equivalent to the Dwyer-Kan simplicial localization L(M);
(ii) A admits all small homotopy colimits, and there is a small subset of
objects of A which are A-small, and which generate A by homotopy colimits.
We call a simplicial category satisfying the conditions of the theorem, an
1-pretopos. If in addition a certain exactness condition is satisfied (see the
statement of Theorem 14 for details) then we say that A is an 1-topos.
The possibility of having a reasonable notion of n-topos was predicted in
[29]. This prediction came about due to the influence of correspondence with
C. Teleman who at the time was telling me about pullbacks of simplicial
presheaves under morphisms of sites. Of course, like most of what we do
here, this idea is very present in spirit throughout [13].
A word about rigour and level of detail in this version of the present paper.
At several places in the argument, we skip verification of some details. These
are mostly details concerning "homotopy-coherent category theory" as done
with Segal categories. They are all generalizations to the "weak-enriched"
setting of classical statements in category theory, so it seems completely
clear that the statements in question are true. It also seems clear that in
the relatively near future, techniques will have sufficiently advanced in order
to cover these questions. Finally, it seems likely that using some of the
other approaches (such as Cordier-Porter [7] or the model category of Dwyer-
Hirschhorn-Kan [8]), a significant number of these details could be verified
relatively easily_the reason I haven't taken that route is lack of familiarity
with those approaches. However, at the time of writing of the present version,
I have not verified the details any further than what is written down below.
One could say that the present paper is premature in this sense, but the
result seemed interesting enough to justify writing it up quickly. In order to
clarify matters, the places where this problem occurs are marked with the
symbol ().
Acknowledgements: I would like to thank very much M. Hovey, C. Rezk,
A. Hirschowitz, P. Hirschhorn, C. Teleman, Z. Tamsamani, B. Toen, and J.
Tapia, for their important contributions to the realization of the idea outlined
in the present paper.
Segal categories
3
A simplicial category means a category C enriched over simplicial sets.
In other words, for every pair of objects x; y 2 ob(C), we have a simplicial
set HomC (x; y). In order to conform with our notations for Segal categories,
we shall denote this simplicial set by C1=(x; y). In the case of a simplicial
category, composition of morphisms is a map of simplicial sets
C2=(x; y; z) := C1=(x; y) x C1=(y; z) ! C1=(x; z);
and this operation is strictly associative. In view of the strict associativity,
we obtain a bisimplicial set (i.e. simplicial simplicial set) by setting
a
Cp= := Cp=(x0; : :;:xp);
x0;:::;xp
with
Cp=(x0; : :;:xp) := C1=(x0; x1) x : :x:C1=(xp-1; xp):
Here we set C0 := C0= := ob(C). This bisimplicial set has the property that
for any m, the "Segal map"
Cm= ! C1=xC0 : :x:C0C1=
is an isomorphism; and conversely any bisimplicial set such that the simplicial
set C0= is a discrete set which we denote C0 or ob(C), and such that the
above Segal maps are isomorphisms, corresponds to a simplicial category.
The composition is obtained by using the third face map C2= ! C1=.
The above presentation of the notion of "simplicial category" motivates
the definition of "Segal category"_a Segal category is just a bisimplicial
set such that the simplicial set C0= is a discrete set which we denote C0
or ob(C), and such that the Segal maps are weak equivalences of simplicial
sets. A simplicial category thus gives rise to a Segal category, and we shall
sometimes call the Segal categories which arise in this way "strict".
Suppose C is a Segal category. As suggested by the previous notation,
for any sequence of objects x0; : :;:xp 2 ob(C) we obtain a simplicial set
Cp=(x0; : :;:xp) defined as the inverse image of (x0; : :;:xp) under the map
(given by the p + 1 "vertex" maps)
Cp= ! C0 x : :x:C0:
4
We think of the simplicial set C1=(x; y) as being the space of maps from x to
y in C. The Segal condition can be rewritten as saying that the morphism
(given by the p "principal edge" maps)
Cp=(x0; : :;:xp) ! C1=(x0; x1) x : :x:C1=(xp-1; xp)
is a weak equivalence. In particular, the "composition of morphisms in C"
is given by the diagram
~=
C1=(x; y) x C1=(y; z) C2=(x; y; z) ! C1=(x; z):
The notion of Segal category is based in an obvious way on Segal's weak-
ened notion of "topological monoid" [27] (which is the case where ob(C)
contains only one element), although Segal himself never seems to have writ-
ten anything suggesting to look at this notion for several objects. This notion
per se first appears in Dwyer-Kan-Smith [11] where they also show the equiv-
alence between Segal categories-up-to-equivalence and simplicial categories-
up-to-equivalence (see below).
This notion later appeared in an ad hoc way in my preprint [28] (I was
unaware of [11] at the time and until fairly recently); and it appears as the
basic idea which is iterated in Tamsamani's definition of weak n-category
[34]. Further occurences are in my preprint [32] and the joint paper [16].
A couple of closely related notions are use by Rezk in [25]. He defines a
notion of Segal space which is a simplicial set satisfying the condition that
the Segal maps are equivalences but not necessarily the condition that C0=
be a discrete set; consequently he includes a "Reedy fibrant" condition in
the definition in order to make sure that the fiber products involved in the
definition of the Segal maps are homotopically correct ones. He also defines
a notion of complete Segal space which basically says that the simplicial set
C0= should itself correspond to the space which is the realization of the sub-
category obtained by only looking at invertible (up-to-homotopy) morphisms
in C. We will state without proof below what should be the relation between
Rezk's notions and our own.
There are also other related notions such as various notions of A1 -
category see for example Batanin [4]; and more generally there are several
definitions of n-category alternative to Tamsamani's definition and which
should also have variants for weak simplicial categories, see Baez-Dolan [3]
5
for example. These other notions should be directly related to our own but
we don't go into that here.
Finally, we note that the above notions should be viewed as substitutes
for the notion of "1-groupic 1-category" i.e. an 1-category in which the
i-morphisms are invertible (up to equivalence) for i 2. We leave it to the
reader to make this notational translation.
We shall use the framework of "Segal categories" throughout the rest of
the paper, although we sometimes speak of the relationship with the classical
notion of simplicial category. The reader is refered to [31] and [16] for any
further details and introductory materiel that we may leave out in our brief
discussion which follows.
By abuse of notation, we may sometimes forget to put in the qualifier
"Segal" and just use the word "category" for "Segal category". In order to
avoid confusion, we will try to systematically use the terminology 1-category
for classical (non-simplicial) categories.
A morphism of Segal categories C ! D is said to be fully faithful if for
every x; y 2 ob(C), the morphism C1=(x; y) ! D1=(x; y) is a weak equivalence
of simplicial sets. This is the natural generalization of the corresponding
notion in category theory; however one should be careful that the separate
notions of "full" and "faithful" don't have reasonable generalizations to the
present theory, because there is no way of decomposing the condition of being
a weak equivalence of simplicial sets, into "injectivity plus surjectivity". For
this reason, huge swaths of the argumentation which is employed in SGA 4 [1]
are no longer available and we are forced to look for more intrinsic reasoning.
If C is a Segal category, define a 1-category denoted ho(C) with the same
objects as C, by setting
ho(C)1=(x; y) := ss0(C1=(x; y)):
We say that a morphism of Segal categories C ! D is essentially surjective
if the resulting morphism of 1-categories
ho(C) ! ho(D)
is essentially surjective. We say that a morphism of Segal categories is an
equivalence if it is fully faithful and essentially surjective.
In the context of simplicial categories this notion of equivalence was in-
troduced by Dwyer and Kan in [9]. (A morphism of simplicial categories is
6
an equivalence if and only if the corresponding morphism of Segal categories
is an equivalence.) In the context of n-categories this notion was called "ex-
ternal equivalence" by Tamsamani in [34]. In his situation of Segal spaces,
this notion was called "Dwyer-Kan equivalence" by Rezk in [25].
