Abstract of “Extending Grothendieck topologies to diagram categories and Serre functors on diagram schemes” by Henning Arn´or Ulfarsson,´ Ph.D., Brown University, May 2009.
We study Serre functors and related constructions. A Serre functor on a triangulated category was defined by Bondal and Kapranov to be an auto-equivalence inducing certain natural dualities on homorphism sets in the category. In the special case of the bounded derived category of complexes of coherent sheaves on a smooth scheme Y the Serre functor is given by twisting by the dualizing sheaf and shifting by the dimension of the scheme. In the work of Lunts it is shown that a Serre functor exists if the scheme Y is replaced by a diagram scheme, which is a collection of schemes connected by morphisms; or more precisely a functor X : D → Schemes where D is some category, often called the shape of the diagram. We will give a description of the Serre functor for certain diagram schemes. Related to the description of the Serre functor for diagram schemes is the study of the category DiagSchemes of diagram schemes and how it inherits properties from the category of schemes. We will consider inheritance of Grothendieck topologies and construct a general method for diagrammatizing any topology in such a way that desirable properties are not lost in the process. We will then consider how to carry prestacks and stacks over along with a Grothendieck topology. A part of this work is joint with Jonathan Wise, at Stanford University. Extending Grothendieck topologies to diagram categories and Serre functors on diagram schemes
by Henning Arn´or Ulfarsson´ M. Sc., Brown University, 2006 B. Sc., University of Iceland, 2004
Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Mathematics at Brown University
Providence, Rhode Island May 2009 c Copyright 2009 by Henning Arn´or Ulfarsson´ This dissertation by Henning Arn´or Ulfarsson´ is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.
Date Dan Abramovich, Director
Recommended to the Graduate Council
Date Dan Abramovich, Reader
Date Thomas Goodwillie, Reader
Date Stephen Lichtenbaum, Reader
Approved by the Graduate Council
Date Sheila Bonde Dean of the Graduate School
iii Vita
Henning Ulfarsson´ was born in 1981 in Reykjav´ık, Iceland. He received his bachelor’s degree in mathematics from The University of Iceland in 2004 and his master’s degree in mathematics from Brown University in 2006. Henning is married to Emil´ıaL´oaHalld´orsd´ottirand together they have three daughters.
iv Acknowledgements
First I thank my wife, Emil´ıa, and my daughters, Salka, Kolka and Hekla, who have given me endless support and joy in my pursuit of becoming a mathematician. I also want to thank my parents who have always encouraged me to follow my dreams. I am grateful also to my advisor, Dan Abramovich, without whose gentle nudging I might never have finished this thesis. I also want to thank the many friends I’ve had through graduate school. In particular, I’m glad to count Conni & Joseph, Panagiotis, Katherine & Jonathan, Michelle & David, Steffen, Noah, Qile and Kevin among the friends I’ve had since I started graduate school. I am also grateful to everyone in Brown’s math department, especially Natalie, Doreen, and Audrey, for all their help.
v Contents
1 Introduction 1 1.1 A short note on the literature ...... 2 1.2 The structure of this thesis ...... 2
I Diagram Categories 4
2 Configuring categories 5
2.1 Constructing the configured category ♦C ...... 5 2.2 2-categorical aspects of ♦C ...... 9
3 Configuring the base of a fibered category 12 3.1 Connections with poset categories ...... 16
4 Configuring sites and mixing topologies 22 4.1 Constructing the necessary fibered products ...... 22 4.2 The configured topologies ...... 26 4.3 Subcanonicity of the configured topology ...... 31
5 Configuring stacks 33
II Serre Functors 36
6 Configurations 37 6.1 Poset schemes and configuration schemes ...... 37 6.2 The order topology on a poset ...... 39 6.3 Poset categories and configuration categories ...... 40 6.4 A note on Serre duality ...... 41 6.5 Building adjunctions ...... 42
vi 7 Constructing the Serre functor for the inclusion of a divisor 45 7.1 Describing the Serre functor in terms of the individual Serre functors ...... 45 7.2 Describing the Serre functor in terms of the dualizing sheaves ...... 47 7.3 Conjectural description of the trace map ...... 51
A Review of some preliminaries 54 A.1 Some category theory ...... 54 A.2 Posets ...... 59
B Auxiliary material 61
B.1 A simpler version of ♦C ...... 61 B.2 Some observations on the topologies on ♦C ...... 62
C Homological algebra of general shapes 66 C.1 Introduction ...... 66 C.2 Replacing the source Z ...... 66 C.3 Generalizing the category of complexes ...... 67
Bibliography 77
vii Chapter 1
Introduction
This project started with my advisor’s question about what the Serre functor [3] looks like for diagram schemes, i.e., diagrams X : S → Sch in the category of schemes1. Any smooth scheme Y has a dualizing sheaf ωY which gives us Serre duality in the bounded derived category of complexes of coherent sheaves on Y, Db(Y ); i.e., for any two complexes E,F in Db(Y) we have a natural isomorphism L ∨ RHom(E,F ) ' RHom(F,E ⊗ ωY [n]) , b b L where n is the dimension of Y . We define a functor S : D (CohY ) → D (CohY ), S(E) = E ⊗ ωY [n], and call it the Serre functor. The Serre functor is a powerful tool for working with coherent sheaves and Lunts proved that a Serre functor exists when the scheme Y is replaced by certain types of diagram schemes in [13], but did not given a concrete description of it. A simple example of a diagram scheme is the inclusion of a divisor into a scheme, D → Y ; this is given by the functor
X : {•1 → •2} → Sch, defined by X(•1) = D, X(•2) = Y and X(•1 → •2) the inclusion of D into Y . I was able to give a description of the Serre functor for this type of diagram scheme with Jonathan Wise and we have a forthcoming preprint on the proof [16]. This makes up Part II of this thesis. Part I contains an investigation of how the category of diagram schemes inherits properties from the category of schemes. In [13] where Lunts proved the existence of the Serre functor he did not construct a category around diagram schemes, so I defined morphisms between diagram schemes and constructed the category DiagSch2. Lunts defined and used poset categories3 to describe sheaves on diagram schemes and I wanted to give a more natural description, in terms of fibered categories. When I had taken care of the above I asked if it were possible to carry Grothendieck topologies and stacks with you when you “diagrammatize”, i.e., if you start with a site (C, T ) and you form the diagram category DiagC, is there an extension of T to DiagC, call it DiagT , such that 1. DiagT restricted to C is equivalent to T . 1Here X is a functor and S is any category, sometimes called the shape of the diagram. 2It turned out that this category had been used in [11, 12], see the note on the literature below, as well as many other places. 3A poset category is a collection of categories “glued together along a poset”.
1 2
2. If T is subcanonical4 then so is DiagT .
3. If F → C is a stack in T , then DiagF → DiagC is a stack in DiagT .
I found two candidate topologies for DiagT , one which satisfies only (1), and another which satisfies (1), (2) and (3). The latter one is constructed in such a way that “both the objects and the arrows in the diagrams are covered”, without going into too much detail.
1.1 A short note on the literature
Diagrams of schemes have been studied before, such as by Illusie in [11], [12], where the a category of sheaves over a fixed configuration scheme is studied as a topos. Here we will be interested in looking at configuration sheaves over many configuration schemes at once and in some sense this should be a topos glued from the individual topoi. We will not delve too deeply into this connection here. See also Remark 2.4.
