Twisted Diagrams
T. H˜ uttemann
Department of Mathematical Sciences, University of Aberdeen, King's College
Aberdeen AB24 3FX, U.K., email: huette@maths.abdn.ac.uk
O. R˜ ondigs
Fakult˜at f˜ur Mathematik, Universit˜at Bielefeld, Postfach 10 01 31
D--33501 Bielefeld, Germany, email: oroendig@mathematik.unibielefeld.de
Abstract
We introduce generalised diagram categories, construct Kan extensions, and establish
various model category structures. Using these, we define ``homotopy sheaves'' and
show that a twisted diagram is a homotopy sheaf if and only if it gives rise to a ``sheaf
in the homotopy category''.
Keywords: Model categories, functor categories
Mathematics Subject Classification (2000): 55U35
Contents:
1 Introduction
2 Foundations
2.1 Adjunction Bundles 2.2 Twisted Diagrams 2.3 Limits, Colimits, Direct and
Inverse Image 2.4 Twisted Kan Extensions 2.5 Construction of Adjunction Bundles
3 Model Structures
3.1 Some Remarks on Model Structures 3.2 The cStructure 3.3 The fStructure
3.4 The gStructure
4 Sheaves and Homotopy Sheaves
4.1 Associated Homotopy Bundle 4.2 Construction of h and •h 4.3 Comparison of
Sheaves and Homotopy Sheaves
1 Introduction
One often encounters constructions which look like diagrams in some category but
cannot be described with that formalism. An important example is the notion of (na˜ve)
spectra, a sequence of pointed spaces X 0 , X 1 , . . . and structure maps #X n
X n+1 .
This almost determines a diagram indexed over N (regarded as a category), and in fact
can be described by a ``twisted diagram'' with ``twists'' given by iterated suspension
functors. Another example (and the origin of the present paper) is the category of
2 T. H˜uttemann, O. R˜ondigs
quasicoherent sheaves on projective spaces as defined by the first author in [H˜u]: a
``sheaf '' is a collection of equivariant spaces, each equipped with an action of a di#er
ent monoid, together with structure maps which are equivariant with respect to the
``smaller'' monoid. A detailed description is contained in the examples in this paper.---
The new formalism also applies, as a special case, to diagram categories in the usual
sense (i.e., functor categories).
To illustrate the general idea, suppose we have two categories C and D and a
functor F : C D which has a right adjoint U . A twisted diagram (with respect to
this data) is a morphism (in D)
F (Y ) f #
Z
where Y is an object of C and Z is an object of D. This gadget should be thought of as
a generalised diagram of the form Y
f
# Z. Since Y and Z live in di#erent categories,
the ``structure map'' f has to act by a ``twist'' given by F .
The paper is divided into three parts. § 2 is devoted to the definition of twisted
diagrams and the development of basic machinery. The basic notion is that of an
adjunction bundle, consisting of a collection of categories and adjoint functor pairs.
It encodes the shape of the diagrams and carries all the necessary information about
twists. We discuss the behaviour of twisted diagrams with respect to morphisms of
adjunction bundles and prove a convenient criterion for completeness. In 2.4 we con
struct a twisted version of Kan extensions. Section 2.5 includes a di#erent description
of twisted diagrams and shows how to construct important examples of adjunction
bundles.
In § 3 we prove the existence of several Quillen closed model category struc
tures on categories of twisted diagrams. This part is based on model category struc
tures for diagram categories as in [Ho]. In more detail, we consider ``pointwise'' weak
equivalences. Depending on properties of the adjunction bundle (the index category is
required to be a ``direct'' or ``inverse'' category), we establish Reedytype model struc
tures using (generalised) latching or matching spaces. If the adjunction bundle consists
of cofibrantly generated model categories, we construct (for arbitrary index categories)
a cofibrantly generated model structure.
Finally, in § 4 we propose definitions of sheaves and homotopy sheaves. Starting
from an adjunction bundle of model categories we construct an associated bundle of
homotopy categories. A twisted diagram over the original adjunction bundle gives rise
to a twisted diagram over the homotopy bundle, and the former is a homotopy sheaf if
and only if the latter is a sheaf.
Required prerequisites for this paper are elementary category theory as presented
in [ML] and basic model category theory ([DS] or [Ho]).
A special case of the results on model structures has been used by the first author
to study the algebraic Ktheory of projective spaces [H˜u]. Twisted diagrams and their
model structures also appear implicitly in [HKVWW]. As the authors learned recently,
Hirschowitz and Simpson obtained related model structures [HS].
Twisted Diagrams 3
2 Foundations
2.1 Adjunction Bundles
Let I be a small category. It will serve as the index category for our diagrams.
Definition 2.1.1. An adjunction bundle B = (C, F, U) over I, or Ibundle, consists
of the following data:
. for each object i # I a category C i ,
. for each morphism # : i j in I a pair of adjoint functors F # : C i
C j and
U # : C j
C i (with F # being the left adjoint),
such that U determines a functor I op Cat, i.e., U id i
= Id C i
, and for each pair of
composable arrows i # j # k, the equality U ### = U # #U # holds. In addition, we
require F id i
= Id C i
. The properties of adjunctions guarantee that there is a canonical
isomorphism F ### # = F # # F # (which will be referred to as uniqueness isomorphism),
since both functors are left adjoint to U ### = U # # U # ([ML, IV.1, corollary 1, p. 83]).
Example 2.1.2. Any category C gives rise to a trivial Ibundle with C i = C for all i,
and all adjunctions being the identity adjunction.
Example 2.1.3. (The nonlinear projective line.)
If M is a monoid, denote by M Top # the category of pointed topological spaces
having a basepointpreserving action of M . A map of monoids f : M M # deter
mines an adjoint functor pair
· #
M
M # : M Top #
oe M # Top # : R f
with R f being restriction along f , and · #M M # being its left adjoint (inducing up).
The integers Z form a monoid under addition, and we have submonoids N+ (non
negative integers) and N- (nonpositive integers). Hence we can form the adjunction
bundle P 1 over I = (+ # 0 oe #
-), consisting of the categories N+ Top # , ZTop #
and N- Top # , and the adjoint pairs ``inducing up'' and ``restriction'' along the inclusions
N+ # Z and N- # Z.
2.2 Twisted Diagrams
Definition 2.2.1. (Twisted diagrams.)
Let B be an adjunction bundle over I. A twisted diagram Y with coe#cients in
B consists of the following data:
. for each object i # I an object Y i # C i ,
. for each morphism # : i j in I a map y #
# : Y i
U # (Y j ) in C i
4 T. H˜uttemann, O. R˜ondigs
such that Y behaves like a functor, i.e., y #
id i
= id Y i
and y #
### = U # (y #
# ) #y #
# for each pair
i # j # k of composable arrows in I. (A reformulation using the left adjoints
will be given below.)
A map f : Y Z of twisted diagrams is a collection of maps f i : Y i
Z i
in C i , one for each object i # I, such that for each morphism # : i j in I the
equality U # (f j ) # y #
#
= z #
# # f i holds. (A reformulation using the left adjoints will be
given below.) The category of twisted diagrams and their maps is denoted Tw (I, B).
For each of the structure maps y #
# : Y i
U # (Y j ) there is a corresponding ad
joint map y #
#
: F # (Y i ) Y j . The idea is to think of the (meaningless) symbol
y # : Y i # Y j as a kind of ``structure map'' having two incarnations as a #type
map (a morphism in C i ) and a #type map (a morphism in C j ).
The definition of twisted diagrams does not make explicit use of the left adjoints
provided by the adjunction bundle. However, the properties of adjunctions will play a
crucial r“ole for the discussion of limits and colimits in Tw (I, B).
Example 2.2.2. (Spectra.)
Let N denote the ordered set of natural numbers, considered as a category. For
each n # N, define C n to be the category S of pointed simplicial sets. If n # m, we have
an adjunction # m-n : S
oe S
:# m-n of iterated loop space and suspension functors.
It is clear that this defines an adjunction bundle Sp over N. A twisted diagram X with
coe#cients in Sp, graphically represented by the ``diagram''
X 0 # X 1 # X 2 # . . . ,
is nothing but a spectrum in the sense of [BF].
Remark 2.2.3.
(1) If B is a trivial Ibundle (2.1.2), we recover the functor category: Tw (I, B) =
Fun (I, C).
(2) If I is a discrete category (i.e., contains no nonidentity morphisms), an adjunction
bundle over I is simply a collection of categories {C i } i#I , and the category of
twisted diagrams is the product category # i#I C i .
(3) Suppose B # = (C # , F # , U # ) is a family of adjunction bundles indexed by I # .
Then we can form the following adjunction bundle
# # B # =: B = (C, F, U)
indexed by the disjoint union I := # # I # : for each i # I there is exactly one # with
i # I # , and we define C i = C #
i (and similarly for the F and U ). It is easy to see
that Tw (I, B) =
# #
Tw (I # , B # ) in this case.
Twisted Diagrams 5
Given twisted diagrams Y, Z # Tw (I, B) and a collection of maps f i : Y i
Z i
in C i , we can form two squares for each morphism # : i j in I
Y i
f i Z i
U # (Y j )
y #
#
?
U# (f j )
U # (Z j )
z #
#
?
and
F # (Y i ) F# (f i ) F # (Z i )
Y j
y #
#
?
f j
Z j
z #
#
?
and the definition of adjunctions imply that the left square commutes if and only if the
right square commutes. Thus the family (f i ) i#I determines a map of twisted diagrams
if and only if z #
# # F # (f i ) = f j # y #
#
.
For later use, we record the following fact:
Lemma 2.2.4. Suppose we have a map y #
# : Y i
U # (Y j ) in C i for each morphism
# : i j in I satisfying y #
id
= id, and denote by y #
# the adjoint map F # (Y i ) Y j .
Let # : j k be another morphism in I. Then if one of the squares
F # # F # (Y i ) # = F ### (Y i )
F # (Y j )
F# (y #
# )
?
y #
#
Y k
y #
###
?
and
Y i ============== Y i
U # (Y j )
y #
#
?