We say that a morphism in C (i.e. a vertex of C1=(x; y)) is an equivalence
if its image in ho(C)1=(x; y) is an isomorphism in ho(C). This corresponds to
what Tamsamani called "internal equivalence" in [34]. The essential surjec-
tivity condition can be expressed as saying that every object of D is equivalent
(in this "internal" sense) to an object coming from C.
We often use the terminology full subcategory for a fully faithful functor
of Segal categories C ! D which is injective on objects. In this case, up to
equivalence in the variable C, we may assume that the morphism is actually
an isomorphism on all of the Cp=(x0; : :;:xp). With this convention, the
intersection of full subcategories is again a full subcategory. Furthermore, we
say that a full subcategory C D is saturated if it satisfies the "saturation
condition" that whenever x 2 ob(C) and y is (internally) equivalent to x,
then y 2 ob(C) too. Again, the intersection of saturated full subcategories is
again a saturated full subcategory.
Strictification
We can now explain the comparison result of Dwyer, Kan, Smith which
was alluded to above. Let splCat denote the 1-category of simplicial cate-
gories, and let SegCat denote the 1-category of Segal categories. Let
Ho(splCat)(resp. Ho(splCat))
denote the Gabriel-Zisman localizations of these categories by inverting the
equivalences. We say that two simplicial categories (or two Segal categories)
are equivalent if they project to isomorphic objects in these homotopy cat-
egories. Dwyer, Kan and Smith in the last few pages of [11] show that the
morphism
Ho(splCat) ! Ho(SegCat )
is an equivalence of categories. Among other things, this says that any Se-
gal category can be "strictified", i.e. made equivalent (in the above sense)
to a simplicial category. We should take this occasion to stress that, as
Ho(splCat) and Ho(splCat) are Gabriel-Zisman localizations, one can have
7
two objects (simplicial categories or Segal categories) which are equivalent
but without there being any actual morphism between the two; the "equiv-
alence" in question might be realizable only as a chain of morphisms which
are equivalences, going in different directions. This situation is improved by
the introduction of closed model structures as we shall explain below (and
in particular if ever it is necessary to go through a chain of equivalences, at
least one can restrict to looking at chains of length 2). In the statement of
Theorem 1, it is the present notion of equivalence which is used.
In view of the strictification result of Dwyer-Kan-Smith, we may at many
places in the present paper assume that the Segal categories we are dealing
with are actually simplicial categories. This can simplify the problem of
composing morphisms and the like.
Closed model structures
There are several possible closed model structures which can be used to
attack the homotopy category Ho(splCat) ~=Ho(splCat). What seems to be
historically the first is that of Dwyer-Hirschhorn-Kan [8]. 1
To introduce the structure of [8], we first point out that Dwyer and Kan
obtained (essentially trivially) a closed model structure on splCat in [9] where
the weak equivalences were the equivalences which induce isomorphisms on
objects. In this structure, the fibrations are the morphisms of simplicial cat-
egories which induce fibrations of the individual simplicial Hom sets. The
cofibrations are closely related to the free resolutions which are used through-
out [9], and the cofibrant objects are just the simplicial categories which are
free at each stage. This closed model structure is not the one which we
are actually interested in (although it can be useful in a preliminary way),
because we are interested in understanding the equivalences which are essen-
tially surjective but not isomorphisms on objects. This problem was rectified
in [8] where a closed model structure on splCat is given, with the following
properties. The cofibrations are the same as in the previous structure; and
the weak equivalences are the "Dwyer-Kan equivalences" as described above.
This leads to a more restrictive notion of fibration than that which occurs
in their first structure. However, the fibrant objects are the same as in the
previous structure, namely the simplicial categories C with C1=(x; y) being
______________________________
1The draft of [8] that I have is dated at approximately the same time as [30*
*] but earlier
versions of [8] had apparently been in limited circulation for some time.
8
fibrant simplicial sets. To sum up what is going on here, we can say that in
order to correctly calculate the morphisms between two simplicial categories
C and D, one must make a replacement D ! Df by an equivalent one in
which the simplicial Hom-sets are fibrant, and one must make a replacement
Cc ! C with Cc cofibrant (which essentially means taking a free resolution).
Now Hom(Cc; Df) contains representatives for all of the homotopy classes
of morphisms from C to D in Ho(splCat).
The main drawback of the Dwyer-Hirschhorn-Kan closed model structure
is that the cofibrant replacement Cc ! C is not compatible with direct prod-
uct. Thus one does not obtain (in any direct way) an internal Hom__(C; D).
This internal Hom__ will be crucial for the arguments in the present paper.
In [30] is given a closed model structure for n-categories. This is essen-
tially the same problem as for Segal categories, and in [32], the closed model
structure for Segal categories was announced with the statement that the
proof is the same as in [30]. A complete proof was written up in [16]. This
closed model structure yields as underlying homotopy category Ho(SegCat ),
and it is "internal" i.e. admits a homotopically correct internal Hom__. We
shall use this structure in the present paper.
Before getting to a more detailed description, we note that Rezk con-
structs a closed model structure for what he calls complete Segal spaces in
[25]. Rezk's closed model structure again yields as underlying homotopy cat-
egory a category which is equivalent to Ho(SegCat ) ~=Ho(SegCat ) (this fact
follows immediately from the statements in [25] plus the strictification result
of Dwyer-Kan-Smith). And Rezk's closed model structure is "internal", in
other words it can be used to calculate Hom__(C; D). Thus it should be pos-
sible to write the present paper using Rezk's structure rather than my own.
The obvious conjecture is that Rezk's structure and my own are Quillen-
equivalent. We don't prove this here, but it is probably an easy consequence
of everything that is said in Rezk's preprints and my own_the only problem
being to digest all of that!
We should also at this point mention another approach, which is the
"homotopy-coherent" approach of Cordier and Porter [7]. They define a sim-
plicial category Coh(C; D) for any two simplicial categories C and D. This
should be equivalent to the Hom__constructed in either my model category or
Rezk's model category, and again it should be possible to write the present
paper using Cordier-Porter's theory (and indeed this might be advantageous
in many places).
9
Invoking the principle that the author of a paper is allowed to choose
which approach he wants to use, we will use the closed model structure of
[30], [32] and [16], which we now describe. The first step is to define a 1-
category of Segal precats denoted SeP C. The objects are the bisimplicial
sets (denoted as above p 7! Xp= with Xp= denoting a simplicial set) such that
X0= is a discrete set which we denote C0 or ob(X). The morphisms in SeP C
are just morphisms of bisimplicial sets. This category admits all small limits
and colimits. We define the cofibrations to be the monomorphisms in this
category, in other words the injections of bisimplicial sets. It remains to be
seen how to define the weak equivalences. For this, note that the category
SegCat is a subcategory of SeP C. The main step (we refer to [30], [32] and
[16] for the details of which) is an essentially unique "projection functor"
SeCat : SeP C ! SegCat SeP C;
together with a natural transformation jX : X ! SeCat(X), such that jX
is an equivalence if X is already a Segal category. This is a variant of the
well-known notion of "monad" in category theory, a variant which uses the
notion of equivalence (rather than isomorphism) in the target subcategory
SegCat. We think of SeCat(X) as being the Segal category generated by
the "generators and relations" X. In [32] the operation X 7! SeCat(X) is
analyzed explicitly and shown to have good effectivity properties.
Now we say that a morphism X ! Y is a weak equivalence if the resulting
morphism of Segal categories SeCat(X) ! SeCat(Y ) is an equivalence in the
sense explained above. This gives rise to the notion of trivial cofibration (a
cofibration which is a weak equivalence) and hence to the notion of fibration
(a morphism which satisfies lifting for all trivial cofibrations). It is shown *
*in
[30] and [16] that SeP C with these three classes of morphisms is a cofibrantly
generated closed model category. One thing to note is that the fibrant objects
of SeP C are themselves Segal categories, i.e.