1.2 The structure of this thesis
Part I
Preliminary material on posets, fibered categories, Grothendieck topologies and stacks can be found in Appendix A. In Section 2 we consider a generalization of Lunts’s configuration schemes by
1. replacing the category of schemes with an arbitrary category C; and
2. replacing the fixed poset S with a category of categories S called the category of shapes.
We define the category of diagrams, denoted ♦C, whose objects are functors X : S → C, where S is an object of S. We give examples of diagram categories and look at some of their basic properties. In Section 3 we replace the category of coherent sheaves on Sch by an arbitrary fibered category
F → C and show in Theorem (3.1) that ♦F → ♦C is also fibered. To get a better description of the fiber ♦F(X) for a diagram X : S → C we define the category of liftings X in Definition 3.2. Lunts uses poset categories to describe these same fibers when S is a (finite) poset and in Theorem (3.13) we show how this is related to our liftings X by using the slightly more general pseudo-poset categories. In Section 4 we fix a Grothendieck topology T on C and ask whether there is a natural configured topology ♦T on ♦C that extends the topology on C and inherits its properties. First we need to take care of a technical detail: if C has all the necessary fibered products the same is true for ♦C. This is taken done in Proposition (4.1). The configured topology ♦T is defined in Definition-Proposition (4.7). We show in Theorem (4.11) that if T is subcanonical then ♦T is also subcanonical. We leave 4A topology is subcanonical if every representable functor is a sheaf. 3 some additional observations on these topologies to Appendix B, since their proofs are rather tedious and we want to get to stacks. In Section 5 we look at prestacks and stacks. In Theorem (5.1) we show that if F → C is a prestack then so is ♦F → ♦C, in the configured topology. In Theorem (5.2) we show that if F → C is a stack then so is ♦F → ♦C, again in the configured topology.
Part II
Here we turn our attention to Serre functors. In Chapter 6 we recall some definitions of [13] compare them to ours. In Section 6.1 we recall Lunts’s definition of a configuration scheme from [13] of them and morphism between them. In Section 6.2 we define the order topology on a poset which allows us to define open and closed embeddings between configuration schemes. In Section 6.3 we see how categories of liftings, section categories and poset categories fit together. We also define coherent and quasi-coherent sheaves on a configuration scheme. In Section 6.4 we recall the definition of a Serre functor from [3] and rephrase it using the Yoneda’s lemma in terms of trace maps. In Section 6.5 we construct adjunction sequences of functors which we will need to describe the Serre functor of a configuration scheme. In Chapter 7 we describe the Serre functor for a configuration scheme which is the inclusion of a divisor in an ambient scheme, up to a non-unique isomorphism; in Section 7.1 in terms of the Serre functors of the individual schemes and then using the dualizing sheaves of the individual schemes in Section 7.2. Finally in Section 7.3 we give a conjectural description of the trace map of the Serre functor, which would give a complete description of the Serre functor.
The Appendix
As mentioned above the Appendix starts with preliminary material in Chapter A. Then comes Chapter B on auxiliary material, not necessary for the main text. Finally there is a chapter exploring how to generalize homological algebra to general diagrams. This is done in Chapter C but see also Example 2.9. Part I
Diagram Categories
4 Chapter 2
Configuring categories
2.1 Constructing the configured category ♦C
Here we will generalize Lunts’s configuration schemes introduced in [13]. But first we look at morphisms of presheaves as motivation:
Definition 2.1. If X is a topological space, a presheaf (of sets) F on X is an assignment:
1. for every open subset U in X we get a set F (U),
2. for every inclusion V ⊆ U of open subsets in X we get a map, called restriction, rUV : F (U) → U F (V ), sometimes written s 7→ s|V , such that the following conditions are satisfied: the restriction rUU : F (U) → F (U) is the identity map idU and given inclusions W ⊆ V ⊆ U we have rVW ◦ rUV = rUW .
We can generalize this definition to a presheaf of objects in a category C, keeping in mind that the restrictions need to be arrows in the category. The definition can also be rephrased using category theory: a presheaf (of objects in a category C) is a functor
op Xcl → C, where Xcl is the category defined in Example (A.12). If we have a two presheaves F and G on the same topological space X, how should we define a morphism φ : F → G? Well, for every open subset U in X we should have a morphism φU : F (U) → G(U) (in the appropriate category) such that for every inclusion V ⊆ U the natural diagram
φU F (U) / G(U)
rUV rUV
φV F (V ) / G(V ) 5 6 commutes. This can also be translated into category theory: A morphism of presheaves is just a op natural transformation between the two associated functors Xcl → C. Note that the components of the natural transformation are the arrows φU . Let’s try to take this a bit further and ask if we can talk about a morphism between presheaves, F op and G, on two different topological spaces X and Y . Here we are dealing with two functors Xcl → C op and Ycl → C with different source categories so we can’t talk about a natural transformation between them. So what can we do? Well, if we had a continuous function f : X → Y we would get a functor −1 fe : Ycl → Xcl defined (on objects) as fe(U) = f (U) and we would get the diagram
Yop cl fe
op G Xcl
# C r F .
(Note that by reversing the arrows in both the source and the target we can view the functor fe as op op having Ycl as its source and Xcl as its target.) Now the functor G and the composition F fe have the same source and target and we can talk about a natural transformation between them—and that is exactly how we define a morphism of the presheaves F on X and G on Y : It consists of a continuous function f : X → Y , giving rise to the functor fe above, and a natural transformation η : G → F fe.
It should be noted that if we have X = Y and take f to be the identity function idX then we arrive back at the usual definition of a morphism of presheaves on the same topological space.
Now we are ready to define the configured category ♦C. We first fix a 2-subcategory S of the 2-category Cat. We will think of the objects of S as the “shapes” we are going to use as the building blocks and we will sometimes refer to S as the category of shapes. If we want to restrict ourselves to configurations in the sense of Lunts we would choose S as FinPos, the category of all finite posets, and place the additional restrictions in Definition 6.1.
Definition-Proposition 2.2. For an arbitrary category C we can construct a new category ♦C, which we call the diagram category, in the following way:
1. The objects of ♦C are functors X : S → C where S is a category in S.
2. An arrow f : X → Y between two objects X : SX → C and Y : SY → C consists of:
S (a) a functor f : SX → SY (in S), and (b) a natural transformation f ν : X → Y f S 7
SX f S
f ν X +3 SY
# C r Y .
S If we need to work with more than one category of shapes we will use the notation ♦C.
Note the similarity with a morphism between two presheaves.
Before proving that ♦C is actually a category let’s write out the definition above in terms of elements: Let α → β be an arrow in SX . Then we have a commutative diagram
f ν X(α) α / Y f S(α)
ν fβ X(β) / Y f S(β) where the unlabeled vertical arrows are the images of the arrow α → β under X and Y f S respectively.
Proof. 1. Composition of arrows and associativity: Consider two arrows f : X → Y and g : Y → S S S ν ν ν Z. Then we have (g ◦ f) = g ◦ f and (g ◦ f)α = gf S(α) ◦ fα, for α in SX . Now assume we have a third arrow h : Z → A. We need to check that the two compositions h ◦ (g ◦ f), (h ◦ g) ◦ f agree. First note that obviously
[h ◦ (g ◦ f)]S = [(h ◦ g) ◦ f]S.