U# (y #
# )
U ### (Y k )
y #
###
?
commutes so does the other (the upper horizontal map in the left square is the uniqueness
isomorphism). In other words, if for all composable morphisms # and # one of the
squares commutes, the objects Y i together with the maps y #
# form a twisted diagram.
Proof. Assume that the square on the left commutes. We want to show that the
square on the right is commutative. The strategy is to divide the square into smaller
pieces which are known to commute.
For each morphism # : i j in I, there exists a natural transformation of func
tors # # : Id U # #F # called unit of the adjunction of F # and U # . Given the structure
map y #
#
: F # (Y i ) Y j , we obtain the corresponding adjoint map y #
#
: Y i
U # (Y j )
as the composite
Y i
# #
Y i
U # # F # (Y i ) U# (y #
# )
U # (Y j )
(cf. [ML], IV.1, p. 80). In particular, the functors F ### and U ### are adjoint with unit
# ### : Id U ### # F ### . But U ### = U # # U # has another left adjoint F # # F # , and
we denote the corresponding unit by •
# ### : Id U # # U # # F # # F # .
6 T. H˜uttemann, O. R˜ondigs
Now we redraw the square on the right with some extra data added:
Y i =========================== Y i ======================= Y i
1 2
U # # F # (Y i )
# #
Y i
? U# (# #
F# (Y i ) )
U # # U # # F # # F # (Y i )
•
# ###
Y i
? # = U ### # F ### (Y i )
# ###
Y i
?
3 4
U # (Y j )
U# (y #
# )
?
U# (# #
Y j
)
U # # U # # F # (Y j )
U##U# #F# (y #
# )
?
U### (y #
# )
U ### (Y k )
U### (y #
### )
?
The outer square is the right hand square of the lemma. Square 1 commutes by the
composition rules for adjunctions and units ([ML], IV.8.1, p. 101). Square 2 commutes
by definition of the uniqueness isomorphism. Square 3 commutes since # # is a natural
transformation of functors, and since U # is a functor. Finally, square 4 commutes by
hypothesis (apply U # # U # = U ### to the left diagram of the lemma).
The other direction of the lemma is proved using similar techniques. We omit the
details. ##
2.3 Limits, Colimits, Direct and Inverse Image
The next proposition says that Tw (I, B) is as complete and cocomplete as all
the C i , and that limits resp. colimits can be computed ``pointwise'' in the categories C i .
For i # I, let Ev i : Tw (I, B) C i denote the ith evaluation functor which maps a
twisted diagram Y to its ith term Y i .
Proposition 2.3.1. (Limits and colimits of diagrams of twisted diagrams.)
Let G : D Tw (I, B) be a functor, and suppose that for all i the limit of Ev i #G
exists. Then lim G exists and the canonical map
Ev i (lim G) lim(Ev i # G)
is an isomorphism. A similar assertion holds for colimits.
Proof. The proof relies on the compatibility of left (resp. right) adjoint functors with
colimits (resp. limits): if F is a left adjoint, and D is a functor, then there is a unique
natural isomorphism colim (F # D) F (colim D), and similarly for right adjoints
and limits ([ML, V.5, theorem 1, p. 114]).
To prove the lemma, we treat the case of colimits only. (For limits one has to use
similar techniques. Since U is supposed to be functorial, this is slightly easier.) Let
G i := Ev i # G, and define C i := colim G i . We claim that the objects C i assemble to a
twisted diagram C, and it is almost obvious that C is ``the'' colimit of G.
Let # : i j denote a morphism in I. The #type structure maps of the twisted
diagrams G(d) (for objects d # D) assemble to a natural transformation
G #
# : F # # G i
G j
Twisted Diagrams 7
of functors D C j . Hence we can define the #type structure map c #
# as the composite
F # (C i ) = F # (colim G i ) # = colim (F # # G i ) f colim G j = C j
with f induced by G #
# . By lemma 2.2.4 we are left to show that the following square
commutes for all composable morphisms # and # in I:
F # # F # (C i ) # = F ### (C i )
F # (C j )
F# (c #
# )
?
C #
#
C k
c #
###
?
(#)
We replace the symbols C # and the structure maps by their definition and obtain the
following bigger diagram:
F # # F # (colim G i ) ======= F # # F # (colim G i ) # = F ### (colim G i )
1 2
F # # colim (F # # G i ) #
# =
?
# = colim (F # # F # # G i )
# =
?
# = colim (F ### # G i )
# =
?
3 4
F # # colim (G j ) #
?
# =
colim (F # # G j )
?
colim (G k )
?
(##)
All the small squares commute: for 1 this is true by uniqueness of the isomorphisms
for commuting left adjoints with colimits. The horizontal maps of 2 are induced by
the uniqueness isomorphism, the vertical maps are induced by the isomorphism for
commuting left adjoints with colimits. By uniqueness, 2 commutes. Both horizontal
maps of 3 are induced by the isomorphism for commuting colimits with F # , and both
vertical maps are induced by the natural transformation G #
# : F # #G i
G j . Hence 3
commutes. Finally, square 4 commutes by lemma 2.2.4, applied componentwise to the
diagrams G # , and by functoriality of colim .
Hence the diagram (##) commutes. But the square (#) is contained in there as the
outer square, thus is commutative as claimed. ##
If I is a small category and C is an arbitrary category, the category of diagrams
Fun (I, C) is the value of an internal hom functor on the category of categories. Hence
it is functorial in both variables (provided the entries in the first variable are small). To
discuss a similar functoriality of the category of twisted diagrams, we have to introduce
some notions.
Definition 2.3.2. (Inverse image of bundles.)
Given a functor # : I J and a J bundle B = (D, G, V ), we define the
inverse image of B under #, denoted # # B, to be the Ibundle (C, F, U) given by
C i := D #(i) , U i := V #(i) and F i := G #(i) .
If # : I J is the inclusion of a subcategory, we write B| I instead of # # B and
call the resulting Ibundle the restriction of B to I.
8 T. H˜uttemann, O. R˜ondigs
Forming inverse images is functorial, i.e., id # C B = B and (# # #) # B = # # # # B.
The inverse image of a trivial bundle is a trivial bundle.
Definition 2.3.3. (Morphisms of bundles.)
Suppose A = (C, F, U) and B = (D, G, V ) are Ibundles. An Imorphism
# : A B consists of two families of functors # i : C i
D i and # i : D i
C i
where i ranges over the objects of I such that # i is left adjoint to # i , and such that for
each morphism # : i j in I we have V # # # j = # i # U # .
Given an Ibundle A and a J bundle B, a morphism of bundles # : A B
is a pair # = (#, #) where # : I J is a functor and # : A # # B is an
Imorphism of Ibundles.
Definition 2.3.4. (Inverse image of twisted diagrams.)
Suppose we have a functor # : I J , a J bundle B, and a twisted dia
gram Y # Tw (J , B). We define the inverse image of Y under #, denoted # # Y , as
the twisted diagram over I with coe#cients in # # B given by (# # Y ) i := Y #(i) and
(# # y) #
# := y #
#(#)
for all objects i # I and all morphisms # # I. We obtain a functor
# # : Tw (J , B) Tw (I, # # B).
Now suppose we have Ibundles A = (C, F, U) and B = (D, G, V ), and an
Imorphism # = (#, #) : A B. The functor inverse image under #, denoted
# # : Tw (I, B) Tw (I, A), assigns to a twisted diagram Y # Tw (I, B) the ob
ject # # Y # Tw (I, A) given by (# # Y ) i := # i (Y i ) with #type structure maps (# # y) #
#
given by the composition
F # ((# # Y ) i ) = F # (# i (Y i )) # = # j (G # (Y i )) # i (y #
# )
# j (Y j ) = (# # Y ) j
for all objects i # I and morphisms # : i j. (We will prove in the next lemma
that # # is welldefined, i.e., that # # Y is a twisted diagram.)
More generally, a morphism # = (#, #) : A B of bundles induces an inverse
image functor # # = # # # # # : Tw (J , B) Tw (I, A).
If # : I J is the inclusion of a subcategory, we write Y | I instead of # # Y
and call the resulting twisted diagram with coe#cients in B| I the restriction of Y
to I. This defines the restriction functor Tw (J , B) Tw (I, B| I ). As a special
case of restriction (if I = {i} is the trivial subcategory consisting of i), we obtain the
evaluation functors Ev i as defined above.
Lemma 2.3.5. Given Ibundles A = (C, F, U) and B = (D, G, V ), an Imorphism
# = (#, #) : A B, and a twisted diagram Y # Tw (I, B), the object # # Y defined
in 2.3.4 is a twisted diagram with coe#cients in A.
Twisted Diagrams 9
Proof. Let # : i j and # : j k be morphisms in I and consider the diagram
F # # F # # # i (Y i ) # = F ### # # i (Y i )
F # # # j # G # (Y i )
# =
?
# = # k # G # # G # (Y i ) # = # k # G ### (Y i )
# =
?
F # # # j (Y j )
F# ## j (y #
# )
?
# =
# k # G # (Y j )
#k #G# (y #
# )
?
#k (y #
# )
# k (Y k )
#k (y #
### )
?
in which all arrows labelled with `` # ='' denote uniqueness isomorphisms. Recall that
the compositions of functors appearing in the upper rectangle are left adjoints to the
functor U # #U # ## k . Thus the upper rectangle commutes by uniqueness. The lower left
square commutes by naturality. The lower right square commutes since Y is a twisted
diagram (lemma 2.2.4) and # k is a functor. Hence the whole diagram commutes and
# # Y is a twisted diagram by another application of lemma 2.2.4. ##
Definition 2.3.6. (Direct image of twisted diagrams.)
Suppose we have a bundle morphism # = (#, #) : A B, where A = (C, F, U)
is an Ibundle, B = (D, G, V ) is a J bundle, # is a functor I J , and
# = {(# i , # i )} i#I is an Imorphism A # # B. Let Y be a twisted diagram with
coe#cients in A. It is straightforward to check that the definition # # (Y ) i := # i (Y i )
yields a twisted diagram with coe#cients in # # B having the structure maps
# # (y) #
# : # i (Y i ) # i (y #
# )
# i # U# (Y j ) = V #(#) # # j (Y j )
for # : i j. In this way we obtain a functor # # Tw (I, A) Tw (I, # # B).
Suppose the right adjoint R# of # # exists. The composition
# # := R# # # # : Tw (I, A) Tw (J , B)
is called the direct image functor.