SeP Cf SegCat :
It follows that
Ho(SeP C) ~=Ho(SeP Cf) ~=Ho(SegCat ):
The closed model category SeP C is "internal", see [30] and [16]. This
basically means that the cartesian product is a monoidal structure in the
10
sense of Hovey et al.. The effect of this property is that we have a notion of
internal Hom__ in SeP C. This is defined by the adjunction property that for
any Segal precat E, a morphism
E ! Hom__(A; B)
is the same thing as a morphism (in SeP C)
A x E ! B:
Now if B is a fibrant Segal category (i.e. a fibrant object in SeP C) then
for A any Segal precat, Hom__(A; B) is again a fibrant Segal category. In the
case where the second variable is fibrant, formation of the internal Hom__ is
compatible with weak equivalences in both variables. We will make heavy use
of this internal Hom__, bearing in mind that whenever it is used, the second
variable has to be made fibrant.
The above discussion leads to the notion of natural transformation be-
tween two functors of Segal categories. If A and B are Segal categories (with
B assumed to be fibrant) and if f; g : A ! B are morphisms, a natural
transformation from f to g is a vertex of the simplicial set
j 2 Hom__(A; B)1=(f; g):
In general for a Segal category C, a vertex of C1=(x; y) is the same thing
as a morphism I ! C (where I is the 1-category with two objects 0; 1 and
one arrow 0 ! 1) such that 0 goes to x and 1 goes to y. Apply this with
C = Hom__(A; B). We get that a natural transformation from f to g is the
same thing as a morphism
j : A x I ! B
such that j|Ax0 = f and j|Ax1 = g.
The internal Hom__ is used in [16] (following the same idea in the case of
n-categories in [30]) to define the Segal 2-category 1SeCAT of all Segal cate-
gories. This has for objects the fibrant Segal categories, and between two ob-
jects A; B one takes as Segal category of morphisms the internal Hom__(A; B).
We get a strict category enriched over fibrant Segal categories, which yields
a Segal 2-category. We refer to [30] and [16] for more details; this will not
be used in the remainder of the present paper.
11
We now indicate a sketch of how one should obtain the relationship be-
tween the above closed model category and Rezk's closed model category [25]
of complete Segal spaces which we shall denote RC for the present discus-
sion. If A is a Segal category, let rf(A) be a Reedy-fibrant replacement of A
as bisimplicial set. Then rf(A) is a Segal space in Rezk's terminology. Now
Rezk has a construction which replaces a Segal space by a complete Segal
space, which we will denote by crf(A). This gives a functor going from the
category of Segal categories to the category of complete Segal spaces. It
descends to the Gabriel-Zisman (or even Dwyer-Kan) localizations where we
divide out by equivalences (Rezk states that his construction takes Dwyer-
Kan equivalences of Segal spaces, to equivalences of complete Segal spaces).
In the other direction, given a complete Segal space X, we can discretize the
space of objects and chop up the other spaces accordingly (in the minimal
way so that the transition morphisms remain continuous). This yields a Segal
category. Again, this construction takes equivalences to equivalences. Thus
we obtain an equivalence of 1-categories between the homotopy category of
Segal categories, and the homotopy category of complete Segal spaces:
Ho(SegCat ) ~=Ho(SeP C) ~=Ho(RC):
Furthermore, on the level of Dwyer-Kan localizations we obtain an equiva-
lence of simplicial categories
L(SeP C) ~=L(RC):
Technically speaking, there is probably some remaining verification to be
done here, for example verifying that the two constructions are really inverses.
It would also be nice to set up a Quillen equivalence between the two model
categories, and to verify that the equivalences are compatible with internal
Hom__.
This last compatibility is already obtained on a homotopy-theoretic level
in the following way: it was observed (e.g. in [16]) that if a closed model
category M is "internal", then its Dwyer-Kan localization L(M) is a simpli-
cial category admitting internal Hom as defined in an appropriate way. In
this case, the internal Hom__(X; Y ) (for X; Y 2 L(M)) may be characterized
in a way which is internal to L(M). This applies both to SeP C and to
Rezk's closed model category RC. Since the two localizations L(SeP C) and
L(RC) are equivalent (by the argument sketched above), this shows that the
12
internal Hom__(X; Y ) are equivalent in L(SeP C) and L(RC). Another way
to recast this remark is to point out that, applying the result of Dwyer-Kan-
Smith [11] we obtain an equivalence with the Dwyer-Kan localization of the
Dwyer-Hirschhorn-Kan model category (we denote the latter by DHK)
L(DHK) ~=L(splCat) ~=L(SegCat ) ~=L(SeP C) ~=L(RC);
and existence of the internal closed model categories SeP C and Rezk's RC
can be viewed as ways of proving that the simplicial category L(DHK)
admits an internal Hom__.
We close this subsection on a slightly more technical note. In many places,
the notation introduced in [31] is crucial for correctly manipulating Segal
categories in our method. We refer to there (or to any of a number of my
more recent preprints where this notation is used) for details and examples.
A rapid overview would say that if E is a simplicial set then we obtain a Segal
precat (E) having two objects denoted 0; 1, and having E as simplicial set
of morphisms from 0 to 1. In the case E = * we recover (*) = I, the 1-
category with objects 0 and 1 and a single morphism 0 ! 1. This has a sort
of universal property: for any Segal precat A, a morphism E ! A1=(x; y) is
the same thing as a morphism
(E) ! A
sending 0 to x and 1 to y.
More generally if E; F are simplicial sets then we obtain 2(E; F ) which
has objects 0; 1; 2 and E as morphisms from 0 to 1; F as morphisms from 1
to 2; and E x F as morphisms from 0 to 2. This latter is useful for dividing
up a square into two triangles: one has the pushout formula
(E) x (F ) ~=2(E; F ) [(ExF) 2(F; E):
Finally, the existence of weak compositions is manifested in the statement
that the inclusion
(E) [{1}(F ) ! 2(E; F )
is a trivial cofibration.
Simplicial sets and cartesian families
13
Let S denote the simplicial category of all fibrant simplicial sets. It has
for objects the fibrant simplicial sets K, and for simplicial Hom sets the
internal Hom__(K; L) of simplicial sets.
Unfortunately, S is not fibrant as a Segal category. Thus we must fix
a fibrant replacement S ! S0 (i.e. an equivalence of Segal categories with
S0 fibrant). Note here that S0 cannot be a strict simplicial category. This
fibrant replacement is a source of most of the technical difficulties which were
encountered in [31] and [16]. The best way to get around these problems, at
least in the context of the theory we are exposing here, is the canonical fibra*
*nt
replacement defined using the notion of "cartesian family" in [33]. This was
constructed in the context of n-categories, giving a fibrant replacement for
the n + 1-category nCAT of all n-categories. We describe here the variant
for obtaining a fibrant replacement for S (note that in the notation of [16],
a simplicial set is a Segal 0-category and S = 0SeCAT ; the variant we are
about to describe is obtained from the discussion in [33] by substituting
"0Se" for "n").
For ease of use in the rest of the paper, we consider "contravariant" carte-
sian families; these will correspond to functors Ao ! S0, and this constitutes
a change with respect to [33] where "covariant" cartesian families were con-
sidered.
Suppose A is a Segal category, considered as a bisimplicial set. A (con-
travariant) precartesian family (of simplicial sets) over A is a morphism of
bisimplicial sets
F ! A
satisfying the "cartesian property" which we now explain. We first establish
some notations: Fp= is the simplicial set obtained by putting p in the first
bisimplicial variable; thus Fp= ! Ap=. For objects x0; : :;:xp 2 ob(A), we
denote by
Fp=(x0; : :;:xp)
the inverse image of Ap=(x0; : :;:xp). It is also the inverse image of (x0; : :*
*;:xp)
under the map Fp= ! A0 x : :x:A0. We do not make the assumption that
F0= is a discrete set, and indeed for x 2 ob(A) the simplicial set F0=(x) is
exactly the one which is considered to be parametrized by the object x. We
have a map of simplicial sets
Fp=(x0; : :;:xp) ! Ap=(x0; : :;:xp) x F0=(xp)
14
given by the projection F ! A and the structural map for F with respect to
the arrow 0 ! p in corresponding to the last vertex. The "(contravariant)
cartesian condition" is that the above map should be a weak equivalence of
simplicial sets. Note that the "covariant cartesian condition" would be the
same but using the structural map to F0=(x0) rather than to F0=(xp).