Furthermore,
ν ν ν [h ◦ (g ◦ f)]α =h(g◦f)S(α) ◦ (g ◦ f)α ν ν ν =h(g◦f)S(α) ◦ gf S(α) ◦ fα ν ν =(h ◦ g)f S(α) ◦ fα ν =[(h ◦ g) ◦ f]α,
so h ◦ (g ◦ f) = (h ◦ g) ◦ f.
S ν 2. Identity arrows: We define idX = idSX and (idX )α = idX(α) and we leave it to the reader to
check that idY ◦ f = f and f ◦ idX = f, for any f : X → Y .
Remark 2.3. As in any category, the isomorphisms in ♦C are those arrows f : X → Y which have a two-sided inverse g : Y → X, i.e., g ◦ f = idX and f ◦ g = idY . In our case this means that S S S S S S S idSX = (g ◦ f) = g ◦ f and idSY = (f ◦ g) = f ◦ g so f is an isomorphism (not just an 8
S ν ν ν equivalence) of the underlying shapes with inverse g . Similarly idX(α) = (g ◦ f)α = gf Sα ◦ fα and ν ν ν ν ν idY (β) = (f ◦ g)β = fgSβ ◦ gβ for any α ∈ SX and β ∈ SY . So we see that fα has the inverse gf Sα and therefore that f ν is a natural isomorphism. Requiring that the functor f S be an isomorphism might be too much to ask and it should be S possible to relax the definition of an arrow in ♦C so that f is only required to be an equivalence.
Remark 2.4. The construction of ♦C is by no means new. It is often used when talking about directed limits in the following sense: A directed system in a category C is just a functor X : I → C where I is a directed set. This is of course just the description of an object in ♦C. A morphism between two directed systems is defined in the same way as we defined a morphism of diagrams in
♦C. The papers [10] and [15] use these categories of directed systems to prove Pontryagin duality for compact abelian groups. The notation for these categories used there is CS. When S consists of a single category I (and just the identity functor idI ) this notation generalizes the notation for the functor category CI , where the objects are functors I → C and the arrows are natural transformations.
Example 2.5. 1. The most trivial example arises when we let S contain only the terminal cat-
egory pt which has only one object and its identity morphism. Then ♦C = C. 2. If the category S contains only the poset α ≤ β (and all possible functors from this poset to
itself) then one sees easily that ♦C has the same objects as the arrow category Arr C, [14], but has more morphisms because it contains degenerate morphisms such as
A / C A C . r9 LLL rrr LL f rr g f LL g rr LLL rrr LL rr LL B r D B /% D
3. If we again let S only contain the poset α ≤ β and only allow the identity functor from this
poset to itself then ♦C will be exactly Arr C. See also Examples 6.3, 6.4 and 6.5.
Definition 2.6. Let X, Y be objects of ♦C and f : X → Y be an arrow.
1. For α ∈ SX we call X(α) a component of X. −→ 2. We define the set of arrows inside X as X := X({a | a ∈ Ob Arr(SX )}).
−→ ν ν −→ 3. We define the set of arrows inside f as f := {fα | α ∈ SX }. A member fα of f is called a
component of f. We also define for any β in SY −→ −→ f −1(β) := {g ∈ f | cod(g) = Y (β)},
the set of arrows inside f with target Y (β) and more generally for any subset S in SY define −→ −→ f −1(S) := {g ∈ f | cod(g) ∈ Y (S)},
the set of arrows inside f with target in S. 9
For future reference we record the following two simple results.
Lemma 2.7. For any element β in SY we have
−→−1 ν S ν S −1 f (β) = {fα | α ∈ SX such that f (α) = β} = {fα | α ∈ (f ) (β)}.
Furthermore for any subset S of SY we have
−→−1 ν S ν S −1 f (S) = {fα | α ∈ SX such that f (α) ∈ S} = {fα | α ∈ (f ) (S)}.
ν S Proof. This is clear since cod(fα) = Y (f (α)).
Lemma 2.8. For two arrows f : X → Y and g : Y → Z we have
−−−−→ ν ν (g ◦ f) = {gf S(α) ◦ fα | α ∈ SX }.
−−−−→ ν ν ν Proof. By definition (g ◦ f) = {(g ◦ f)α | α ∈ SX } = {gf S(α) ◦ fα | α ∈ SX }
Example 2.9. Let S consist of the poset Z along with the identity functor idZ : Z → Z. Then ♦C will be the category of complexes of objects of C [4], without the requirement that the composition of two adjacent maps be zero – this is no surprise since in a general category there are no special “zero morphisms”. If C is an abelian category then we can add the requirement that given any
α < β < γ in Z the composition X(α) → X(β) → X(γ) is zero for any object X of ♦C. Then we end up with exactly C(C), the category of complexes of objects from C. I believe there is a more natural way, involving auto-equivalences of the source of the functor X, to enforce this condition. This is pursued further in Appendix C
2.2 2-categorical aspects of ♦C
Configuring a category C is only one part of a 2-functor (see Definition A.1) ♦ : Cat → Cat defined as follows:
1. Given an object C of Cat (i.e., a 0-arrow in Cat) we have described ♦(C) = ♦C above.
0 2. Given a functor F : C → C (i.e., a 1-arrow in Cat) we get a new functor ♦F : ♦C → ♦C0 defined as follows:
(a) The action of ♦F on an object X in ♦C is ♦F (X) = F ◦ X. S (b) Recall that an arrow f : X → Y in ♦C consists of a functor f : SX → SY and a natural ν S transformation f : X → Y f . We get a new arrow ♦F (f): ♦F (X) → ♦F (Y ) whose underlying functor is still f S and whose underlying natural transformation has component ν F (fα) at α ∈ SX . It is easy to see that ♦F is actually a functor. 10
3. Given a natural transformation τ : F → F 0 between two functors F,F 0 : C → C0 (i.e., a
2-arrow in Cat) we get a natural transformation ♦τ : ♦F → ♦F 0 defined as follows: For an arrow f : X → Y in ♦C we get a diagram
(♦τ )X ♦F (X) / ♦F 0 (X)
♦F (f) ♦F 0 (f)
(♦τ )Y ♦F (Y ) / ♦F 0 (Y )
where we need to define the horizontal arrows. We do that in two steps: On the left we see what is happening on the level of the shapes in S and on the right we see what is happening on the ν-level
idS τX(·) X 0 SX / SX FX / F X
f S f S F f ν (·) F 0f ν (·)
idS τY (·) Y S 0 S SY / SY F Y f / F Y f .
This completes the definition of ♦ but we need to make sure that it satisfies the conditions for being a 2-functor, i.e., we must check:
1. For any category C we have = id . This is obvious. ♦idC ♦C
2. For the identity transformation id : F → F we have = id . This is easy to see. F ♦idF ♦F
0 0 00 3. For any two composable functors F : C → C and G : C → C we have ♦F ◦G = ♦F ◦ ♦G: This is easy to prove.
4. Vertical compostion of natural transformations: Given three functors F, G, H : C → C0 and natural transformations τ : F → G and σ : G → H we get a natural transformation σ ◦ τ :
F → H and we need to show that ♦σ◦τ = ♦σ ◦ ♦τ : ♦F → ♦H .
Proof. This follows from the fact that (σ ◦ τ)X(·) = σX(·) ◦ τX(·), when the natural transfor- mations are composed vertically.