We will see below that if the bundle B consists of complete categories, the func
tor R# exists and can be constructed by twisted Kan extension. Using this, we can
prove:
Corollary 2.3.7. Let # = (#, #) : A B be a bundle morphism, with B consisting
of complete categories. Then the functor # # (inverse image under #) has a right adjoint
# # (direct image under #).
Proof. Since R# is right adjoint to # # by assumption, it remains to show that # # is
right adjoint to # # . However, this is true, because # # is pointwise right adjoint to # # ,
and it can be checked that adjoining pointwise respects maps of twisted diagrams. We
omit the details. ##
10 T. H˜uttemann, O. R˜ondigs
2.4 Twisted Kan Extensions
Assume that B is a trivial bundle over J , consisting of the category C (and identity
functors), and # : I J is a functor. In this case, the inverse image of B under #
is the trivial bundle over I (consisting of C and identity functors), and # # is the functor
Fun (J , C) Fun (I, C) mapping Y to Y # #. If C is complete, the functor # # has
a right adjoint given by right Kan extension along # ([ML, X.3, corollary 2]).
It is possible to construct Kan extensions in our framework. We consider only left
Kan extensions, the other case being similar (and easier).
Let # : I J be a functor, B = (C, F, U) a J bundle, and Y a twisted
diagram over I with coe#cients in # # B = (D, G, U ). First, we have to define a
twisted diagram L(Y ) over J with coe#cients in B. (Later, we will convince ourselves
that the assignment Y L(Y ) is a functor which is left adjoint to # # .) Let j # J be
given, and let # # j denote the category of objects #over j. Its objects are maps of the
form # : #(i) j # J (for i an object of I). The morphisms from # : #(i) j
to # : #(i # ) j are morphisms # : i i # # I satisfying # # #(#) = #. Consider
the assignment
D Y
j : # # j C j , (#(i) # j) ## F # (Y i )
This is welldefined because Y i is an object of D i = C #(i) by definition of # # B, so F # (Y i )
is an object of C j .
The assignment D Y
j is in fact a functor, as one can deduce as follows. Let pr I
denote the obvious projection functor # # j I mapping the object #(i) j
to i, and define pr J := ##pr I . Using the equality pr # J B = pr # I
(# # B), we get a functor
pr # I : Tw (I, # # B) Tw(# # j, pr # J B). Let {j} denote the subcategory of J given
by the object j (and no nonidentity morphism) and consider the category C j as a (triv
ial) bundle over {j}. Then we have a morphism of bundles # : C j
pr # J B consisting
of the functor # # j {j} and the (# # j)morphism # from pr # J B to the trivial
bundle with #component the adjunction F # : C #(i)
oe C j : U # (for # : #(i) j).
The inverse image under # is a functor # # : Tw(# # j, pr # J B) Fun (# # j, C j ).
Tracing the definitions shows D Y
j
= # # pr # I
(Y ).
Now assume that the bundle B consists of cocomplete categories. Define L(Y ) j
as the colimit of D Y
j
. To prove that the L(Y ) j assemble to a twisted diagram, we
construct for each # : j k a structure map l #
#
: F# (L(Y ) j ) L(Y ) k and apply
lemma 2.2.4.
Since F# is a left adjoint, we have a unique isomorphism
u# : F# (colim D Y
j ) # = colim (F # # D Y
j ) .
Let #(i) # j be an object of # # j. Then # # # is an object of # # k, and there
is a canonical map F### (Y i ) colim D Y
k
= L(Y ) k (since F### (Y i ) appears in the
Twisted Diagrams 11
diagram D Y
k
). The composition with a uniqueness isomorphism yields a map
t # : F# # F # (Y i ) L(Y ) k .
The t # 's assemble to a natural transformation from F# # D Y
j to the constant diagram
with value L(Y ) k (a proof involves the uniqueness of the uniqueness isomorphisms and
the naturality of the canonical maps mentioned above; we omit the details). By taking
colimits, this determines a map
v # : colim (F # # D Y
j
) L(Y ) k ,
and we set l #
# := v # # u# .
Now we have to check that, for j # k # l # J , the square
F # # F# (L(Y ) j ) # = F ### (L(Y ) j )
F # (L(Y ) k )
F# (l #
# )
?
l #
#
L(Y ) l
l #
###
?
(#)
commutes. First of all, the diagram
F # # F# (L(Y ) j ) # = F ### (L(Y ) j )
colim (F # # F# # D Y
j )
# =
?
# =
colim (F ### # D Y
j )
# =
?
consisting of uniqueness isomorphisms commutes because of their uniqueness. By the
universal property of the colimit and the definition of the structure maps, we are left
to show that for every # : #(i) j the diagram
F # # F# # F # (Y i ) # = F ### # F # (Y i )
F # # F### (Y i )
# =
?
# =
F ##### (Y i )
# =
?
colim (F # # D Y
k )
c###
?
F # (L(Y ) k )
# =
?
l #
#
L(Y ) l
c #####
?
commutes, where the maps c ### and c ##### are canonical maps to the colimit, and all
maps labelled with ` # =' are uniqueness isomorphisms. The upper square commutes by
uniqueness, and the lower square commutes by definition of l #
#
. This implies that the
square (#) commutes, and 2.2.4 shows that L(Y ) is a twisted diagram as claimed.
12 T. H˜uttemann, O. R˜ondigs
Theorem 2.4.1. (Left Kan extensions.)
Let B be a J bundle consisting of cocomplete categories, # : I J a functor
and Y a twisted diagram with coe#cients in # # B. The assignment Y ## L(Y ) described
above is the object function of a functor L# : Tw (I, # # B) Tw (J , B) which is
left adjoint to # # .
Proof. Abbreviate L# by L and keep the notation used in the construction of L(Y ).
We start by describing the e#ect of L on morphisms. Let f : Y Z be a
map of twisted diagrams with coe#cients in # # B, and fix an object j # J . For each
# : #(i) j, the maps F # (f i ) form a natural transformation from D Y
j to D Z
j , because
the uniqueness isomorphisms are natural, f is a map of twisted diagrams and F # is a
functor. This defines a map on the colimits L(f) j : L(Y ) j
L(Z) j .
We claim that the maps L(f) j assemble to a map L(f) of twisted diagrams. For
# : j k in J , consider the diagram
F# (L(Y ) j ) F# (L(f) j )
F# (Z j )
L(Y ) k
l #
#
?
L(f)k
L(Z) k
m #
#
?
where l and m denote the structure maps of L(Y ) and L(Z). It commutes if and only
if for each object # : #(i) j of # # j, the diagram
F# # F # (Y i ) F##F# (f i )
F# # F # (Z i )
F### (Y i )
# =
?
F### (f i )
F### (Z i )
# =
?
L(Y ) k
?
L(f)k
L(Z) k
?
commutes. The isomorphisms are uniqueness isomorphisms, which are natural, hence
the upper square commutes. The lower vertical arrows denote the canonical map to
the colimit, and the naturality of these make the lower square commute.
Having checked that L(f) is indeed a map of twisted diagrams, it is clear that L is
a functor, because maps of twisted diagrams are defined pointwise, and L j is defined
as the composition of functors colim ## # # pr # I , (with # and pr I being explained below
the definition of D Y
j ). To prove that L is left adjoint to # # , we construct natural
transformations # : Id # # # L and # : L # # # Id satisfying the triangular
identities ([ML, IV.1], theorem 2 (v)).
For Y # Tw (I, # # B), the Y component # Y is given (pointwise) as the canonical
map to the colimit Y i
# # (L(Y )) i = L(Y ) #(i) which corresponds to the identity
Twisted Diagrams 13
id : #(i) #(i) (an object of # # #(i)). We check that # Y is a map of twisted
diagrams. Let # : i j # I be given and consider the diagram
F #(#) (Y i ) F #(#) ((#Y ) i )
F #(#) (L(Y ) #(i) )
Y j
y #
#
?
(#Y ) j
L(Y ) #(j)
l #
#(#)
?
with the structure map y #
# starting from G# (Y i ) = F #(#) (Y i ) by definition of # # B.
Since the structure map l #
#(#)
is defined via the canonical maps to the colimit
F #(#) # F # (Y k ) # = F #(#)## (Y k ) L(Y ) #(j)
(for # : #(k) #(i) an object of # # #(i)), the composition l #
#(#) # F #(#) ((# Y ) i )
coincides with the canonical map to the colimit c : F #(#) (Y i ) L(Y ) #(j) (the special
case # = id #(i) ). Hence we have to show that the triangle
F #(#) (Y i )
Y j
y #
#
?
(#Y ) i
L(Y ) #(j)
c
commutes. But this is true by the definition of L(Y ) #(j) as the colimit of D Y
#(j)
. The
naturality of # can be explained as follows. For i # I, the canonical maps to the colimit
Y j
L(Y ) #(i) for varying # : j #(i) are a natural transformation of diagrams
(with shape # # #(i)). In particular, the #(i)component, being the map (# Y ) i , is
natural. We turn to the definition of # : L # # # Id. For Z # Tw (J , B), the map
# Z is given pointwise as follows: for every j # J and every # : #(i) j in # # j, the
structure maps F # (# # (Z) i ) = F # (Z #(i) ) y #
# Z j assemble to a natural transformation
from D # # Z
j to the constant diagram with value Z j (this follows from lemma 2.2.4 and
the fact that Z is a twisted diagram). By the universal property of the colimit, this
natural transformation defines a unique map
(# Z ) j : L(# # (Z)) j
Z j .
To prove that # Z is a map of twisted diagrams, let # : j k # J and consider the
following diagram:
F# (L(# # (Z)) j ) F# ((# Z ) j ) F# (Z j )
L(# # (Z)) k
m #
#
?
(# Z ) k
Z k
z #
#
?
14 T. H˜uttemann, O. R˜ondigs
Using the universal property of the colimit, the definition of # Z and the definition of
the structure map m #
# , we are left to show that, for each # : #(i) j, the diagram
F# (F # (Z i )) F# (z #
# ) F# (Z j )
L(# # (Z)) k
?