A cartesian family corresponds to a weak functor Ao ! S in much the
same way as the Segal condition encodes the notion of weak category: the
action of the space of morphisms A1=(x; y) is given by the diagram
~=
F0=(y) x A1=(x; y) F1=(x; y) ! F0=(x);
the second morphism being the structural morphism for the map 0 ! 1 in
corresponding to the first vertex. The higher Fp= encode homotopy-coherent
associativity of this action.
In [33] the notion of cartesian family is defined by saying that it is a
precartesian family which satisfies a certain quasi-fibrant condition. This
quasi-fibrant condition (which is analogous to the classical notion of quasi-
fibration and is somewhat similar to Rezk's notion of "sharp map" [26]) is
designed to guarantee that cartesian families over Segal precats can be glued
together. This glueing property ensures representability of the associated
functor of Segal precats, and allows us to define a Segal category S0 with
the property that a morphism Ao ! S0 is exactly the same thing as a con-
travariant cartesian family over A. In [33] it is shown that there is a natural
morphism S ! S0, that this is an equivalence of Segal categories, and that S0
is fibrant; thus S0 is a canonical fibrant replacement for S. This fact means
that "weak families" of simplicial sets parametrized by a Segal category A,
i.e. weak functors Ao ! S, may be viewed as cartesian families. The proofs
in [33] are given in the context of n-categories but the same work in the Segal
category context (or more generally for Segal n-categories [16]).
In practice, there is no essential difference between the notion of pre-
cartesian family and the notion of cartesian family. Generally speaking, the
natural constructions that one can make are precartesian but not cartesian;
then one should make a fibrant replacement (which is consequently quasi-
fibrant) to get a cartesian family. We will systematically ignore this point
in the remainder of the paper, and speak only of precartesian families but
use the terminology "cartesian family". The reader should note that in order
to be precise, one must make fibrant replacements sometimes. Since these
15
are essentially unique (i.e. unique up to coherent homotopy) this doesn't
pose any homotopy-coherence problems. Of course one should check that
the previous phrase is true ().
Segal categories of presheaves
The fundamental construction underlying SGA 4 [1] is the Yoneda em-
bedding of a category into the category of presheaves over itself. We have
the same thing for Segal categories. For this section I should acknowledge
the suggestion of A. Hirschowitz who pointed out that it would be interesting
to look at the notion of representable functor in the context of n-categories.
And J. Tapia who pointed out to me that this was the fundamental thing in
SGA 4; he is working on an altogether different generalization of it.
Let S be the simplicial category of fibrant simplicial sets, and let S0 be
its replacement by an equivalent fibrant Segal category. If A is any Segal
category, put
Ab:= Hom__(Ao; S0):
Recall that Ao is the "opposite" Segal category, with the same objects as A
and obtained by putting
Aop=(x0; : :;:xp) := Ap=(xp; : :;:x0):
The first step is that we would like to construct a natural morphism
hA : A ! bA:
In view of the definition of the internal Hom__(Ao; S0) (see above), constructi*
*ng
the morphism hA is equivalent to constructing the "arrow family"
ArrA : Ao x A ! S0:
We give two discussions of the construction of ArrA . Both of these construc-
tions were done for n-categories in [33]. We should also note that in the
simplicial case, the "arrow family" is certainly very classical; among other
things it occurs in Cordier-Porter [7].
The easy case is when A is a strict simplicial category with fibrant sim-
plicial Hom sets. In this case, the formula
ArrA (x; y) := A1=(x; y)
16
defines in an obvious way a morphism of strict simplicial categories
Ao x A ! S:
There is a canonical fibrant replacement within the category of simplicial
sets, compatible with direct product (namely taking the singular complex of
the topological realization of a simplicial set), so we obtain a way of replaci*
*ng
any simplicial category by one whose simplicial Hom sets are fibrant. This
can be composed with the Dwyer-Kan strictification described above, so if A
is any Segal category then we can replace A by an equivalent strict simplicial
category with fibrant Hom spaces and then apply the construction of ArrA
given in the present paragraph. Thus this construction technically speaking
suffices in order to define the morphism hA and the reader wishing to avoid
technicalities may skip the subsequent paragraph.
The more complicated case is to treat directly the case where A is a Segal
category. This has the advantage of avoiding a number of equivalences used
in the previous paragraph; however it makes use of the notion of "cartesian
family" described above (and for which the reader must refer to [33]). We
choose for fibrant replacement that S0 which was obtained using the notion
of cartesian family. Thus, in order to define the morphism
ArrA : Ao x A ! S0;
we have to define a contravariant cartesian family over A x Ao. We do this
by first defining a natural precartesian family F, then replacing by a fibrant
replacement F0. The precartesian family F has the very simple formula
Fp=((x0; y0); : :;:(xp; yp)) := A2p+1=(x0; : :;:xp; yp; : :;:y0):
Note that
(A x Ao)p=((x0; y0); : :;:(xp; yp)) = Ap=(x0; : :;:xp) x Ap=(yp; : :;:y0):
The Segal condition for A implies that the map
Fp=((x0; y0); : :;:(xp; yp)) ! Ap=(x0; : :;:xp) x Ap=(yp; : :;:y0) x A1=(xp; y*
*p)
is an equivalence. This is the cartesian condition for F, so F is a precartesian
family. The morphism ArrA is defined by choosing a fibrant replacement F0
for F.
17
In the above discussion, the Segal category A must be small. For a
"big" Segal category (by which we always mean one in which the objects
can form a class, but in which the Ap=(x0; : :;:xp) are still sets), it doesn't
seem to be reasonable to define Ab. However, we will run across the following
intermediate situation: suppose
C ! A
is a morphism from a small Segal category C to a "big" Segal category A.
Then we still obtain a morphism
i : A ! bC:
Define this by exhausting A by small Segal categories Afi, and on each of
these define i as the composition
Afi! bAfi! bC:
Here is the statement of our main "Yoneda-type" theorem.
Theorem 2 If A is any small Segal category then the morphism
hA : A ! bA
is fully faithful.
Proof: We prove the following more general statement: if G 2 bAand if x 2 A
then there is a natural equivalence
Ab1=(hA (x); G) ~=G(x)
(which is required to be compatible with hA , see below).
We first point out how to go from here to the statement of the theorem:
for x; y 2 ob(A), apply the above to G := hA (y). We get
Ab1=(hA (x); hA (y)) ~=hA (y);
but hA (y) ~=A1=(x; y) by construction (recall that hA comes from the arrow
family). Thus
bA1=(hA (x); hA (y)) ~=A1=(x; y):
18
This equivalence will be compatible with the morphism hA : A ! Ab, so it
shows that hA is fully faithful.