5. Horizontal composition of natural transformations: Given functors F,G : C → C0 and H,I : C0 → C00 and natural transformations τ : F → G and σ : H → I we get a natural transforma-
tion σ ∗ τ : HF → IG and we need to show that ♦σ∗τ = ♦σ ∗ ♦τ : ♦HF → ♦IG. 11
Proof. When natural transformations are composed horizontally the components are described as follows
(σ ∗ τ)(·) = σG(·) ◦ Hτ(·).
Therefore for a fixed X ∈ ♦C we have
( ∗ ) = ( ) ◦ ( ) ♦σ ♦τ X(·) ♦σ ♦GX(·) ♦H ♦τ X(·)
= (♦σ)GX(·) ◦ ♦H τX(·)
= σGX(·) ◦ H(τX(·))
= (σ ∗ τ)X(·)
= (♦σ∗τ )X(·).
Which is exactly what we need.
The category ♦C inherits a lot of its structure from the shape category S in the sense that:
1. the objects S of C give us the underlying shape of an object X : S → C in ♦C;
2. the arrows SX → SY of C are functors that give us arrows X → Y in ♦C
SX f S
f ν X +3 SY
# C r Y .
Since S is a category of categories it is inherently a 2-category where the 2-arrows are natural transformations of functors. These give ♦C the structure of a 2-category: A 2-arrow between 1-arrows S S f, g : X → Y is a natural transformation τ (in S) between the underlying functors f , g : SX → SY giving the diagram
f S SX gS Y τ ( SY ν X ν SY `h f g 6>
Y , C r Y .
We leave the details of showing that this makes ♦C a 2-category to the reader. We will only regard ♦C as a 1-category in this thesis. Chapter 3
Configuring the base of a fibered category
In [13] Lunts investigates the category of sheaves over a fixed configuration scheme. Here we ask what the category of all sheaves over all configuration schemes looks like. We start by looking at an arbitrary fibered category [9, SGA I, VI.6.1] p : F → C, configure the base category C and ask whether the category F can “be carried along” to give a new fibered category over ♦C. The answer is as good as we could hope for. (See Definition A.6 for the definition of a fibered category).
Theorem 3.1. If p : F → C is a fibered category then ♦p : ♦F → ♦C is also a fibered category.
Before we give the proof we describe the fibers ♦F(X) over a diagram X from ♦C: Definition 3.2. Let S, C and F be arbitrary categories and assume we have functors X : S → C and p : F → C. We define a new category X, the category of liftings of X (with respect to p) whose objects are liftings, or diagrams of sections, of X along p, i.e., functors x : S → F such that px = X. An arrow between two liftings x and x0 is a natural transformation τ : x → x0 such that p(τ) = idX : X → X i.e., the components of τ project to identity arrows in C,
F F ? 5 I x0 x ^fDD p DDDD p τ DD x S / C S / C . X X Note that we do not limit ourselves to cartesian sections [9, SGA I, VI.5.2]. See also Remark A.3. It is easy to see that this is in fact a category.
Proposition 3.3. The fiber over a diagram X is the category of liftings of X with respect to p, i.e.,
♦F(X) = X. 12 13
Proof. Fix a diagram X : S → C. We break the proof into two steps:
• Objects: We have ξ ∈ ♦F(X), if and only if ♦p(ξ) = X, i.e., ξ is a functor
ξ : S → F,S ∈ S,
such that ♦p(ξ) = X, i.e., p ◦ ξ = X. This is precisely the description of an object in X.
S • Arrows: An arrow φ : ξ → η is in ♦F(X) if an only if ♦p(φ) = idX , i.e., φ = idS and ν p(φα) = idξ(α) for all α ∈ S. This is precisely the description of an arrow in X.
Proof of the Theorem. We need to show that given a morphism f : X → Y and an object y over Y we can find a cartesian arrow φ : f ∗y → y with pφ = f.
F ? y p S S Y f ? JJ Y EM JJ ν JJ f J$ SX / C X
We proceed in three steps:
∗ ∗ ν ∗ S 1. Constructing f y: Given α ∈ SX we define f y(α) = (fα) y(f α). Note that with this comes ∗ S ν a cartesian arrow φα : f y(α) → y(f α) such that pφα = fα. Similarly for an arrow a : α → β we define f ∗y(a) as the unique dashed arrow in the diagram below (it is unique since the arrow
φβ is cartesian).
φ ν ∗ S α S (fα) y(f α) / y(f α) _ ? ?_ ? ? ?? ? ?? ? ?? ? ?? ? ? φ ? ν ∗ S β S (fβ ) y(f β) / y(f β) _ _
X(α) / Y (f Sα) f ν ? α ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? S X(β) ν / Y (f β) fβ
Now by definition we have that p(f ∗y) = X and it is easy to verify that f ∗y is indeed a functor
SX → F.
∗ S S 2. φα defines an arrow φ : f y → y: First of all it is clear that we should set φ := f and ν φα := φα. If φ turns out to qualify as an arrow then pφ = f is obvious. We just need to show 14
that φν is a natural transformation, but that is exactly what the top square on the cube above is telling us!
3. φ is cartesian: Assume we have an object z over Z along with arrows ψ : z → y and g : Z → X, such that the diagram of solid arrows below commutes. We need to produce a unique arrow θ so that everything commutes. z _ ? ψ ? ? ? θ ? ? ? f ∗y /# y _ φ _ Z ?? ?? h ?? ?? g ?? ?? ? # XY/ f
To that end take an arrow a : α → β in SZ and consider the diagram
z(β) ν ψβ _TT _ T _ T _ θβ T _ _ T _ T z(a) _ T ' f ∗y(gSβ) / y(f SgSβ) T φν T _TT gSβ J_ T T T z(α) ν T ψα _ _ T _ T ∗ S _ T f y(g a) θα T _ T _ _ T $ f ∗y(gSα) / y(f SgSα) φν _ gSα _
Z(α) ?? ? hν ?? α gν α ?? ?? ? $ X(gSα) / Y (f SgSα) f ν gSα * * ** Z(β) ** ?? * ? ** ?? * ? hν * ν ?? β * gβ ? * ?? * ? ' * X(gSβ) / Y (f SgSβ) f ν gSβ 15
where the arrows θα and θβ arise as the unique arrows making the inner and outer diagrams commutative. The only part of the diagram that is not guaranteed to be commutative is the “squiggly” square. But we have
ν ∗ S ν φgSβ ◦ f y(g a) ◦ θα = φgSβ ◦ θβ ◦ z(a),
ν and since φgSβ is cartesian, this implies that
∗ S f y(g a) ◦ θα = θβ ◦ z(a),
i.e., everything is commutative and we have defined an arrow θ : z → f ∗y with θS = gS and such that θν is a natural transformation.
By examining the proof of Theorem 3.1 we have proved one direction in the Proposition below.