(# Z ) k
Z k
z #
#
?
commutes, where the left vertical map is the composition of the uniqueness isomorphism
and the canonical map to the colimit F### (Z i ) L(# # (Z)) k . However, the definition
of # Z implies that the diagram above commutes since Z is a twisted diagram. To prove
the naturality of #, let f : Y Z be a map in Tw (J , B). For j # J and every
# : #(i) j in # # j, the maps F # (f # (i)) : F # (Y # (i)) F # (Z # (i)) assemble to a
natural transformation D # # f
j
of functors on # # j making the diagram
D # # Y
j
Y j
D # # Z
j
D # # f
j ?
Z j
f j
?
commute. The horizontal maps are the ones appearing in the definition of #. Since the
colimit functor is left adjoint to the ``constant diagram'' functor, the square
L(# # (Y )) j
(# Y ) j
Y j
L(# # (Z)) j
L(# # f)
? (# Z ) j
Z j
f j
?
commutes, proving the naturality of #.
It remains to prove that the composites
L L# L # # # # L #L L and # # ## #
# # # L # # # # # #
# #
are identity natural transformations.
In the first case, let Y # Tw (I, # # B) and j # J . The map
L(# Y ) j : L(Y ) j
L(# # (L(Y ))) j
is defined via the canonical maps to the colimit F # (Y i ) F # (L(Y ) #(i) ) (for mor
phisms # : #(i) j). The definition of # then implies that it su#ces to prove the
commutativity of the triangle
F # (Y i ) F # (L(Y ) #(i) )
L(Y ) j
l #
#
?
Twisted Diagrams 15
for each # : #(i) j, where the two arrows in the middle denote canonical maps to
the colimit. The definition of l #
# gives the desired result.
In the second case, let Z # Tw (J , B) and i # I. We have to show that the
triangle
Z #(i)
L(# # (Z)) #(i)
Z #(i)
(# Z ) #(i)
?
id
commutes, where the upper horizontal map is the canonical map to the colimit (corre
sponding to id #(i) ). But this is obvious from the definition of #. ##
The right adjoint of # # , obtained by the corresponding twisted version of right
Kan extension along #, will be denoted R#. By the dual of theorem 2.4.1 it exists
if B consists of complete categories.
Recall the functor Ev i defined as the restriction along {i} J . If the bundle
B consists of cocomplete categories, its left adjoint F r i : C i
Tw (J , B) exists by
theorem 2.4.1. It is the analogue of the free diagram at i and will be needed later in
the construction of a cofibrantly generated model structure. We call F r i (K) the free
twisted diagram generated by K # C i .
Example 2.4.2. (Spectra, continued.)
Let Sp be the bundle defined in 2.2.2 which leads to ordinary spectra. The nth
evaluation functor maps a spectrum to its nth term, and the corresponding nth free
twisted diagram of a pointed simplicial set K is the spectrum
# # # # . . . # # # K # #K # # 2 K # . . .
with K appearing at the nth spot and all #type structure maps being identities except
for the map #(#) = # K.
2.5 Construction of Adjunction Bundles
We think of twisted diagrams as generalised diagrams. However, there is an alter
native approach using fibred and cofibred categories in the sense of Grothendieck.
For definitions and notation the reader may wish to consult [Q1].
Let us recall the Grothendieck construction
#
Gr (U) of a contravariant functor U
defined on I with values in the category of (small) categories. The objects of
#
Gr (U)
are the pairs (i, Y ) with i an object of I and Y an object of U(i). A morphism
(i, Y ) (j, Z) consists of a morphism i # j in I and a morphism Y A U(#)(Z)
in U(i). Composition is given by the rule
(#, B) # (#, A) := # # # #, U(#)(B) # A # .
This construction comes equipped with a functor
#
Gr (U) I.
16 T. H˜uttemann, O. R˜ondigs
Remark 2.5.1. An adjunction bundle determines a functor U : I op Cat, hence
a functor
#
Gr (U) I. The existence of the left adjoints F # make
#
Gr (U) a cofibred
category over I op , even a bifibred bundle in the sense of the next definition.
Definition 2.5.2. Given a functor # : E A, we call E a bifibred bundle over A if
the following conditions are satisfied (using notation from [Q1]):
(1) The functor # is fibred, and for all composable morphisms # and # in A, the
natural isomorphism # # # # # (# # #) # is the identity.
(2) The functor # is cofibred, and for all morphisms # # A the functor # # is right
adjoint to # # .
In this situation, a functor f : I A determines an Iindexed adjunction bundle
f ## # = I ## A E which sends the object i # I to the category # -1 (f(i)) and the
morphism µ # I to the adjoint pair f(µ) # and f(µ) # .
Remark 2.5.3. ( M. Brun's reformulation of twisted diagrams.)
Recall from remark 2.5.1 the functor # :
#
Gr (U) I associated to an adjunction
bundle. A straightforward calculation which we omit shows that Tw (I, B) is the
category of sections of #.
More generally, given a bifibred bundle # and an adjunction bundle f ## # as
in 2.5.2, the category of twisted diagrams Tw (I, f ## #) is the category of lifts of f
to E , i.e., the category of functors g : I E satisfying # # g = f .
Example 2.5.4. Let Mod Rng denote the canonical functor from the cate
gory of all modules over all rings to the category of rings. (The objects of Mod are
pairs (R, M) with R a ring and M an Rmodule. A morphism (R, M) (S, N)
consists of a ring map f : R S and an fsemilinear additive map M N .)
This defines a bifibred bundle.
A toric variety determines a functor into Rng , hence (by 2.5.2) an adjunction
bundle. In fact, a fan # of a toric variety can be regarded as a poset, hence as a
category, and we obtain a functor
# op Rng , # ## C[Ÿ# # M ]
where Ÿ
# is the dual cone of # and M is the dual lattice (see [O] for details). Thus the
toric variety X(#) determines the adjunction bundle # op ## Rng Mod .
This example can be generalised to obtain an adjunction bundle from a diagram
of monoids and a cocomplete category D. We proceed with a construction.
It is well known that we can consider any monoid M as a category with one object
and morphisms corresponding to the elements of M . A morphism of monoids then is
a functor between two such categories. Suppose that D is a cocomplete category. We
define the category of Mequivariant objects in D, denoted M D, as the category of
functors M D. A monoid homomorphism f : M M # induces the ``restriction''
Twisted Diagrams 17
functor f # = R f : M # D M D (given by precomposing with f ). Since D is
cocomplete, this functor has a left adjoint f # = · #M M # : M D M # D. For
composable monoid homomorphisms we have the relations (g # f) # = f # # g # and
(g # f) # # = g # # f # . Moreover id # = id, and we choose id # = id.
Let EqD denote the category of equivariant objects in D. Objects are the pairs
(M, D) where M is a monoid and D is a functor M D. A morphism from
(M, D) to (M # , D # ) is a pair (#, #) where # : M M # is a monoid homomorphism
and # is a natural transformation of functors D D # # #. The forgetful functor
# : EqD Mon into the category of monoids make EqD into a bifibred bundle in
the sense of 2.5.2. The fibre over the monoid M is the category M D of Mequivariant
objects in D.
Definition 2.5.5. Suppose we have a (small) category I and an Iindexed diagram G
of monoids, i.e., a functor G : I Mon. For a cocomplete category D we define the
Iindexed adjunction bundle AdDG = (C, F, U) by
AdDG := I ## Mon EqD .
Explicitly, for an object i # I we let C i := G(i)D, the category of G(i)equivariant
objects in D, and for a morphism # # I we define F # := G(#) # and U # := G(#) # .
This definition is clearly natural in G, i.e., given a natural transformation of
diagrams of monoids G G # we obtain an Imorphism of adjunction bundles
AdDG # AdDG.
Example 2.5.6. (Nonlinear projective spaces.)
This generalises the nonlinear projective line (2.1.3). Let [n] denote the set
{0, 1, . . . , n}, and write #n# for the category of nonempty subsets of [n]; morphisms
are given by inclusion of sets. For A # [n], define the (additive) monoid
M A := # (a 0 , . . . , a n ) # Z n+1
# # #
n
# 0
a i = 0 and #i /
# A : a i # 0 # .
These monoids assemble to a functor G : #n# Mon. Let EqTop # denote the
category of equivariant spaces as constructed above. (Objects are pairs (M,T ) where
M is a monoid and T is a pointed topological space with a basepoint preserving
Maction. Maps are semiequivariant continuous maps of pointed topological spaces.)
This category is a bifibred bundle over the category of monoids. Thus we are in
the situation of definition 2.5.5 (with I = #n#); denote the resulting adjunction bun
dle Ad Top #
G = #n# ## Mon EqTop # by P n (G). The category of twisted diagrams
Tw # #n#, P n (G) # is nothing but the category pP n (G) of Gequivariant quasicoherent
presheaves as defined in [H˜u, 6.1].
18 T. H˜uttemann, O. R˜ondigs
3 Model Structures
3.1 Some Remarks on Model Structures
The terminology concerning model categories is taken from [DS] and [Ho], the
proofs are mostly modifications of the corresponding proofs in [Ho]. The term ``model
category'' is always to be understood in the sense of [DS], which is slightly more general
than the definition given in [Ho]. The di#erences are the following: In [Ho], it is required
that a model category has all small limits and colimits (instead of just finite ones),
and the factorizations have to be functorial and are part of the structure (instead of
assuming that they simply exist).
Definition 3.1.1. Let B = (C, F, U) be an adjunction bundle over I. We call B
an adjunction bundle of model categories if all the C i are model categories, and all
the F # preserve cofibrations and acyclic cofibrations. In other words, we require the
pair (F # , U # ) to form a Quillen adjoint pair.---If in addition all the C i are left proper
model categories, B is called left proper , and similarly for ``right proper'' and ``proper''.
Note that the inverse image of an adjunction bundle of model categories B is again an
adjunction bundle of model categories, which is as proper as B.
Example 3.1.2. The projective space bundles P n (G) (for G a cofibrant topological
monoid, 2.1.3) and spectra Sp (cf. 2.2.2) are examples of proper adjunction bundles of
model categories. The model structure defined on M Top # (for M a monoid) has weak
equivalences and fibrations on underlying spaces, the model structure on the category
of pointed simplicial sets is the usual one.
Before defining the model structures on twisted diagrams, we make a technical
observation.