Now we show how to prove the more general statement. We can view G
as being a cartesian family over A. In order to define a morphism
G(x) ! bA1=(hA (x); G)
we need to define a morphism
(G(x)) ! bA
or equivalently a morphism
[(G(x)) x A]o ! S0
restricting over 0 x Ao to G, and restricting over 0 x Ao to G. This latter
morphism corresponds to a contravariant cartesian family
F ! (G(x)) x A;
with F restricting as above to ArrA (-; x) and G on the endpoints. In order
to define the family F, given that we already know its restrictions to 0 x A
and 1 x A, it suffices to define
Fp=(u0; : :;:ua; v0; : :;:vb) := Gp+1=(u0; : :;:ua; v0; : :;:vb; x):
for a; b 0 and a + b + 1 = p. Here ui; vj 2 ob(A) and the variables
ui indicate objects considered in 0 x A; the variables vj indicate objects
considered in 1 x A. The simplicial restriction maps are obtained by those of
G whenever the sequence of objects still contains an object of 0xA, otherwise
it is obtained by composing with the morphism G ! A. The structural
morphism to (G(x)) x A will be seen in the upcoming verification. We
check the cartesian condition:
Gp+1=(u0; : :;:ua; v0; : :;:vb; x) ~=Ap+1=(u0; : :;:ua; v0; : :;:vb; x) x G(*
*x)
~= Ap=(u0; : :;:ua; v0; : :;:vb) x hA (x)(vb) x G(x)
~=[(G(x)) x A]p=(u0; : :;:ua; v0; : :;:vb) x hA (x)(vb):
19
Thus F is a precartesian family. As said previously, we are ignoring the
difference between cartesian and precartesian families. Thus we have defined
our morphism
G(x) ! bA1=(hA (x); G):
The next step is to define a morphism in the other direction:
Ab1=(hA (x); G) ! G(x):
For this, note that the restriction along {x} ! A gives a morphism
Ab! S0:
We obtain a morphism
bA1=(hA (x); G) ! S01=(hA (x)(x); G(x)):
On the other hand, the identity element gives a morphism * ! hA (x)(x) =
A1=(x; x), and "composing" with this gives
S01=(hA (x)(x); G(x)) ! S01=(*; G(x)) ~=G(x):
As usual this "composition" requires inverting some equivalences which come
up in the notion of Segal category. We don't write out the details of that here
(although this neglect doesn't actually merit a ). We get our morphism
Ab1=(hA (x); G) ! G(x):
To complete the proof, we have to say that these two morphisms are in-
verses up to homotopy. In one direction it is basically easy (modulo struggling
with the details of the weak compositions everywhere) that the composition
G(x) ! bA1=(hA (x); G) ! G(x)
is homotopic to the identity of G(x). For this direction, one way to proceed
would be to note that, for an appropriate Dwyer-Kan-Smith strictification
and then Dwyer-Hirschhorn-Kan cofibrant replacement, A can be assumed
to be a strict simplicial category and G a strict diagram A ! S. In this setup
we obtain (by just simplicially-enriching the easy discussion for 1-categories)
a sequence
G(x) ! Hom(A; S)1=(hA (x); G) ! G(x)
20
whose composition is the identity of G(x) on-the-nose. In this formula, the
simplicial category Hom(A; S) is not necessarily the "right" one but it maps
into Ab, and this is sufficient to check that the above composition that we are
interested in, is homotopic to the identity. Note that the morphism in the
strictified setup is homotopic to the morphism we have constructed in the
original weak situation.
It is somewhat more problematic to see why the composition
Ab1=(hA (x); G) ! G(x) ! bA1=(hA (x); G)
is the identity. This is because it is not clear (to me at least) whether all
of Ab1=(hA (x); G) can in some way_and after appropriate replacements of
A and G_be supposed to consist entirely of strict natural transformations
between strict diagrams.
Instead, we again make a more general statement, namely the naturality
of the morphism
G(x) ! bA1=(hA (x); G)
in the variable G. This says that if F and G are objects in Ab then the
diagram
F (x) x bA1=(F; G) ! G(x)
# #
bA1=(hA (x); F ) x bA1=(F; G)! bA1=(hA (x); G)
commutes up to homotopy.
For this statement and its proof, we first take note of the following remark:
if U and V are diagrams in bAthen a morphism E ! bA1=(U; V ) is by definition
a morphism
(E) ! Hom__(Ao; S0)
or equivalently a contravariant cartesian family over
A x (E)
restricting to V on A x 0 and to U on A x 1 (in this last reduction we use the
natural isomorphism (E)o ~=(E) which interchanges 0 and 1). It is easy
to see that a precartesian family over A x (E), with restrictions U and V ,
is exactly the same thing as a precartesian family over AxI with restrictions
V on A x 0, and U x E on A x 1.
21
With this remark in mind, we can return to the above diagram and (ap-
plying the remark to the vertical arrows) note that it is the same thing as
giving a diagram in Abof the form
hA (x) x F (x) x bA1=(F; G)! hA (x) x G(x)
# #
F x bA1=(F; G) ! G:
Again applying the remark of the previous paragraph (but to the horizontal
arrows this time ) with E = bA1=(F; G) we get that the above diagram is the
same thing as a precartesian family over
A x I x (E)
whose restrictions to the corners are respectively:
A x 0 x 0 : G
A x 0 x 1 : F
A x 1 x 0 : hA (x) x G(x)
A x 1 x 1 : hA (x) x F (x):
The restrictions to the edges A x I x 0 and A x I x 1 should be the cartesian
families constructed above for G and F respectively; the restrictions to Ax0x
(E) and Ax1x(E) should be the tautological families. The construction
of this cartesian family is done in the same way as the previous construction
for the morphism hA (x) x G(x) ! G, but starting with the tautological
cartesian family over A x (E) corresponding to the morphism F x E ! G.
We leave it to the reader to write down the details (). This gives the
homotopy-commutative diagram of naturality.
Let's now look at how to go from the above naturality statement to the
fact that our composition of morphisms is homotopic to the identity. For
this, apply the naturality statement with F = hA (x) and G as given. Then
the naturality statement is a diagram
A1=(x; x) x bA1=(hA (x); G) ! G(x)
# #
bA1=(hA (x); hA (x)) x bA1=(hA (x);!G)Ab1=(hA (x); G):
22
Plugging in the identity from x to x we get a diagram
Ab1=(hA (x); G) ! G(x)
# #
{1hA(x)} x bA1=(hA (x); G)! bA1=(hA (x); G):
The composition along the top followed by the right is the morphism we are
interested in; the other composition is the identity. Therefore, homotopy-
commutativity of the square shows that the composition in question
Ab1=(hA (x); G) ! G(x) ! bA1=(hA (x); G)
is the identity. This completes the proof of the theorem. ===
A lemma which will be used in several places below (and indeed, which
is at the origin of the statement of Theorem 14) is the following calculation
of bC. Recall that the Heller model category of simplicial presheaves [14] was
the precursor to the now standard Joyal-Jardine model category [20] [19];
Heller's result was the special case of a category with trivial Grothendieck
topology (which is the case we need here).
Lemma 3 Suppose C is a small 1-category. Then Cb is equivalent to L(M)
where M = SC is the Heller model category of simplicial presheaves over C.
Proof: This is a special case of Theoreme 12.1 of [16]. To obtain the special
case, replace n by 0 in the statement of that theorem, and note that (in the
notation of [16]) a "Segal 0-category" is the same thing as a simplicial set.
One should also refer to Theoreme 11.11 of the same reference.
This statement is also given by Rezk in [25], and the proof Rezk gives
uses some results of Dwyer-Kan. (The results of Dwyer-Kan were of course
much prior to [16]). ===
Adjoint functors
There is a notion of adjunction between functors of simplicial categories
or Segal categories, which is a direct generalization of the classical notion of
adjunction of functors. In making this generalization, it is best to specify
only one of the adjunction transformations and impose the condition that
23
it induces an equivalence between the appropriate simplicial Hom sets. If
one tried to specify both of the classical adjunction transformations, this
would run into the homotopy-coherence problem that it would be necessary
(in order to obtain a well-behaved notion) to specify higher order homotopy
coherencies.
The basic historical reference for this section is Cordier and Porter [7],
who treat the case of adjunctions of homotopy-coherent functors between
simplicial categories. This should be completely equivalent to what we say
here. Furthermore, refering to their approach might allow easy removal of
the many which appear in the following discussion.
Suppose A; B are Segal categories (which we suppose fibrant) and suppose
F : A ! B and G : B ! A are functors. Suppose j : 1B ! F G is a natural
transformation; technically speaking, this means
j 2 Hom__(B; B)1=(1B ; F G);
which in turn means that j is a morphism of Segal categories
B x I ! B
restricting to 1B on B x0 and to F G on B x1. Here as throughout, I denotes
the category with two objects 0; 1 and a single (non-identity) morphism 0 !
1. Generally we consider I as a Segal category.
We obtain the following morphisms:
(Gox1)* o 0
Hom__(Ao x A; S0) ! Hom__(B x A; S );
and
(1xF)* o 0
Hom__(Bo x B; S0) ! Hom__(B x A; S ):
In particular, we have two elements
(Go x 1)*(ArrA ); (1 x F )*(ArrB ) 2 Hom__(Bo x A; S0):
These represent respectively
(x; y) 7! A1=(Gx; y)
and
(x; y) 7! B1=(x; F y):
24
In the same way as for the classical 1-category case, the natural transforma-
tion j gives rise to a morphism adj(j) in the Segal category Hom__(BoxA; S0)
relating the above two elements; this morphism arises as a morphism I xBox
A ! S0 restricting to (Gox 1)*(ArrA ) over 0 2 I and to (1 x F )*(ArrB ) over
1 2 I.