Proposition 3.4. An arrow φ in ♦F is cartesian if and only if each of its components is cartesian. Proof. Define X = px and Y = py. We have seen above that if every component of φ : x → y is cartesian then φ itself is cartesian. Now assume that φ is cartesian, fix α in Sx and consider one of ν its compents φα and the diagram (of solid arrows)
zα _ ? ψα ? ? ? ? . θα ? ? $ x(α) / y(φSα) φν _ α _ Zα ? ?? ?? hα ?? gα ?? ?? ? $ S X(α) ν / Y (φ α) p(φα
S S ν We can define z : {α} → F by z(α) = zα, Z = pz, ψ : z → y by ψ (α) = φ α and ψα = ψα. S ν Similarly we can define h : Z → Y and finally g : Z → X by g (α) = α and gα = gα. The diagram above becomes z _ ? ψ ? ? ? θ ? ? ? x /# y _ φ _ Z ?? ?? h ?? ?? g ?? ?? ? # XY/ pf 16
and since φ is cartesian we get an arrow θ : z → x which gives us exactly the arrow θα : zα → x(α) we are looking for (and shows that it is unique).
Using this we can now prove:
Proposition 3.5. If we have two fibered categories p : F → C, q : G → C and F : F → G is a functor of fibered categories then we also get a functor of fibered categories
♦F : ♦F → ♦G.
Proof. There are two things we need to show:
1. That ♦q♦F = ♦p. But from Section 2.2 we know that ♦ is associative on functors.
2. We also need to show that ♦F sends cartesian arrows to cartesian arrow, but that follows from Proposition (3.4) and how the action of ♦F on arrows in defined; again see Section 2.2.
Proposition 3.6. For a fixed shape S in S we have a commutative diagram of functors
iS F / ♦F
p ♦p
iS C / ♦C where the horizontal functor iS sends an object U to the diagram which sends all objects of S to U and all arrows of S to the identity arrow idU . The action of iS on arrows is defined similarly.
Proof. This is clear.
3.1 Connections with poset categories
Here we will relate the fibers X = ♦F(X) above to poset categories, defined and studied by Lunts in [13].
Poset categories
Below is a slightly modified version of Lunts’s definition of poset categories from [13], Section 2.1. The modification lies in how we treat posets as categories: Lunts defines an arrow α → β if α ≥ β while we have an arrow in the opposite direction.
Definition 3.7. Let S be a poset, which we view as a category, and let X be a contravariant functor from S to the category of all categories (viewed as a 1-category). That is for each α ∈ S we have a category X (α) and if α ≤ β we have a functor φβα : X (β) → X (α). Since there is either a unique arrow between two elements in a poset or no arrow at all we also have that if α ≤ β ≤ γ then 17
φγα = φβα ◦ φγβ. The pair (S, X ) defines a new category M(S, X ), called a poset category, which may be considered as a gluing of the categories X (α) along the functors φβα. Namely:
1. The objects are collections
F = (Fα ∈ X (α))α∈S
together with structure morphisms for any α ≤ β
σβα(F ) = σβα : φβα(Fβ) → Fα
such that σγα = σβα ◦ φβα(σγβ).
2. The arrows f ∈ Hom(F,G) are collections
f = (fα ∈ Hom(Fα,Gα))α∈S,
which are compatible with the structure morphisms σβα.
Example 3.8. Here is an (almost) example of poset category: Let X : S → Sch be a functor from a poset to the category of schemes. For α ≤ β in S we get a morphism fαβ : X(α) → X(β) of schemes ∗ that induces the inverse image functor fαβ : Coh(X(β)) → Coh(X(α)) between the categories of coherent sheaves over X(β) and X(α). ∗ We would hope to obtain a functor X : S → Cat such that X (α) = Coh(X(α)) and φβα = fαβ in which case we could form a poset category M(S, X ). But this doesn’t quite work because given composable morphisms f and g of schemes, the functors (gf)∗, f ∗g∗ are not equal, they are only isomorphic up to a unique isomorphism. There are two possible ways to fix this problem: Replace the fibered category of coherent sheaves Coh by an equivalent split fibered category; or try to modify the definition of a poset category to make it apply to this example. We choose the latter route below.
Pseudo-Poset Categories
Definition 3.9. Let S be a poset, which we view as a category, and let X be a pseudo-functor [9, SGA I, VI.6.1] (see Definition A.9) from S to the category of all categories (which we now view as a 2-category). That is for each α ∈ S we have a category X (α) and if α ≤ β in S we have a functor
φβα : X (β) → X (α) as well as some natural transformations associated to the pseudo-functor. Then we construct a new category M(S, X ), called a pseudo-poset category, in the following way:
1. The objects are collections
F = (Fα ∈ X (α))α∈S
together with structure morphisms for any α ≤ β
σβα(F ) = σβα : φαβ(Fβ) → Fα 18
which behave well under pulling back: Given elements α ≤ β ≤ γ in S we have three functors
φγβ : X (γ) → X (β),
φβα : X (β) → X (α),
φγα : X (γ) → X (α), ' related by a natural isomorphism ναβγ : φβα ◦ φγβ → φγα (γ) X φγβ
' φγα +3 X (β) ναβγ
! X (α) v φβα and we require that the following diagram
(ναβγ )Fγ φγα(Fγ ) / φβαφγβ(Fγ ) ? ?? ?? ?? φβα(σγβ ) ?? ?? ?? ?? φβα(Fβ) σγα ?? ?? ?? ?? σβα ?? ?? ? Fα is commutative.
2. The arrows f ∈ Hom(F,G) are collections of arrows (fα : Fα → Gα)α∈S which satisfy the following compatibility condition: If α ≤ β then the square
σβα Fα o φβα(Fβ)
fα φβα(fβ )
Gα o 0 φβα(Gβ) σβα commutes. Note that since any contravariant functor X : S → Cat is trivially a pseudo-functor we see that any poset category is trivially a pseudo-poset category. Example 3.10. The functor we were trying to construct in Example 3.8 is actually a pseudo-functor so given a functor X : S → Sch we can glue the fibers of Coh over the individual schemes in X into a pseudo-poset category. In exactly the same way we can glue the fibers of QCoh, the category of quasi-coherent sheaves, over the individual schemes in X into a pseudo-poset category. 19
Pseudo-poset categories are fibers
We will be interested in these two constructions when they arise from a fibered category p : F → C (e.g., the category of coherent sheaves on schemes as in Examples 3.8 and 3.10) in the following way:
Let S be a poset and let X : S → C be a functor, i.e., an element of ♦C. If the fibered category is split we have a functor [9, SGA I, VI.6.1] Φ : C → Cat and we can consider the diagram F
p
S / C/ Cat . X Φ Using the composition ΦX we can define the poset category M(S, ΦX). However, if the fibered category is F not split but comes equipped with a general cleavage then we have a pseudo-functor Φ : C → Cat [9, SGA I, VI.6.1] which we can compose with X and get another pseudo-functor ΦX : S → Cat. This allows us to construct the pseudo-poset category M(S, ΦX).
One might hope that the fiber ♦F(X) would be equivalent to this category but that isn’t quite true. We must first make a slight modification to the fibered category F:
Definition-Proposition 3.11. Let p : F → C be a fibered category. Then we define a new fibered category p : F◦ → C (abusing notation and still using p to denote the functor) over C in the following way:
1. The objects of F◦ are the same as the objects of F.
2. An arrow between two objects ξ and η of F◦ is a pair (f, σ) where f is an arrow f : pξ → pη in C and σ is an arrow σ : f ∗η → ξ in the fiber of F over pξ.
Proof. We need to check that F◦ is actually fibered over C, i.e., given an arrow f : U → V in C and an object η over V we need to construct a cartesian arrow φ : ξ → η over f. We set ξ = f ∗η (note ∗ ∗ that this choice is only unique up to a unique isomorphism) and let σφ : f η → f η be the identity.