Remark 3.1.3. Suppose C =
# # C # is the product of model categories C # . Then there
is a product model structure on C where a map is a weak equivalence (resp. fibration,
resp. cofibration) if its image under the canonical projection is a weak equivalence
(resp. fibration, resp. cofibration) in C # for all # (see [Ho, 1.1.6]). If all the C # are left
proper, C is a left proper model category, and similarly for ``right proper''.
3.2 The cStructure
The first model structure on Tw (I, B) we want to consider has pointwise weak
equivalences and pointwise fibrations. The price one has to pay for the simple definition
of fibrations is that the description of cofibrations is rather involved. Moreover, we have
to restrict to ``nice'' indexing catergories.
Twisted Diagrams 19
Definition 3.2.1. (Direct categories.)
A category with degree function is a (small) category I together with a Zvalued
function d, defined on the objects, such that whenever there is a nonidentity morphism
i j we have d(i) #= d(j). (We say that all nonidentity arrows change the degree. In
particular, objects have no nontrivial endomorphisms.) The category is called bounded
if d is bounded below, and it is called locally bounded if each connected component
is bounded. Without restriction, the degree of a bounded category has values in an
honest ordinal, namely N. If nonidentity arrows always increase the degree and the
category is (locally) bounded, we say that I is a (locally) direct category .
All finite dimensional categories (i.e., categories with finite dimensional nerve)
admit degree functions and can be made into direct categories. A disjoint union of
locally direct categories is locally direct. If I is (locally) direct, so are subcategories,
under and over categories formed with I. In particular, the full subcategory I n of
objects of degree less than or equal to n is (locally) direct. A finite product of direct
categories is direct (with degree given by sum of partial degrees).
In what follows, B = (C, F, U) is an adjunction bundle of cocomplete model cate
gories over I. Let Y be a twisted diagram with coe#cients in B and i an object of I.
To describe the cofibrations in the model structure we are going to construct, we have
to introduce the latching object of Y at i. Recall that for a diagram Z (untwisted
case) the latching object at i is defined as the colimit over all components Z j which
map to Z i . For a twisted diagram Y , we mimick this construction, using the ``twisting''
functors F # to push everything into the category C i . The colimit is to be taken with
respect to the #type structure maps of Y .
Technically, we can describe the latching spaces as follows. For each object i # I,
let I # i denote the category of objects over i. Let I # i denote the full subcategory
of I # i which consists of all objects # : j i with # #= id i . There are it is a
good acyclic ccofibrationcanonical functors # : I # i # I # i (the inclusion) and
pr : I # i I (the projection (# : j i) ## j). Set P I#i := pr # # and denote the
trivial bundle over I # i with value C i by C i again. We define an I # imorphism of
bundles # : C i
(P I#i ) # B as follows: For # : j i, the adjoint pair
F # : C j
oe C i : U #
is the #component of #, and it is obvious from the definitions that # is in fact a bundle
morphism. Hence we have a functor # # : Tw (I # i, (P I#i ) # B) Fun (I # i, C i ).
Define G i : Tw (I # i, (P I#i ) # B) C i as the composition G i := colim ## # .
Definition 3.2.2. The latching object of Y at i is defined as L i Y := G i # (P I#i ) # (Y ).
It is an object of C i .
Remark 3.2.3. Note that L i is a composition of functors, hence itself a functor. The
structure maps y #
# : F # (Y j ) Y i for # : j i define a natural transformation
L i
Ev i . If a map L i Y Y i is mentioned, it is always this natural map.
20 T. H˜uttemann, O. R˜ondigs
Example 3.2.4. If X is a spectrum and n > 0, the latching object of X at n is the
pointed simplicial set #X n-1 , and the natural map #X n-1
X n of 3.2.3 is the
(#type) structure map of the spectrum.
Example 3.2.5. Let Y = (Y +
y#
# Y 0
oe y#
# Y- ) be a twisted diagram with coe#cients
in the projective line bundle P 1 (cf. 2.1.3). The latching objects of Y at + and at - are
the initial objects in N+ Top # and N- Top # , respectively. The latching object at 0 is
the Zequivariant pointed space (Y + #N+ Z) # (Y - #N- Z). The #type structure maps
induce a map to Y 0 .
Definition 3.2.6. (The cstructure.)
Let f : Y Z be a map in Tw (I, B). We call f a weak equivalence if f i is
a weak equivalence in C i for every object i # I. We call f a ccofibration if for all
objects i of I, the induced map Y i #L i Y L i Z Z i is a cofibration. We call f a
cfibration if all f i are fibrations in C i .
To prove that the cstructure is a model structure, we concentrate on the lifting
axiom first. Call a map f # Tw (I, B) a good acyclic ccofibration if for all objects i
of I, the induced map Y i #L i Y L i Z Z i is an acyclic cofibration. Later, we will
prove that the class of good acyclic ccofibrations coincides with the class of acyclic
ccofibrations.
Lemma 3.2.7. Let I be a direct category, and let B be an adjunction bundle of cocom
plete model categories over I. Good acyclic ccofibrations have the left lifting property
with respect to cfibrations. Similarly, ccofibrations have the left lifting property with
respect to acyclic cfibrations.
Proof. We treat the first case only, the other is similar. Let
A g X
B
f
?
h
Y
p
?
be a commutative diagram in Tw (I, B) such that f is a good acyclic ccofibration
and p is a cfibration. We will construct the desired lift by induction on the degree of
objects of I.
Since I is direct, the degree function d has a minimum k. If i is an object in I of
degree k, then L i is the constant functor with the initial object as value. By definition
of a good acyclic cofibration, the map f i is an acyclic cofibration in C i . Hence we can
find a lift l i in the following diagram:
A i
g i
X i
B i
f i #
?
?
h i
... ... .. ... ... ...
l i
Y i
p i
? ?
Twisted Diagrams 21
Since the full subcategory I k of objects of degree k is discrete, the lifts l i for the
various i # I k assemble to a map l| Ik : B| Ik
X| Ik in Tw (I k , B| Ik ).
Now let n > k, and assume that we have constructed a lift in the diagram
A| In-1
g| I n-1 X| In-1
B| In-1
f | I n-1
?
h|I n-1
... .. .. ... .. .. .. ... .. .
l| I n-1
Y | In-1
p| I n-1
?
making it a commutative diagram in Tw (I| n-1 , B| In-1 ). If i is an object of degree n
and # : j i an object of I # i, the map F # (B j ) F# (l j )
F # (X j ) x #
# X i is part of
a natural transformation # : L i B X i such that the diagram
L i A A i
L i B
L i f
?
#
X i
g i
?
commutes. Hence we get a diagram
A i #L i A L i B X i
B i
# ?
?
h i
Y i
p i
? ?
in which, by hypothesis, the left vertical map is an acyclic cofibration and the right
vertical map is a fibration. Thus a lift l i : B i
X i exists, and it is straightforward
to check that these maps l i , together with the morphism l| In-1 , define a map of twisted
diagrams l| In : B| In
X| In such that the diagram
A| In
g| In X| In
B| In
f | In ?
h|In
........... ......
l| In
Y | In
p| In
?
commutes. This completes the induction. ##
Let # : I J be a functor and A an adjunction bundle of cocomplete model
categories over J . Obviously, the functor # # : Tw (J , A) Tw (I, # # A) pre
serves weak equivalences and cfibrations. The question is whether # # also preserves
ccofibrations. Under certain conditions (which are satisfied in the case of interest) we
can give a positive answer.
22 T. H˜uttemann, O. R˜ondigs
Suppose the functor # : I J is injective at identities, i.e., whenever #(#) is
an identity morphism, so is #. (For example, a faithful functor is injective at identities.)
Then # induces a functor
# # i : I # i J # #(i)
which sends # : k i to #(#) : #(k) #(i). This construction is compatible
with the projection functors, i.e., we have # # P I#i = P J##(i) # # # i.
Recall that a functor F : C D is called final if for each A # D the category
A # F of objects F under A is nonempty and connected.
We say that the functor # satisfies the finality condition if it is injective at iden
tities, and the functor # # i is final for all objects i # I.
Lemma 3.2.8. Let # : I J be a functor, B an adjunction bundle of cocomplete
model categories over I and i an object of I. Denote by L i the ith latching object
functor of Tw (I, # # B), and by L # #(i)
the #(i)th latching object functor of Tw (J , B).
If # satisfies the finality condition, then there is a natural isomorphism L i ## # # = L # #(i)
.
Proof. The functor L i is defined as the composition colim ## # # P #
I#i , with # being
an I # imorphism with #component given by the adjunction
F #(#) : C #(j)
oe C #(i) : U #(#)
where # : j i is an object of I # i. On the other hand, L # #(i)
is the composition
L # #(i)
= colim ## # # P #
J##(i)
, with # having the #component given by the adjunction
F # : C j
oe C #(i) : U #
where # : j #(i) is an object of J # #(i). It is straightforward to check that the
equality L i # # # = colim #(# # i) # # # # # P #
J##(i)
holds. Hence the ith latching object
of # # (A) is given by
L i (# # (A)) = colim #(# # # P #
J##(i)
(A)) # (# # i) .
The functor # # i induces a map L i (# # (A)) L # #(i)
(A) which is an isomorphism
by [ML, IX.3.1] since # # i is final. ##
Corollary 3.2.9. If # satisfies the finality condition, then # # preserves ccofibrations
and good acyclic ccofibrations.
Proof. This follows immediately from 3.2.8 since the maps L i (# # A) A #(i) and
L # #(i) A A #(i) correspond under the isomorphism. ##
Remark 3.2.10. The functor P I#i satisfies the finality condition because (P I#i ) # #
is an isomorphism of categories for each object # # I # i.
Twisted Diagrams 23
Lemma 3.2.11. Let I be direct. For each i # I, the latching object functor L i maps
ccofibrations to cofibrations and good acyclic ccofibrations to acyclic cofibrations.
Proof. Recall that L i was defined as the composite G i # (P I#i ) # . By remark 3.2.10
and corollary 3.2.9, we are left to show that G i maps ccofibrations to cofibrations
and good acyclic ccofibrations to acyclic cofibrations. However, G i has a right adjoint
V i := # # # #, where # : C i
Fun (I # i, C i ) denotes the constant diagram functor
and # # is the direct image under the I # imorphism # having #component
F # : C j
oe C i : U #
where # : j i is an object of I # i. It is easy to see that V i maps (acyclic) fibrations
to (acyclic) cfibrations. Hence the statement follows from lemma 3.2.7 and the fact
that C i is a model category. ##
Corollary 3.2.12. If f is a (good acyclic) ccofibration, all its components are (acyclic)
cofibrations in their respective categories. In particular, a good acyclic ccofibration is
an acyclic ccofibration.