In the case where F and G are strict morphisms of strict simplicial cate-
gories and j is a strict natural transformation between them, the adjunction
morphism adj (j) is easy to describe; it is just given by exactly the same
formula as in the classical case.
The paragraph which follows contains a more technical description of
how to construct the morphism refered to above, in our framework of Segal
categories. This construction in turn relies on the explicit construction of
a certain cartesian family F which is left to the intrepid reader. The less
intrepid who are willing to accept that everything works as usual, may skip
the following paragraph.
Note that (B x I)o ~=Bo x I using Io ~=I (an involution which switches
0 and 1). Look in Hom__(I x Bo x A; S0) at
(j x F )*(ArrB ):
Over 0 2 I this restricts to (1B x F )*(ArrB ). Over 1 this restricts to
((F G)o x F )*(ArrB ):
Essentially speaking, this means that we have a natural transformation
B1=(F Gx; F y) ! B1=(x; F y):
Note that (F G)o x F is the composition
ox1 o FoxF o
Bo x A G! A x A ! B x B:
Thus
((F G)o x F )*(ArrB ) = (Go x 1)*((F ox F )*ArrB ):
The morphism of functoriality for F is a natural transformation
A1=(x; z) ! B1=(F x; F z);
25
which translates in our language to a morphism
F : I x Ao x A ! S0;
restricting over 0 to ArrA , and over 1 to (F oxF )*ArrB . Technically speaking,
F needs to be constructed as a cartesian family (recall that ArrA and ArrB
are themselves cartesian families). We leave this construction to the reader
(). Now look at
(1 x Go x 1)*(F ) : I x Bo x A ! S0:
Heuristically it is the natural transformation
A1=(Gx; y) ! B1=(F Gx; F y):
We can "compose" this with the previous transformation to obtain a natural
transformation
A1=(Gx; y) ! B1=(F Gx; F y) ! B1=(x; F y):
Technically speaking, this means using the above two morphisms to give the
01 and 12 edges which can be filled in to a morphism
2(*; *) x Bo x A ! S0;
the third (02) edge of which is a morphism
adj(j) : I x Bo x A ! S0
restricting on the endpoints to (1 x G)*(ArrA ) and (F ox 1)*(ArrB ) respec-
tively. This is the technical description of how we get from the natural
transformation j : 1B ! F G to a natural transformation
adj (j)(x; y) : A1=(Gx; y) ! B1=(x; F y):
Now getting back to our discussion of adjoint functors, we say that j is
an adjunction between F and G if the natural transformation adj (j) is an
equivalence between (Go x 1)*(ArrA ) and (1 x F )*(ArrB ) (by "equivalence"
here we mean internal equivalence in the Segal category Hom__(Bo x A; S0)).
26
Remark: In order to check the adjunction condition, it suffices to check
that for every pair of objects x 2 ob(B) and y 2 ob(A), the morphism
adj (j)(x; y) : A1=(Gx; y) ! B1=(x; F y)
is a weak equivalence of simplicial sets. This is a general fact about natural
transformations between functors of Segal categories: being a levelwise equiv-
alence implies being an equivalence. It is Corollary 2.5.8 of [31] (which was
stated for n-categories but which works the same way for Segal categories);
a similar early result was shown in [28].
Lemma 4 Suppose F : A ! B, G : B ! A are functors of fibrant Segal
categories, and j : B x I ! B is a natural transformation 1B ! F G which
is an adjunction. Suppose that C is another Segal category. Let FC ; GC be
the induced functors between Hom__(C; A) and Hom__(C; B), and let jC denote
the functor
Hom__(C; B) x I ! Hom__(C; B)
defined by the composed morphism
j
Hom__(C; B) x I x C = C x Hom__(C; B) x I ! B x I ! B:
Then jC is a natural transformation
1Hom_(C;B)! FC GC ;
which is an adjunction between FC and GC .
Proof: After the details of how to define everything, we will end up with a
natural transformation
adj (jC )(u; v) : Hom__(C; A)1=(Gu; v) ! Hom__(C; B)1=(u; F v):
According to the previous remark, we have to show that this is an equivalence
for every u : C ! B and v : C ! A. To check this, note that
Hom__(C; A)1=(Gu; v)
is calculated by a homotopy-coherence calculation using the
A1=(Gu(c); v(c0))
27
for c; c0 in C (something like a coend, see Cordier-Porter [7]). Similarly,
Hom__(C; B)1=(u; F v)
is calculated by the samehomotopy-coherence calculation using
B1=(u(c); F v(c0)):
The fact that the adjunction induces an equivalence
A1=(Gu(c); v(c0)) ~=B1=(u(c); F v(c0))
for any c; c02 ob(C), implies that the two calculations give the same answer;
thus adj(jC )(u; v) is an equivalence. This completes the proof, but a number
of details need to be followed through (). ===
Construction: We can apply this to the case where C = A, where u = F
and where v = 1A . We obtain an equivalence
~=
adj (jA )(F; 1A ) : Hom__(A; A)1=(GF; 1A ) ! Hom__(A; B)1=(F; F ):
In particular, there is an essentially unique element
i 2 Hom__(A; A)1=(GF; 1A )
which goes to 1F under the above equivalence. (To be more precise, what
is essentially unique_i.e. parametrized by a contractible space_is the pair
consisting of i plus a path between the image of i and 1F ).
We leave it to the reader () to check that i is an adjunction morphism
going in the other direction between F and G (reversing the appropriate
things in the above discussion/definition). We will use this construction of
the other adjunction morphism, at some point in the argument below.
Lemma 5 With the above notations, the composed morphisms
jF() F(i)
F ! F GF ! F
and
G(j) iG()
G ! GF G ! G
are homotopic to the identity natural transformations of F and G respectively.
28
We don't give a proof of this here ().
For the above places where details are left out in our discussion of adjunc-
tion, the necessary arguments can probably be obtained from Cordier-Porter
[7].
Homotopy colimits
It would be impossible to give a complete list of references to everything
pertaining to homotopy colimits (and limits). A non-exhaustive list includes
[6] [37] [38] [12] [15] [8] . . . .
Recall the notion of homotopy colimit in a simplicial category or Segal
category. If A is a Segal category (which we suppose fibrant) and if J is a
small Segal category, then we can form the category of diagrams Hom__(J; A).
This is the "homotopically correct" one if A is fibrant. There is a morphism
cJ : A ! Hom__(J; A) induced by the projection J ! *; thus cJ(x) is the
constant diagram with values x. Suppose F : J ! A is a diagram. If x is
an object of A and f : F ! cJ(x) is a morphism, then we say that x is the
homotopy colimit of the diagram F and write
(x; f) = colimJ(F )
(or just x = colimJ(F ) if there is no confusion about f), if for any object
y of A, the morphism of "composition with f", which can be seen as the
composition
A1=(x; y) ! Hom__(J; A)1=(cJ(x); cJ(y)) ! Hom__(J; A)1=(F; cJ(y));
is an equivalence of simplicial sets. Here the second morphism is essentially
well-defined as "composition" in the Segal category Hom__(J; A), see above.
Note that we never speak of actual limits or colimits in a simplicial cate-
gory, so the notation colim means homotopy colimit. If we forget to include
the qualifier "homotopy" in front of the word "colimit" in the text below, the
reader will insert it. However, for homotopy limits or colimits of simplicial
sets, we keep the classical notation holim or hocolim so as not to confuse
these with 1-limits or 1-colimits in the 1-category of simplicial sets.