We claim that φ = (f, σφ) is the cartesian arrow we are looking for: Let ζ be an arbitrary element over W with an arrow ψ = (h, σψ): ζ → η. Set h = pψ and assume we have an arrow g : W → U such that fg = h. We need to show that there is a unique choice for the arrow θ = (g, σθ) below, making everything commute. ζ _ ? ψ ? ? ? θ ? W ? h ?? ? φ # ?? ξ / η ?? _ _ ?? g ?? ?? ? # UV/ f 20
If we pull everything back to the fiber over W then we are asking that there is a unique arrow ∗ ∗ ∗ ∗ ∗ σθ : g f η → ζ such that νη ◦ σθ = σψ, where νη :(fg) η → g f η is the unique isomorphism. But −1 then we are forced to set σθ = νη ◦ σψ and we are done.
Remark 3.12. 1. Note that F◦ is not the opposite category of F.
2. If we choose a cleavage for F then what we are essentially doing when we construct F◦ from F is
the following: Let ΦF : C → Cat be the associated pseudo-functor of F. Then the composition
op ◦ ΦF (where op is the (covariant) pseudo-functor that sends a category to its opposite category) is the associated pseudo-functor of F◦, i.e., we are reversing the arrows in each fiber.
3. It is clear that F◦◦ = F.
Here is the connection between the pseudo-poset categories and the fibers we were looking for:
Theorem 3.13. Let p : F → C be a fibered category, with a chosen cleavage, and denote its associated pseudo-functor with ΦF. Let X : S → C be a functor from a poset S to the category C and denote the category of liftings of X into F◦ by X◦. Then there is an equivalence of categories
◦ X ' M(S, ΦFX).
Equivalently, there is an equivalence of categories
X ' M(S, ΦF◦ X).
Proof. Note that if we have the first equivalence, then the second one follows:
◦◦ X = X ' M(S, ΦF◦ X).
We therefore only prove the first equivalence, by showing that the two categories in question have the same objects and arrows:
◦ ◦ • An object of X is a functor x : S → F such that px = X. This is equivalent to F = (Fα ∈ F(X(α)), α ∈ S). The structure morphisms of F come from the arrows x(α → β), α ≤ β in S. The fact that these structure morphisms behave well under pulling back comes from x being a functor.
• An arrow of X◦ is natural transformation τ : x → x0, so for any α ≤ β in S we have the usual commutative square. Pulling this square back to the fiber over X(α) yields the compatibility condition with the structure morphisms.
The Theorem implies that any pseudo-poset category (in particular, any poset category) is equivalent to a fiber in some fibered category: Given a pseudo-functor X : S → Cat we can 21 consider the diagram F
p
SS/ / Cat idS X where F is the fibered category associated to X . Then the Theorem says that the pseudo-poset ◦ category M(S, X ) is equivalent to the category of liftings of idS into F , i.e., the fiber over idS in the fibered category ♦F◦ → ♦S. Poset categories and pseudo-poset categories can be viewed as taking separate categories for each element of the poset and gluing them along the relations in the poset. The Theorem allows us to replace the poset S by an arbitrary category and perform a similar gluing construction. Chapter 4
Configuring sites and mixing topologies
4.1 Constructing the necessary fibered products
Before we try to configure a site C (i.e., a category with a Grothendieck topology) [2, SGA3] (see also Definition A.11) we will need to make sure that when the necessary fibered products [14] exist in C the same will be true in ♦C:
Proposition 4.1. Let C be an arbitrary category and consider objects X, Y and Z in ♦C along with arrows f : X → Z, g : Y → Z. The fibered product X ×Z Y exists in ♦C if the following two conditions are satisfied:
1. the fibered product SX ×SZ SY exists in S.
πSX SX ×SZ SY / SX
S πSY f
gS SY / SZ
2. for each (α, β) ∈ SX ×SZ SY the fibered product
X(α) ×Zf Sα Y (β) = X(α) ×ZgSβ Y (β),
22 23
exists in C. πX(α) X(α) ×Zf Sα Y (β) / X(α)
π ν Y (β) fα
ν gβ Y (β) / Zf Sα ZgSβ
and then the fibered product is the functor SX ×SZ SY → C given on objects by
(α, β) 7→ X(α) ×Zf Sα Y (β) = X(α) ×ZgSβ Y (β)
0 0 0 0 and on arrows it sends f :(α, β) → (α , β ) consisting of f1 : α → α and f2 : β → β to u, the unique arrow that fits into the solid arrow diagram
X(α) ×Zf Sα Y (β) ? ? X(f )◦π ? 1 X(α) ? u ? ? ? ? π 0 0 0 X(α ) # 0 X(α ) ×Zf Sα0 Y (β ) / X(α ) .
Y (f2)◦πY (β)
π 0 ν Y (β ) fα0
ν gβ0 Y (β0) / Zf Sα0
The projections πX : X ×Z Y → X and πY : X ×Z Y → Y are defined as follows:
S ν πX := πSX , (πX )(α,β) := πX(α), S ν πY := πSY , (πY )(α,β) := πY (β), and we have the following cartesian square
πX X ×Z Y / X .
πY f
g YZ/ 24
Proof. Given any object W in ♦C with arrows τX : W → X and τY : W → Y making the diagram commute W + SSS + SSS + SSS τX + SSS + SSS ++ SSS + πX SSS + X ×Z Y /) X ++ τY ++ ++ + πY f ++ ++ ++ g YZ/ we automatically get the commutative diagram of solid arrows
SW + ?SSS + SS S SS τX + ? a SSS ++ ? SSS + ? SSS + πS SS + X SS) +SX ×SZ SY / SX . ++ S + τY + ++ S + πSY f ++ ++ ++ gS SY / SZ
By the universal property of the fibered product in S there is a unique functor a : SW → SX ×SZ SY making the diagram commute. For a fixed δ in SW we have
(X ×Z Y )(a(δ)) = (XπS a(δ)) × S (Y πS a(δ)) X Zf πSX a(δ) Y and we can consider the diagram of solid arrows
W (δ) ) ? UUU ) UUUU ) ? UUU ν UU (τ )δ )) ? bδ UUUU X ) ? UUUU ) ? UUUU ) UUUU ) ? πXπ a(δ) UUU ) SX UU* S ) (X ×Z Y )(a(δ)) / Xτ δ . )) X )) )) ν ) (τY )δ ) )) )) ) ν πY π a(δ) f S ) SY τ δ )) X )) )) )) )) ) gν τS δ S Y S S Y τY δ/ Zf τX δ 25
By the universal property of the fibered product in C there is a unique arrow bδ : W (δ) → (X ×Z
Y )(a(δ)) making the diagram commute. Now we need to show that these bδ’s give us a natural 0 transformation W → (X ×Z Y ) ◦ a: Take an arrow h : δ → δ in W and consider the diagram
bδ W (δ)(/ X ×Z Y )(a(δ))
W (h) (X×Z Y )(a(h))
b 0 0 δ 0 W (δ )(/ X ×Z Y )(a(δ )) which will commute for the following reason: The diagram below is commutative as it is stands and will stay commutative if we add in the arrow W (h): W (δ) → W (δ0) or the arrow
0 (X ×Z Y )(a(h)) : (X ×Z Y )(a(δ)) → (X ×Z Y )(a(δ )).