Proof. Let f : A B be a ccofibration. By 3.2.11, the map L i f : L i A L i B
is a cofibration in C i , hence its cobase change A i
A i #L i A L i B is a cofibration.
Observe that f i factors as this last map followed by A i #L i A L i B B i . Since the
latter is a cofibration by hypothesis, we conclude that f i is a cofibration.---The other
case is similar. ##
Theorem 3.2.13. Suppose I is a locally direct category, and B is an adjunction
bundle of cocomplete model categories over I.
(1) The cstructure is a model structure.
(2) A map f of twisted diagrams is an acyclic ccofibration if and only if for all objects
i # I, the induced map Y i #L i Y L i Z Z i is an acyclic cofibration in C i .
(3) If B is a left resp.right proper bundle, the cstructure is left resp.right proper.
Proof. Let (I # ) denote the family of path components of I. Then I =
# I # , and each
of the I # is a direct category. Since Tw (I, B) =
# #
Tw (I # , B| I# ), it is enough to
show that the cstructure is a model structure for each of the categories Tw (I # , B| I# );
by 3.1.3 we can equip Tw (I, B) with the product model structure. Consequently, we
can assume that I is direct.
We use the axioms for model categories as given in [DS]. First we note that the class
of weak equivalences is closed under composition since weak equivalences are defined
pointwise. Similarly, the composition of two cfibrations is a cfibration again.
Now assume we have two composable ccofibrations A f B g C. To show
that g #f is a ccofibration, we have to prove that for all objects i # I the induced map
A i #L i A L i C C i
24 T. H˜uttemann, O. R˜ondigs
is a cofibration in C i . But we can factor this map as
A i #L i A L i C # = A i #L i A L i B #L i B L i C
x B i #L i B L i C
y C i
where x is induced by f , and y is induced by g. But both of these maps are cofibrations
(since they are cobase changes of cofibrations), hence so is their composite.
It is obvious that each of the classes above contains all identities.
Axiom MC1: existence of finite limits and colimits is guaranteed by 2.3.1 since
they exist in all C i .
Axiom MC2: the ``2of3'' property for weak equivalences is satisfied since weak
equivalences are defined pointwise and MC2 holds in all the categories C i .
Axiom MC3: the class of weak equivalences is closed under retracts since weak
equivalences are defined pointwise, and in each category C i a retract of a weak equiva
lence is a weak equivalence. Similarly, the class of fibrations is closed under retracts.
Suppose g : Y Z is a retract of f : A B and f is a ccofibration. We
have to show that for all objects i # I n , the map L i Z #L i Y Y i
Z i induced by g
is a cofibration in C i . But by functoriality of pushouts and latching objects, this map
is a retract of the map L i B #L i A A i
B i induced by f , which is a cofibration by
hypothesis. Since MC3 is valid in C i , the former map is a cofibration. Hence g is
a ccofibration as claimed. This argument also shows that the class of good acyclic
ccofibrations is closed under retracts.
Axiom MC5: let f : A X be a map in Tw (I, B). We will construct in
ductively a factorization of f as a good acyclic ccofibration followed by a cfibration.
(The other factorization axiom is proved in a similar manner). Let k be the mini
mum of the degree function on I, and let i be of degree k. Then f i factors in C i as
A i
g i
#
T i
p i
X i , with g i being an acyclic cofibration and p i being a fibration.
The collection of these factorizations (where i ranges through all objects of degree k)
yields a factorization of f | Ik in Tw (I k , B| Ik ) as g| Ik : A| Ik
T | Ik followed by
p| Ik : T | Ik
X| Ik .
Let n > k, and assume we have already constructed a factorization of f | In-1 in
Tw (I n-1 , B| In-1 ) as the composite A| In-1
g| I n-1 T | In-1
p| I n-1 X| In-1 . Let i be
of degree n. The canonical functor P I#i : I # i I factors through the inclusion
# : I n-1 # I as # : I # i I n-1 since I is direct. Recall the functor
G i : Tw (I # i, (P I#i ) # B) C i
appearing in the definition of the ith latching object functor L i (3.2.2). By defi
nition, L i = G i # P I#i = G i # # # # # # , hence G i # # # (A| In-1 ) = L i A. The maps
Twisted Diagrams 25
F # (T j ) F# (p j )
F # (X j ) x #
# X i for the di#erent objects # : j i of I # i induce a
map G i # # # (T | In-1 ) X i which makes the diagram
G i # # # (A| In-1 )= L i A A i
G i # # # (T | In-1 )
G i ## # (g| I n-1 )
?
X i
f i
?
commute. Now factor the induced map A i # L i (A) (G i # # # )(T | In-1 ) X i as an
acyclic cofibration h i : A i # L i (A) (G i # # # )(T | In-1 ) # T i followed by a fibration
p i : T i
X i in C i . The collection of the T i 's for the di#erent objects i of degree n,
together with T | In-1 define a twisted diagram in Tw (I n , B| In ). The new structure
maps for # : j i are the compositions
F # (T j ) G i # # # (T | In-1 ) A i # L i (A) (G i # # # )(T | In-1 ) h i
#
T i
where the first two maps are the canonical ones. If we define g i as the composition of
the canonical map A i
A i # L i (A) (G i # # # )(T | In-1 ) with h i , it is straightforward to
check that we get a factorization f | In = p| In # g| In in Tw (I n , B| In ). This completes
the induction.
We end up with a factorization of f as A g T p X. The object T | In we
constructed in the induction step coincides with the restriction of T , and similarly for
the maps g and p. It is clear that p is a cfibration in Tw (I, B). To complete the
proof of axiom MC5, it remains to show that the map g is a good acyclic ccofibration.
However, if i is of degree k = mind, the map A i #L i A L i T = A i
g i
T i is an acyclic
cofibration in C i , and if i is of degree n > k, the map A i #L i A L i T T i coincides
with the map h i : A i # L i (A) (G i # # # )(T | In-1 ) T i which is an acyclic cofibration
in C i . Hence g is a good acyclic ccofibration.
We prove part (2) of the theorem. We have already seen that every good acyclic c
cofibration is an acyclic ccofibration (3.2.12). To prove the converse, let f : A X
be an acyclic ccofibration. Factor f as a good acyclic ccofibration g : A T
followed by a cfibration p : T X, and note that p is an acyclic cfibration by
axiom MC2. The map f is in particular a ccofibration, so we can find a lift in the
diagram
A g T
X
f
?
idX
X
p
?
which expresses f as a retract of g. Since good acyclic ccofibrations are closed under
retracts, we are done.
26 T. H˜uttemann, O. R˜ondigs
Knowing (2), we see that axiom MC4 is an immediate consequence of lemma 3.2.7.
This finishes the proof of (1).
Finally, recall from proposition 2.3.1 that pushouts and pullbacks are calculated
pointwise. Since the components of a weak equivalence (cfibration, ccofibration)
are weak equivalences (fibrations, cofibrations) in the respective categories (use corol
lary 3.2.12 for the ccofibrations), assertion (3) follows. ##
Remark 3.2.14. The definition of a direct category can be extended to more general
degree functions having arbitrary ordinals as values (cf. [Ho]). The two inductive proofs
of 3.2.7 and 3.2.13 can be completed with a discussion of the ``limit ordinal case'', thus
giving the cstructure for a larger class of indexing categories.
3.3 The fStructure
The construction of the cstructure can be dualized. There is a notion of a (locally)
inverse category, and matching objects allow us to define an fstructure with pointwise
cofibrations and weak equivalences.
In the following, let B = (C, F, U) be an adjunction bundle of complete model
categories over I. Denote by i # I the full subcategory of the under category i # I
consisting of objects # : i j with # #= id i . Again we have a canonical functor
# : i # I I. Consider C i as a trivial bundle over i # I, and let # : # # B C i
be the i # Imorphism of bundles with #component given by the adjunction
F # : C i
oe C j : U #
for # : i j. Define H i : Tw (i # I, # # B) C i as the composition lim ## # . In
fact, H i coincides with the direct image functor # # where # is the bundle morphism
given by the pair (#, i # I {i}) (here {i} is the trivial category).
Definition 3.3.1. Let Y be a twisted diagram with coe#cients in B. The matching
object of Y at i is defined as M i Y := H i # # # (Y ).
Remark 3.3.2. The structure maps y #
# : Y i
U # (Y j ) for # : i j define a
natural transformation Ev i
M i . If a map Y i
M i Y is mentioned, it is always
this natural map.
Definition 3.3.3. (The fstructure.)
Let f : Y Z be a map in Tw (I, B). We call f a weak equivalence if f i is
a weak equivalence in C i for every object i # I. We call f an f fibration if for all
objects i # I, the induced map Y i
Z i ×M i Z M i Y is a fibration. We call f an
fcofibration if all f i are cofibrations in C i .
Definition 3.3.4. A category with degree function is called a (locally) inverse category
if its opposite category (with the same degree function) is (locally) direct (3.2.1).
Twisted Diagrams 27
Theorem 3.3.5. Suppose I is a locally inverse category, and B is an adjunction
bundle of complete model categories over I.
(1) The fstructure is a model structure.
(2) If f is an f fibration, all its components are fibrations in their respective categories.
(3) A map f : Y Z of twisted diagrams is an acyclic f fibration if and only if for
all objects i # I, the induced map Y i
Z i ×M i Z M i Y is an acyclic fibration
in C i .
(4) If B is a left resp.right proper bundle, the fstructure is left resp.right proper.
##
Remark 3.3.6. In fact, it is possible to construct a model structure on Tw (I, B)
if I is a Reedy category and B consists of complete and cocomplete model categories.
One has to combine the construction of the cstructure and the fstructure. The weak
equivalences are pointwise weak equivalences, the fibrations and cofibrations are more
complicated to define. In the case of diagram categories, this is done in section 5.2 of
[Ho], and the proof given there applies to our situation as well.