Note that
Hom__(J; A)1=(F; cJ(y)) ~=holimj2JA1=(F (j); y)
29
where the holim on the right is the homotopy limit of simplicial sets. Thus,
we can rewrite the condition for being a homotopy colimit as saying that
for any object y, the composition morphism with f gives an equivalence of
simplicial sets
~=
A1=(x; y) ! holimj2JA1=(F (j); y):
In this sense, the homotopy colimit is in a certain sense dual to the homotopy
limit on the level of the simplicial Hom-sets of A (i.e. the A1=(-; -)). In
particular, we can verify certain formulae for homotopy colimits by verify-
ing the dual formulae for homotopy limits of simplicial sets. For example
it follows from [38] that homotopy colimits commute with other homotopy
colimits.
There is an analogous definition of homotopy limit which we leave to the
reader to write down in our current language.
We now remark that colimits over a Segal category J can be transformed
into colimits over a 1-category J0; thus, in the above discussion, there would
be no loss of generality in considering the indexing category J to be a 1-
category. This remark follows from the following statement, which we isolate
as a lemma because it will also be used in the proof of the main theorem.
Lemma 6 If C is a Segal category (which we assume fibrant), then there
is a 1-category D and a morphism D ! C such that for any fibrant Segal
category A, the induced morphism
Hom__(C; A) ! Hom__(D; A)
is fully faithful. Furthermore, we can assume that D is a "Reedy poset", i.e.
a poset with a Reedy structure such that the Reedy function is compatible with
the ordering.
Proof: It suffices to construct a 1-category D and a subcategory W D
with a morphism D ! C sending the arrows of W to equivalences in C, such
that this morphism induces an equivalence
~=
L(D; W ) ! C:
To see that this suffices, recall from ([16] Proposition 8.6-Corollaire 8.9)
which in turn comes from ([31] Theorem 2.5.1), that
Hom__(L(D; W ); A) Hom__(D; A)
30
is the saturated full Segal subcategory consisting of the morphisms D ! A
which send the morphisms of W to equivalences in A. (In the case A = S0
this result is basically the same as the result of Dwyer-Kan in [10]).
Now for the construction of D and W , we refer to [16] Lemmes 16.1,
16.2. These basically say that one can construct D and W using barycentric
subdivision and the Grothendieck construction in the style of Thomason. ===
Caution: One must be careful in combining this lemma with the Yoneda
result of Theorem 2. In effect, one obtains (in the situation of the lemma
with A = S0) a sequence of three morphisms
D ! C ! bC! cD:
The last two morphisms are fully faithful. The Yoneda morphism D ! cD
is also fully faithful. However, the composition of these three morphisms
is not in general the Yoneda morphism of D, so one cannot conclude that
D ! C must be fully faithful (which visibly it isn't, in general). In fact,
the composition of the above three morphisms is homotopic to the Yoneda
morphism for D if and only if the original morphism D ! C is fully faithful.
Corollary 7 If J is a Segal category (which we may assume fibrant) and if
F : J ! A is a morphism to another Segal category, then there is a strict
1-category (which we may assume to be a Reedy poset) J0 and a morphism
g : J0 ! F such that if colimAJ0F O g exists then colimAJF exists and the two
colimits are equivalent.
Proof: Choose g : J0 ! J (with J0 a Reedy poset) so that
Hom__(J; S0) ! Hom__(J0; S0)
is fully faithful. Now note that if G : J ! S0 is a simplicial set diagram over
J, we have 0
holimSJG ~=Hom__(J; S0)1=(*; G):
The same holds for J0. Therefore the fully faithful property implies that
0 ~= S0
holimSJG ! holimJ0G O g:
Now the fact that homotopy colimits in A are dual to homotopy limits of the
simplicial Hom sets, implies that for any diagram F : J ! A,
colimAJ0F O g ! colimAJF
31
is an equivalence, and in fact existence of the first colimit implies existence
of the second one. For the statement about existence we use the fully faithful
property of the lemma (for target A this time) to say that
Hom__(J; A)1=(F; cJ(colimAJ0F O g)) ! Hom__(J0; A)1=(F O g; cJ0(colimAJ0F O g))
is an equivalence, so there exists a morphism of J-diagrams from F to
colimAJ0F O g restricting to the colimit morphism over J0; now we can ap-
0
plly the previous discussion about holimS to get that this morphism is a
J-colimit. ===
We say that A admits all small homotopy colimits if for any small Segal
category J and for any diagram J ! F , the homotopy colimit exists. From
the previous corollary, it suffices to check the existence of colimits over 1-
categories J which we can furthermore assume are Reedy posets.
Lemma 8 Suppose A ! B is a fully faithful morphism of Segal categories.
Suppose F : J ! A is a diagram. If
colimBJ(F )
is in A, then the natural morphism
colimBJ(F ) ! colimAJ(F )
is an equivalence.
Proof: The facts that colimBJ(F ) is in A and that the inclusion of A in B is
fully faithful imply that we have a morphism of J-diagrams in A
F ! cJ[colimBJ(F )]:
Therefore there is up to homotopy a unique morphism
colimAJ(F ) ! colimBJ(F )
whose composition with the canonical morphism of diagrams for the colimAJ,
is the above morphism. In the other direction, we have a morphism of B-
diagrams
F ! cJ[colimAJ(F )]:
32
Again we get an essentially unique morphism
colimBJ(F ) ! colimAJ(F )
(which is the morphism in the statement of the lemma). Essential uniqueness
implies that the compositions in both directions are homotopic to the identity,
thus our morphism is an equivalence. ===
Lemma 9 If C is a small Segal category, then homotopy colimits in Cbexist
and are calculated object-by-object.
Proof: According to Lemma 6, there is a small 1-category D and a morphism
D ! C such that this induces a fully faithful morphism
Cb! cD:
Furthermore, from the proof of Lemma 6, we may assume that there is a
subcategory W D such that C is equivalent to the localization L(D; W ).
This implies that Cb is the full subcategory of cD consisting of diagrams X :
Do ! S0 such that for any arrow w 2 W , X(w) is an equivalence. This
situation is identical to that of Dwyer-Kan in [10], and all of the elements
going into here are due to [10] in this case.
Next we recall that cDis equivalent to L(M) where M is the Heller closed
model category SD of simplicial presheaves over D (Lemma 3). Now, suppose
we have a diagram F : J ! Cb, which we may also consider as a diagram
in cD. We may assume that J is a Reedy poset. The argument of [16] (see
chapter 18 for example) allows us to "strictify" and assume that F is the
projection of a diagram F 0: J ! M. Furthermore we may assume that F 0
is Reedy-cofibrant in the variable J. Then
colimbDJ(F ) = colimL(M)J(F )
exists and is calculated by taking the 1-colimit of F 0in M (see the discussion
at the proof of (i) ) (ii) in Theorem 14 below). This 1-colimit is calculated
object-by-object over D (recall that M is the category of simplicial presheaves
on D). On the other hand, the Reedy cofibrant condition for F 0also holds
object-by-object. Therefore for any d 2 ob(D), the 1-colimit of F 0(j)(d) over
33
j 2 J, is also the homotopy colimit. This shows that the homotopy colimit
is calculated object-by-object, i.e.
0
colimbDJ(F )(d) = colimSj2JF (j)(d) = hocolimj2JF (j)(d)
for d 2 ob(D). On the other hand, the fact that F is a diagram in Cbmeans
that for arrows w in W , and for any j 2 J, we have that F 0(j)(w) is an
equivalence of simplicial sets. Homotopy-invariance of the 1-colimit of a
Reedy cofibrant diagram [15] implies that the arrow
colimbDJ(F )(w)
is an equivalence for any arrow w in W . It follows that
colimbDJ(F ) 2 ob(Cb):
Now by Lemma 8,
colimbCJ(F ) ~=colimbDJ(F )
including the statement that the homotopy colimit in Cb exists. Finally, we
have 0
colimbCJ(F )(d) = colimbDJ(F )(d) = colimSj2JF (j)(d):
This completes the proof. ===
Smallness and rearrangement of colimits
Recall from [15] the notion of sequential colimit. This is a colimit indexed
by an ordinal fi (where the ordered set fi is considered as a category with
morphisms going in the increasing direction) with the additional property
that if i 2 fi is a limit element then the i-th object Xi is equivalent to the
colimit of the Xj for j < i. A diagram giving rise to a sequential colimit
will be called a sequential diagram. In giving these definitions for a Segal
category A, the notion of colimit which occurs is the notion of homotopy
colimit as defined above.