Note that we are not (yet) claiming that we can add in both arrows and keep the diagram commu- tative. W (δ0) ) ? UUU ) ? UUUU ? UU ν ) ? UU (τ ) 0 ) ? bδ0 UUU X δ ) ?? UUUU )) ?? UUUU ) ?? UUUU ) ? π 0 UUU XπS a(δ ) UU )) 0 X * S 0 )(X ×Z Y )(a(δ )) / X(τX δ ) ) ? ν ) (τ ) 0 ) Y δ ) W (δ) )) ) ? UUUU ) ) ? UUU ) ) ? UUU ν ) S ? b UU (τ )δ ) X(τ h) ) ?? δ UUUU X ) X ) ? UU ) ν ) ? UU πY π a(δ0) f S 0 ) ? UUU SY τ δ ) ? UUU X ) ?? π UUU XπS a(δ) UU ) X U* S ) (X ×Z Y )(a(δ)) / X(τ δ) )) ) X ) )) (τ ν ) ) ) Y δ ) ) ) ) η ) ) g ) τS δ0 ) Y (τ S δ0) Y / Z(f Sτ S δ0) )) Y X ) ? ? ) ν πY π a(δ) f S ) SY τ δ )) X ) )) ) Y (τ S h) Z(f Sτ S h) ) Y X ) )) ) gν τS δ S Y S S Y (τY δ) / Z(f τX δ)
S 0 S 0 Furthermore, looking at the arrows W (δ) → X(τX δ ) and W (δ) → Y (τY δ ) in the diagram we 0 should get a unique arrow W (δ) → (X ×Z Y )(a(δ )), making everything commutative. Therefore the two compositions bδ0 ◦ W (h), (X ×Z Y )(a(h)) ◦ bδ agree. 26
We have the obvious Corollary:
Corollary 4.2. If the categories S and C have all fibered products then the same is true for ♦C.
Convention 4.3. From here on we will assume that the categories S and C have all fibered products, unless explicitly stated otherwise.
4.2 The configured topologies
In the case where C is a site we will introduce two topologies on ♦C, making it a site as well. The first topology, the discrete configured topology should be thought of as a middle step in the direction of the topology that we really want, the configured topology. We make the following definition for notational convenience:
Definition 4.4. For any set of arrows U = {fi : Xi → X}i∈I in C we define the incidence set associated to α in SX as
[ −→−1 [ −→ U|α := fi (α) = {g ∈ fi | cod(g) = X(α)}. i∈I i∈I
This is the collection of all arrows inside any fi with target X(α).
The discrete configured topology
Definition-Proposition 4.5. Let C be a site with a Grothendieck topology denoted by T and d define the discrete configured topology ♦T on ♦C as follows: A collection U = {fi : Xi → X} is a covering if for any element α ∈ SX the incidence set U|α is a covering in T .
Proof. We need to check conditions GT1–GT3 from Definition (A.11).
GT1: Let f : X1 → X be an isomorphism, fix an element α ∈ SX and consider the incidence S set U|α for U = {f}. Since f is an isomorphism, f will be an isomorphism of categories (not just S an equivalence). Therefore there is a unique object α1 in X1 with f (α1) = α and U|α consists of a single arrow f ν : X (α ) → X(α) which is an isomorphism, since every component of f is an α1 1 1 isomorphism.
GT2: Let U = {fi : Xi → X} be a covering and f : Y → X an arbitrary arrow. By Proposi- i tion (4.1) the fibered products Xi ×X Y exist. Let V = {πY : Xi ×X Y → Y }, fix an element β ∈ SY and consider the incidence set V|β. We have
i V|β = {πY (β) : Xi(αi) ×X(f Sβ) Y (β) → Y (β) | S S αi ∈ SXi such that fi αi = f β},
S S but this is exactly what you get when you take the covering U|f Sβ = {Xi(αi) → X(f β) | fi αi = S ν S f β} and take its fiber product with fβ : Y (β) → X(f β). 27
GT3: Let U = {fi : Xi → X} be a covering and for each index i let Vi = {gij : Yij → Xi} be a covering and consider the collection of composites W = {fi ◦ gij : Yij → Xi → X}. Fix an element
α ∈ SX and look at the incidence set W|α. By definition [ −−−−−→ W|α = {h ∈ (fi ◦ gij) | cod(h) = X(α)}.
By Lemma (2.7) we have
−−−−−→ ν ν (fi ◦ gij) = {(f ) S ◦ (g )β | βij ∈ SY }, i gij βij ij ij ij
S S and the condition cod(h) = X(α) is equivalent to the condition (fi ◦ gij)(βij) = α, and therefore
[ ν ν S S W|α = {(f ) S ◦ (g )β | βij ∈ SY such that (f ◦ g )(βij) = α}. i gij βij ij ij ij i ij
S If we define αij := gijβij then
[ ν ν S W|α = {(fi )αij ◦ (gij)βij | βij ∈ SYij , fi αij = α} and this is the composition of the covering U|α with the collection of coverings
S {Vi|αi | fi αi = α for some fi : Xi → X};
ν note that the condition on αi means that we have an arrow (fi )αi : Xi(αi) → X(α) in U|α.
Remark 4.6. We call the above topology “discrete” since a diagram X can be covered by its individual components X(α), α ∈ SX .
The discrete configured topology is not the topology we really want and here are two good reasons d why: If T is subcanonical then ♦T is not necessarily subcanonical as shown in Section 4.3; and even d worse, if F is a stack on (C, T ) then ♦F is not necessarily a stack on (♦C, ♦T ) – we leave it to the reader to construct an example showing this. We revisit the discrete configured topology in Appendix B and show that it does preserve the subordinate property on topologies in Proposition B.9.
The configured topology
With the discrete configured topology we are disregarding the arrows inside the diagrams which in some sense are more important than the objects and this will result in ♦F → ♦C failing to be a stack when F → C is a stack. The missing ingredient in the discrete configured topology is condition (2) below.
Definition-Proposition 4.7. Let C be a site with a Grothendieck topology T and define the configured topology ♦T on ♦C as follows: A collection U = {fi : Xi → X} is a covering if
1. for any element α ∈ SX the incidence set U|α is a covering in T , 28
2. for any arrow s : X(α) → X(β) inside X there are subcovers Uα ⊆ U|α and Uβ ⊆ U|β such ν that for any (fi )αi in Uα the diagram on the left can be completed to the diagram on the right ν S with (fi )βi in Uβ, the arrow Xi(αi) → Xi(βi) inside Xi, and fi (αi → βi) = α → β.
ν ν (fi )αi (fi )αi Xi(αi) / X(α) Xi(αi) / X(α)
ν (fi )βi X(β) Xi(βi) / X(β)
We say that the collections Uα, Uβ satisfy the square completion property (with respect to s), abbre- viated SCP.
Note that the collections Uα and Uβ are allowed to depend on the arrow s : X(α) → X(β) so if s s we need to be precise we will write Uα and Uβ, respectively.
Remark 4.8. We should note that condition (2) in the definition above introduces a type of anti- symmetry which we will encounter in the proof of GT3 below.
Proof. Condition (1) above is the same condition as in Definition-Proposition (4.5) and therefore we only need to check condition (2).