3.4 The gStructure
In this section we consider a cofibrantly generated model structure* with pointwise
weak equivalences and pointwise fibrations. (In particular, the gstructure coincides
with the cstructure provided both are defined.) Terminology is taken from [Ho].
Definition 3.4.1. An Ibundle B of cocomplete model categories is called a cofibrantly
generated adjunction bundle if for all objects i # I the model category C i is cofibrantly
generated.
Examples of cofibrantly generated adjunction bundles include the spectrum bun
dle Sp of example 2.2.2 and the projective space bundle P n (G) of 2.5.6. The inverse
image of a cofibrantly generated adjunction bundle is cofibrantly generated.
Since C i has all colimits, the ith evaluation functor Ev i : Tw (I, B) C i has
a left adjoint F r i : C i
Tw (I, B), the ith free twisted diagram functor obtained
by twisted left Kan extension (theorem 2.4.1). Explicitly, for an object A of C i the
jcomponent of F r i (A) is given by the coproduct
#
##homI (i,j)
F# (A)
and the structure maps are given in the following way: if # : j k is a morphism
* The dual case of fibrantly generated structures seems to be irrelevant in practice,
hence is omitted from the discussion.
28 T. H˜uttemann, O. R˜ondigs
in I, the map F r i (A) #
#
is the composition
F # # F r i (A) j # = F # # #
##homI (i,j)
F# (A) # # = #
##homI (i,j)
F # # F# (A) # = #
##homI (i,j)
F ### (A)
#
##homI (i,k)
F # (A)
where the last map is the canonical map induced by the identity on each summand,
mapping the #summand of the source into the # # #summand of the target.
Define M to be the set of maps in Tw (I, B) of the form F r i (f) with i some object
of I and f a generating cofibration in C i . Define N to be the set of maps in Tw (I, B)
of the form F r i (f ), with i some object of I and f a generating acyclic cofibration in C i .
Note that M and N are sets because I is small.
Definition 3.4.2. (The gstructure.)
Let f : Y Z be a map in Tw (I, B). We call f a weak equivalence if f i is a
weak equivalence in C i for every object i # I. We call f a gfibration if f has the right
lifting property with respect to the set N . We call f a gcofibration if f has the left
lifting property with respect to every gfibration which is also a weak equivalence.
Lemma 3.4.3. A map has the right lifting property with respect to the set N (resp. M)
if and only if all its components are fibrations (resp. acyclic fibrations).
Proof. This follows from the adjointness of F r i and Ev i , and the fact that B is
cofibrantly generated. ##
Lemma 3.4.4. The domains of the maps of M are small relative to Mcell. The
domains of the maps of N are small relative to Ncell.
Proof. This follows from the adjointness of F r i and Ev i , and the fact that B is
cofibrantly generated. We give a detailed argument for the case of M . Let A be the
domain of a map in M , so A is of the form F r i (X) for some i # I, with X being the
domain of a generating cofibration in C i . Denote the set of generating cofibrations in
C i by J and recall that X is #small relative to the class Jcell for some cardinal #,
because C i is cofibrantly generated. We will prove that A = F r i (X) is #small relative
to the class M cell.
Let # be a #filtered ordinal and B : # Tw (I, B) be a functor such that the
map B #
B #+1 is in M cell for all # with # + 1 < #. We have to prove that the
canonical map
colim Tw (I, B)(A, B # ) Tw (I, B)(A, colim B)
is an isomorphism. The adjointness of F r i and Ev i provides that this map is isomorphic
to the composite
colim C i (X, Ev i (B # )) C i (X, Ev i # colim B) # = C i (X, colim Ev i # B)
Twisted Diagrams 29
(where the isomorphism is the one from proposition 2.3.1). This composite is the
canonical map, and X is #small relative to Jcell. By [Ho, 2.1.16], X is then even
#small relative to the class of cofibrations in C i . Hence we are done if for all # with
# + 1 < # the map Ev i (b) : Ev i (B # ) Ev i (B #+1 ) is a cofibration. However, since
the maps in M are in particular pointwise cofibrations, and the class of pointwise
cofibrations is closed under cobase changes and transfinite compositions, every map in
M cell is a pointwise cofibration. This finishes the proof. ##
Theorem 3.4.5. Let B be a cofibrantly generated bundle over I. The gstructure is a
model structure on Tw (I, B) which is cofibrantly generated by the sets M and N .
Proof. We use Theorem 2.1.19 of [Ho], which applies also for model categories in the
sense of [DS]. The weak equivalences clearly define a subcategory which is closed under
retracts and satisfies MC2, so condition 1 holds. Lemma 3.4.4 implies conditions 2
and 3, and lemma 3.4.3 implies conditions 5 and 6, and one half of condition 4. It
remains to prove that every map in N cell is a weak equivalence. Since every map
in N is pointwise an acyclic cofibration, and the class of pointwise acyclic cofibrations
is closed under pushouts and transfinite compositions, every map in N cell is pointwise
an acyclic cofibration, so in particular a weak equivalence. ##
Remark 3.4.6. From the general theory of cofibrantly generated model structures,
we know that a morphism f of twisted diagrams is a gcofibration if and only if it is a
retract of a transfinite composition of cobase changes of maps in M . Similarly, acyclic
gcofibrations can be characterised using the set N .
4 Sheaves and Homotopy Sheaves
Let C be a model category, and suppose the diagram category Fun (I, C) carries a
model structure with pointwise weak equivalences as described in one of the previous
sections. There is a canonical functor
h : Fun (I, C) Fun (I, Ho C)
which replaces each structure map of a diagram by its homotopy class (or, more pre
cisely, by its image under the localization functor C Ho C). In particular, the
structure maps of a diagram Y are weak equivalences if and only if the structure maps
of h(Y ) are isomorphisms.
The functor h factors through a functor
•h : HoFun (I, C) Fun (I, Ho C)
which is, in general, not an equivalence of categories.
30 T. H˜uttemann, O. R˜ondigs
In this section, we construct such functors h and •h for twisted diagrams. Unfortu
nately, this is not as straightforward as one could expect since formation of total derived
functors is not functorial. One way to explain this is the following: if the functor U
preserves weak equivalences between fibrant objects, its total right derived RU exists,
and RU(Y ) is given by evaluating U on a fibrant replacement of Y . Thus for U = id C
we see that if C contains objects which are not fibrant, the total right derived of the
identity functor id C is isomorphic to, but di#erent from, the functor id Ho C .
We remedy this by focussing on the full subcategory C f of fibrant objects in C.
This is possible since by a theorem of Quillen, the localization of C f with respect to
weak equivalences is equivalent to the homotopy category of C.
4.1 Associated Homotopy Bundle
Let C denote a model category, and denote by C f the full subcategory of fibrant
objects. The homotopy category Ho C is the localization of C with respect to the class
of weak equivalences. We will use the rather explicit model described in [HS]: the
objects of Ho C are the objects of C, and morphisms are homotopy classes of maps
between cofibrantfibrant replacements. Let Ho f C denote the full subcategory of Ho C
generated by the fibrant objects.
Lemma 4.1.1. The following diagram commutes:
C f C
Ho f C
# f
?
Ho C
#
?
The vertical arrows are localizations with respect to the class of weak equivalences, the
horizontal arrows are full embeddings. The lower horizontal arrow is an equivalence of
categories.
Proof. The image of the composition C f C # Ho C lies inside Ho f C since # is
the identity on objects. Thus # induces the functor # f by restriction.
Since every object of Ho C is isomorphic (in Ho C) to a fibrant object, the inclusion
Ho f C Ho C is dense, hence is an equivalence of categories.
Let C f [W -1 ] denote the localization of C f with respect to the class of weak equiv
alences (this category exists by [Q2], I.1, theorem 1). By construction, # f maps weak
equivalences to isomorphisms, hence there is a canonical map # : C f [W -1 ] Ho f C.
The composite of this map with the inclusion Ho f C Ho C is an equivalence of
categories by [Q, loc. cit.]. We conclude that # is an equivalence, showing that # f is a
localization functor as claimed. ##
Now choose, for each object X # C, a cofibrant replacement p X : X c # X.
If X is cofibrant itself, we choose p X = id X . Similarly, we choose fibrant replacements
q X : X # X f with q X = id X for fibrant X.
Twisted Diagrams 31
Proposition 4.1.2. Suppose U : C D is right Quillen with left adjoint F .
(1) The total right derived RU : Ho C HoD exists and is given by RU(X) :=
U(X f ) on objects. Moreover, the functor RU has a left adjoint LF .
(2) The image of the functor RU lies inside Ho f D, hence RU induces (by restriction)
a functor R f U : Ho f C Ho f D.
(3) Every map # : X Y in Ho f C is represented by a diagram X oeoe # X c f Y
in C.
(4) The functor R f (U) is given by the identity on objects and, using the description
of (3), by R f (U)(#) = # f (U(f)) # (# f (U(pX ))) -1 on morphisms.
(5) We have R f U ## C
f
= # D f #U . Moreover, the functor R f (U) is a left Kan extension
of U along # f .
(6) The equalities R f id C f = id Ho f C and R f (V # U) = R f V # R f U hold.
(7) The functor R f U has a left adjoint, denoted L f F , given (on objects) by the formula
L f (X) := F (X c ).
Proof. This is a standard exercise in model category theory; we indicate a proof briefly.
Claims (1) and (2) are part of [DS], theorem 9.7. The representation of maps in (3)
can be obtained from [DS], proposition 5.11 (note that Y is fibrant by assumption).
It follows by direct calculation that the functor R f U has the form given in (4). The
first part of (5) is true by construction, the second half follows from 4.1.1 and the fact
that RU is the left Kan extension of U along #. Basically by construction (6) holds.
Finally (7) follows from (1) and 4.1.1. ##
In view of the previous lemma, parts (2) and (5) mean that R f U is a ``good''
substitute for RU . Moreover, by parts (6) and (7), the following definition makes
sense:
Definition 4.1.3. (Associated homotopy bundle.)
If B = (C, F, U) is an Iindexed adjunction bundle of model categories, we define
its associated homotopy bundle of fibrant objects Ho f B = (Ho f C, L f F, R f U) as the
Iindexed adjunction bundle given by i ## Ho f C i for objects i # I and # ## L f F # and
# ## R f U # for morphisms # # I.
4.2 Construction of h and •h
Suppose B = (C, F, U) is an Iindexed adjunction bundle of model categories.