A diagram or colimit is essentially sequential if it satisfies the sequen-
tial condition at sufficiently large points. In what follows we shall make no
34
distinction between sequential and essentially sequential (an essentially se-
quential diagram can be replaced by a sequential one which gives the same
colimit, by starting out with a constant diagram in low degrees).
Suppose A is a Segal category admitting all small colimits.An object
z 2 ob(A) is said to be fi-small in A if for any ordinal ffi of size fi and any
sequential diagram X : ffi ! A, the natural morphism
hocolimi2ffiA1=(z; Xi) ! A1=(z; colimAffiX)
is an equivalence. We say that z is small in A if there is a cardinal fi such
that z is fi-small in A.
Example: Lemma 9 shows that the objects of C are small in Cb. On the
other hand, every object of bCcan be expressed as a small homotopy colimit
of objects of C (see Lemma 11 below). From this it follows easily that every
object in bCis small in bC(although of course there is no bound uniform over
the class ob(C)).
We will now treat some aspects of colimits which are useful in connection
with the notion of smallness.
Suppose J is a small 1-category, and F : J ! A is a diagram. Choose
a well-ordering of the objects of J, in other words choose an ordinal fi and
an isomorphism ob(fi) ~=ob(J). We assume that fi is the first ordinal of its
cardinality. For each i 2 fi let Ji denote the full subcategory of objects < i.
Put
Xi := colimJi(F |Ji):
Then the Xi form a sequential diagram, and we have
colimJ(F ) = colimfi(Xi):
This can be proved by noting the dual property for homotopy limits of sim-
plicial sets.
We call the above expression a normalized reindexing of the colimit. The
word "normalized" refers to the condition that fi be the first ordinal of its
cardinality. With this condition, we get that each Xiis a colimit of size < |fi|
(this latter notation is the cardinality of fi).
Now we discuss another aspect of rearranging colimits. Let A be a Segal
category admitting small colimits and let C A be a small full subcategory.
For any ordinal fi we will define a full subcategory
A(< fiC) A;
35
by the following prescription: it is the smallest saturated full subcategory of
A containing C and closed under colimits of size < |fi|.
It is clear that if there is an ordinal fi0 < fi of the same cardinality, th*
*en
A(< fi0C) = A(< fiC).
The following lemma is our main statement giving a normal form for
successive colimits.
Lemma 10 Suppose fi is the first ordinal of its cardinality. There are two
cases.
(1) If fi is a limit of ordinals of strictly increasing cardinality, then
[
A(< fiC) = A(< flC):
fl ffi. Express*
* U
and V as sequential colimits
U = colimbCfiUi
and
V = colimbCfiVi;
with the Ui and Vi of size |i| < fi. The Ui are fi-small in Cbwhich implies
that, after possibly reindexing the second colimit, we can assume that the
map U ! V comes from a collection of maps Ui ! Vi.
By assumption, U ! V is an equivalence in A The fact that pre-
serves colimits means that the morphisms
colimAfi Ui ! U
and
colimAfi Vi ! V
are equivalences. The fact that fi-colimits agree in A and Cbmeans that the
morphisms
colimbCfi Ui ! U:
and
colimbCfi Vi ! V:
are equivalences. A similar argument shows that these colimits are essentially
sequential, so (by restricting our attention to big enough indices i) we may
assume that they are sequential.
Furthermore by Lemma 9 the above colimits in Cbare calculated object-
by-object. Thus, for every z 2 C the morphism
hocolimSfi( Ui)(z) ! hocolimSfi( Vi)(z)
is a weak equivalence of simplicial sets. This implies that there are subse-
quences ik and jk in fi (which are again indexed by an ordinal which we
denote even though it is isomorphic to fi), such that
Ujk(z) ! ( Vik)(z) [( Uik)(z)( Ujk)(z)
55
are equivalences. This fact about simplicial sets comes from Jardine's argu-
ment [19]. Furthermore, since fi is big with respect to the cardinality of C,
we can assume that these same subsequences work for all z 2 C. Therefore
the morphism
Ujk! ( Vik) [ Uik( Ujk)
is an equivalence in Cb.
Now apply Lemma 8 which says that since the Cb-coproduct
( Vik) [ Uik( Ujk)
is in A (because it is equivalent by the previous paragraph to Ujk) then
this is also the coproduct in A. Now the fact that commutes with colimits
means that
( Vik) [ Uik( Ujk) ~= (Vik[UikUjk):
Therefore we finally get that the morphism
Ujk! (Vik[UikUjk)
is an equivalence. Thus, the morphism
Ujk! Vik[UikUjk
is -trivial. Defining
U0k:= Ujk; Vk0:= Vik[UikUjk;
we still have U = colim U0kand V = colim Vk0, but now the morphisms
U0k! Vk0are -trivial._The U0kand Vk0have size |jk| < fi, so these_ -trivial
morphisms are in F . It follows that the morphism U ! V is in F .
The proof of (iv) ) (i)
Let N be the Heller model category of simplicial presheaves on C [14].
Thus L(N) ~=Cb (Lemma 3).
Let M be the model category with the same underlying category as N,
and the same class of cofibrations, but where a morphism is said to be a
weak equivalence if its image in Cb is -trivial. As generating set of triv-
ial cofibrations, we can take a generating set for N plus a set of cofibrant
56
representatives for our small generating set given in the hypothesis of (iv).
It is easy to see that a morphism is a fibration in M if and only if it is a
fibration in N, and if its image in Cb is a -fibration. This implies that the
given generating set indeed generates the trivial cofibrations (that is, lifting
for the given generating set is equivalent to being a fibration i.e. to lifting
for all trivial cofibrations).
We use the criterion of [16] Lemma 2.5 to obtain a closed model structure
for M (the numbers in the present paragraph refer to the conditions in that
lemma). As a historical point, note that this lemma is just a synopsis of the
techniques of Dwyer, Kan and Hirschhorn [8] [15]. Start by noting that M =
N admits small limits and colimits (0). Since M is a category of simplicial
presheaves, any small subset is adapted to the small object argument so
conditions (4) and (5) are automatic. The three for two condition (2) is
automatic in view of the definition of weak equivalence. A morphism which
satisfies lifting for all cofibrations is an equivalence in N already so it is
an equivalence in M; this gives (3). The cofibrations are the same as for
N so condition (6) comes from that of N. Condition (7), that the trivial
cofibrations are stable under coproduct and sequential colimit, comes from
the same property for -trivial morphisms in Cb. In effect, a coproduct or
sequential colimit of cofibrations, calculated in M, is a homotopy colimit (cf
[8] [15] [16]), in other words it is a colimit in bC; and the -trivial morphis*
*ms
in Cb are stable under coproduct and sequential colimit because preserves
colimits by hypothesis. Finally, for condition (1) note that cofibrations are
stable under retracts because they are the same as for N. As for weak
equivalences, a morphism is by definition a weak equivalence if and only if
it is an equivalence in A, and this condition is equivalent to saying that it
projects to an isomorphism in ho(A). The class of isomorphisms in ho(A) is
closed under retracts, so this implies that the class of weak equivalences in
M is closed under retracts. This gives (1).
Therefore by Lemma 2.5 of [16], we obtain a cofibrantly generated closed
model structure M.
To complete the proof of (i) we just have to show that L(M) ~= A. For
this, note that L(M) is obtained from L(N) by inverting the images of M-
weak equivalences (since L(N) was obtained from N = M by inverting a
subset of the weak equivalences). We have that L(N) ~=Cb, and the images
of the M-weak equivalences are exactly the morphisms of bCwhose image by
is an equivalence in A. The hypothesis of (iv) says that this localization
57
gives exactly A. This completes the proof of (iv) ) (i).
We have now finished the proof of Theorem 14. ===
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