We assume that we can equip Tw (I, B) with a model structure with pointwise weak
equivalences (this is certainly possible if I is locally direct or locally inverse, or if
B is a cofibrantly generated bundle). We want to associate to each twisted diagram
Y # Tw (I, B) a corresponding twisted diagram h(Y ) # Tw (I, Ho f B).
Assume for the moment that Y is a twisted diagram with fibrant components.
Let Z denote the following diagram:
32 T. H˜uttemann, O. R˜ondigs
i ## # f (Y i ) = Y i
# ## # f (y #
# ) : Y i
U # (Y j ) = R f U # (# f (Y j ))
We need to check the commutativity condition: if i # j # k are composable
morphisms in I, the following diagram is supposed to commute:
# f (Y i ) # f (y #
# ) R f U # (# f (Y j ))
R f U ## (# f (Y k ))
# f (y #
## )
?
===== R f U # # R f U # (# f (Y k ))
R f U# (# f (y #
# ))
?
Using 4.1.2 (5) we see that this is just the corresponding diagram for Y after application
of # f , hence commutes as desired.
A morphism f : Y •
Y between pointwise fibrant twisted diagrams induces a
map g : Z •
Z with components g i = # f (f i ) as can be shown using 4.1.2 (5) and
functoriality of # f .
Now we use this construction to define the actual functor h (or •h). We discuss
three cases in order of increasing di#culty.
Case 1: All the model categories C i used in the bundle B consist of fibrant objects
only. Then C f
i = C i and R f U # = RU # . The assignment Y ## Z defines (the object
function of) a functor h. By construction it maps weak equivalences to isomorphisms,
hence descends to a functor •h : HoTw (I, B) Tw (I, Ho f B).
Case 2: Suppose that fibrant objects of Tw (I, B) are pointwise fibrant. Suppose
moreover that Tw (I, B) has a fibrant replacement functor Y ## Y f . Then we can
apply the above construction to Y f instead of Y , and the composite Y ## Y f
## Z
defines (the object function of) a functor h. By construction it maps weak equivalences
to isomorphisms, hence descends to a functor •h : HoTw (I, B) Tw (I, Ho f B).
Case 3: Suppose that fibrant objects of Tw (I, B) are pointwise fibrant. Let K de
note the category with objects the fibrant and cofibrant twisted diagrams in Tw (I, B),
and morphisms the homotopy classes of maps between such objects. By [DS, 5.6] the
inclusion # : K HoTw (I, B) is an equivalence of categories. Thus it su#ces to
construct a functor # : K Tw (I, Ho f B); then we can define •h by the composition
of an inverse of # with #.
An object Y # K is in particular a pointwise fibrant twisted diagram. Hence the
construction preceding case 1 applies, and we can define #(Y ) := Z.
A morphism f : Y •
Y in K can be represented by a map •
f : Y •
Y
in Tw (I, B) by [DS, 5.7], and •
f induces a morphism #(f) : #(Y ) #( •
Y ) with
components #(f) i = # f
•
f i . To show that #(f) does not depend on the choice of •
f , recall
Twisted Diagrams 33
that homotopy is an equivalence relation for maps Y •
Y by [DS, 4.22]. More
over, the evaluation functors Ev i (given by Y Y i ) commute with products and
preserve weak equivalences. Hence they preserve path objects and right homotopies.
Thus if •
f and • g are homotopic, so are •
f i and • g i . Since the localization functor identifies
homotopic maps, this proves that #(f) is well defined.
Since homotopy is compatible with composition ([DS, 4.11, 4.19]), and since the
identity morphisms in K are represented by identity maps, # is a functor as required.
4.3 Comparison of Sheaves and Homotopy Sheaves
Definition 4.3.1. (Left strict sheaves.)
Given an Iindexed adjunction bundle B, we call an object Y # Tw (I, B) a left
strict sheaf if the #type structure map y #
# : F # (Y i ) Y j is an isomorphism for all
morphisms # : i j of I. We write Shv (I, B) for the full subcategory of Tw (I, B)
generated by left strict sheaves.
There is also a dual notion of a right strict sheaf requiring that all #type structure
maps are isomorphisms.
Example 4.3.2. (Quasicoherent sheaves on toric varieties.)
Recall the adjunction bundle # op ## Rng Mod associated to a toric variety X with
fan #, cf. 2.5.4. We claim that the category Shv(# op , # op ## Rng Mod ) is equivalent
to the category of quasicoherent sheaves on X. To see this, recall that a cone # # #
corresponds to an open a#ne subscheme U # of X. Given a quasicoherent sheaf F , the
associated twisted diagram is given by # ## F(U # ) with #type structure maps given
by restriction maps. Conversely, a left strict sheaf Y defines quasicoherent sheaves ”
Y #
on the subschemes U # which can be glued via the #type structure maps to give a
quasicoherent sheaf on X. The details are left to the reader.
Definition 4.3.3. (Left homotopy sheaves.)
Suppose that B is an adjunction bundle of model categories. We call an object
Y # Tw (I, B) a left homotopy sheaf if for all morphisms # : i j of I there is
an acyclic fibration •
Y i
# Y i in C i with •
Y i cofibrant such that the adjoint to the
composite
•
Y i
#
Y i
y #
# U # (Y j )
is a weak equivalence in C j . We write hShv (I, B) for the full subcategory of Tw (I, B)
generated by left homotopy sheaves.
Theorem 4.3.4. (Comparison of strict sheaves and homotopy sheaves.)
Let B denote an Iindexed adjunction bundles of model categories. Assume that
we have a map •h as given by one of the cases of § 4.2. An object Y # Tw (I, B) is
a left homotopy sheaf if and only if •h(Y ) # Tw (I, Ho f B) is a left strict sheaf. In
34 T. H˜uttemann, O. R˜ondigs
particular, if Y # Z is a weak equivalence of twisted diagrams, Y is a left homotopy
sheaf if and only if Z is.
Proof. Fix a morphism # : i j of I, and define Z := •h(Y ). By construction, z #
# is a
morphism in Ho f C i which is isomorphic, in Ho C i , to a morphism k # : Y i
RU # (Y j ).
The isomorphism is given by the fibrant replacement used in the construction of •h. If Y i
and Y j happen to be fibrant, the maps z #
# and k # agree.
There is a commutative diagram of categories and functors
Ho f C i
oe R f U# Ho f C j
HoC i
?
oe
RU#
Ho C j
?
where both vertical arrows are equivalences. Hence z #
#
is isomorphic (in Ho C j ) to the
adjoint k # : LF # (Y i ) Y j of k # . In particular, the morphism z #
# is an isomorphism
if and only if k # is.
Choose a cofibrant replacement q i : Y c
i
# Y i of Y i and a fibrant replacement
p j : Y j
# Y f
j of Y j . Let # # denote the composite map
Y c
i
#
q i
Y i
y #
# U # (Y j )
U# (p j )
U # (Y f
j
) .
By the proof of [DS, 9.7] we know that k # is isomorphic to # i (# # ) where # i : C i
Ho C i
denotes the localization functor. Similarly, k # is isomorphic to # j (# # ), where # j denotes
the localization functor for C j , and # # is adjoint to # # . In particular, k # is an isomorphism
if and only if # # is a weak equivalence. But # # factors as F # (Y c
i
) Y j
#
p j
Y f
j
which
shows that # # is a weak equivalence if and only if the homotopy sheaf condition (``at #'')
holds for Y .
The second assertion follows immediately since •h maps weak equivalences to iso
morphisms and the property of being a left strict sheaf is clearly invariant under iso
morphism. ##
Example 4.3.5. Recall the adjunction bundle P n (G) from 2.5.6. This is an ad
junction bundle of model categories. The resulting category hShv # #n#, P n (G) # is the
category P n (G) of Gequivariant quasicoherent sheaves as defined in [H˜u, 6.3].
The index category #n# is direct with degree function d(A) := #A. Hence the
cstructure exists. Moreover, all objects of Tw # #n#, P n (G) # are cfibrant. Thus we can
use the construction of case 1 in § 4.2, and theorem 4.3.4 applies.
Acknowledgements
The authors have to thank M. Brun for helpful comments and suggestions. All
diagrams were typeset with P. Taylor's macro package [T].
Twisted Diagrams 35
References
[BF] A. K. Bousfield, E. M. Friedlander: Homotopy theory of # spaces, spectra, and bisimplicial
sets, Springer Lecture Notes in Mathematics 658, pp. 80--130, SpringerVerlag 1978
[DS] W. Dwyer, J. Spalinski: Homotopy Theory and Model Categories, in: Handbook of Algebraic
Topology (edited by I. M. James) Elsevier Science B. V., 1995
[HKVWW] T. H˜ uttemann, J. Klein, W. Vogell, F. Waldhausen, B. Williams: The
``Fundamental Theorem'' for the Algebraic KTheory of Spaces: I , J. Pure Appl. Algebra 160
(2001), pp. 21--52
[Ho] M. Hovey: Model Categories, Mathematical Surveys and Monographs 63, American
Mathematical Society 1999
[HS] A. Hirschowitz, C. Simpson: Descentes pour les nchamps, Preprint arXiv:math.AG/9807049;
available via internet from xxx.lanl.gov
[H˜u] T. H˜ uttemann: Algebraic KTheory of NonLinear Projective Spaces, to appear in J. Pure
Appl. Algebra
[ML] S. MacLane: Categories for the Working Mathematician, Springer Graduate Texts in
Mathematics 5, SpringerVerlag 1971
[O] T. Oda: Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer
Grenzgebiete 15, SpringerVerlag 1988
[Q1] D. G. Quillen: Higher algebraic Ktheory: I , in: Springer Lecture Notes 341, SpringerVerlag
1973, pp. 85147
[Q2] D. G. Quillen: Homotopical Algebra, Springer Lecture Notes in Mathematics 43, Springer
Verlag 1967
[T] P. Taylor: Commutative Diagrams in T E X , available via internet (anonymous FTP) from
theory.doc.ic.ac.uk as /tex/contrib/Taylor/tex/diagrams.tex