GT1: Let f : X1 → X be an isomorphism and fix an arrow X(α) → X(β) inside X. If we examine the proof of (4.5) it is clear that we must take U = U| = {f ν } and U = U| = {f ν } α α α1 β β β1 S where α1 and β1 are the unique objects in SX1 that f sends to α and β respectively. It is also clear that there is an arrow (unique in fact) X1(α1) → X1(β1) that completes the necessary square.
GT2: Let U = {fi : Xi → X} be a covering and f : Y → X an arbitrary arrow. We need to i show that V = {πY : Xi ×X Y → Y } is a covering, so fix an arrow Y (α) → Y (β) inside Y . We have the square f ν Y (α) α / X(f Sα)
ν fβ Y (β) / X(f Sβ) and we can choose coverings
Uf Sα ⊆ U|f Sα, Uf Sβ ⊆ U|f Sβ of X(f Sα) and X(f Sβ), respectively, that satisfy the SCP. These pull back to coverings of Y (α) and
Y (β) which are subcollections of V|α and V|β, respectively. Let’s call them Vα and Vβ and show that they satisfy the SCP. So take any arrow π1 : Y (α) ×X(f Sα) Xi(αi) → Y (α) in Vα and consider 29 the diagram (of solid arrows)
π1 Y (α) ×X(f Sα) Xi(αi) / Y (α) ? ? ? ?? ?? ? ν ? π2 ? fα ?? ?? ?? ?? ?? ?? ? ν ? ψ (f )α i i S Xi(αi) / X(f α) . π1 Y (β) ×X(f Sβ) Xi(βi) ______/ Y (β) ? ? ?? ? ? f ν π2 ? β ? ?? ? ?? ? ?? ? ν ? (f )β i i S Xi(βi) ______/ X(f β)
We can complete the front face because Uα and Uβ satisfy the SCP. Then we complete the bottom S by taking the fiber product of Y (β) and Xi(βi) over X(f β). The only thing missing is the arrow ψ. But the universal property of the fiber product ensures that it exists (and is unique). Finally, ν π1 : Y (β) ×X(f Sβ) Xi(βi) → Y (β) is in Vβ since it is the pullback of the arrow (fi )βi , which is in
Uf Sβ; and ψ is inside Y ×X Xi, so this shows that Vα and Vβ satisfy the SCP.
GT3: Let U = {fi : Xi → X} be a covering and for each index i let Vi = {gij : Yij → Xi} be a covering and consider the collection of composites W = {fi ◦ gij : Yij → Xi → X}. Fix an arrow X(α) → X(β) inside X. We have subcollections Uα ⊆ U|α and Uβ ⊆ U|β which are covers of
X(α) and X(β) satisfying the SCP. For each arrow a : Xi(αi) → Xi(βi) inside Xi with Xi(αi) in U and X (β ) in U we can choose subcollections Va ⊆ V | and Va ⊆ V | covering X (α ) α i i β i,αi i αi i,βi i βi i i and Xi(βi), respectively, and satisfying the SCP. Now define the unions [ [ W = Va ◦ (f ν ) , W0 = Va ◦ (f ν ) , α i,αi i αi β i,βi i βi a a
0 taken over arrows a as described above. It is easy to see that Wα is a covering inside W|α. Wβ might fail to be a covering because it is not guaranteed that all objects of the cover Uβ are a target 0 of an arrow with a source in Uα. We remedy this by adding more arrows to Wβ and let [ W = W0 ∪ Vid ◦ (f ν ) . β β i,βi i βi
Xi(βi)∈Uβ
0 Now Wβ is a covering inside W|β. Note that the arrows in Wβ\Wβ are only added to ensure that
Wβ is a covering and we will not need to manipulate them in the rest of the proof.
I claim that Wα and Wβ also satisfy the SCP. To see that we go back to our arrow X(α) → X(β) 30
and choose an arbitrary object Yij(αij) in Wα and look at the diagram
ν ν (g )α ij ij (fi )αi Yij(αij) / Xi(αi) / X(α)
X(β) .
The arrow Xi(αi) → X(α) comes from Uα so we can complete the square on the right with Xi(βi) →
X(β) in Uβ and a : Xi(αi) → Xi(βi) inside Xi.
ν ν (g )α ij ij (fi )αi Yij(αij) / Xi(αi) / X(α)
a
ν (fi )β Xi(βi) / X(β)
Now Y (α ) → X (α ) comes from Va which has the SCP w.r.t. Va so we can complete the ij ij i i i,αi i,βi square on the left with Y (β ) → X (β ) in Va and Y (α ) → Y (β ) inside Y . ij ij i i i,βi ij ij ij ij ij
ν ν (g )α ij ij (fi )αi Yij(αij) / Xi(αi) / X(α)
a
ν (g )β ν ij ij (fi )β Yij(βij) / Xi(βi) / X(β)
The only thing left to notice is that the composition Yij(βij) → Xi(βi) → X(β) is indeed in Wβ and this completes the proof.
Example 4.9. Assume we have a collection of arrows U = {fi : Xi → X} such that
1. for any element α ∈ SX the incidence set U|α is a covering in T ,
2. for any arrow α → β in SX we can find a collection of arrows {αi → βi} from the SXi such S ν ν that fi (αi → βi) = α → β and the collections {(fi )αi } and {(fi )βi } are coverings in T .
d Then U is clearly a covering in ♦T but I claim that a general covering from ♦T need not satisfy the 0 second property. Here is an example: Let SX = {α → β} and SX1 = {α1 → β1, β1}. Then take two topological spaces A and B and define
0 X(α) = A, X(β) = A t B and X1(α1) = A, X1(β1) = A, X1(β1) = B. 31
Then let X(α → β) be the obvious inclusion and let X1(α1 → β1) = idA. There is an obvious morphism X1 → X between these diagrams which is a covering in ♦T but fails to satisfy the second property above.
See also Example 6.8 Although more complicated than the discrete configured topology it is better in the sense that it preserves subcanonicity and stacks. It also has some other nice, but less interesting, properties which we inspect in Appendix B.
4.3 Subcanonicity of the configured topology
We would hope that if we start with a subcanonical site [2, SGA3] (see also Definition A.14) C then d ♦C would also be subcanonical. This is not the case if we use the discrete configured topology, ♦T , as the following example shows.
Example 4.10. Let SX be a category with two objects α and β with two distinct arrows a and b between them. Define X(α) = A3, X(β) = A1, X(a):(x, y, z) 7→ y3 and X(b):(x, y, z) 7→ z2.
Then let SY be a category with two objects α and β with a single arrow c between them. Define Y (α) = A1, Y (β) = A1 and Y (c): t 7→ t6. Now consider a morphism f : Y → X, defined as S ν 2 3 ν S f (c) = a, fα : t 7→ (t, t , t ) and fβ : t 7→ t; and another morphism g : Y → X with g (c) = b, ν ν ν ν gα = fα and gβ = fβ . The diagram Y can be covered by its components in the discrete topology, as shown below.
1 ______/ 1 / 3 A t7→t A t7→(t,t2,t3) A
t7→t6 (x,y,z)7→y3 (x,y,z)7→z2
Ò 1 1 1 . A t7→t ______/ A t7→t / A It is clear that f and g are equal when pulled back to both objects in the cover, but f 6= g so d Hom(·,X) is not a sheaf in ♦T .
If we use the configured topology ♦T then we are in better shape as the following Theorem shows.
Theorem 4.11. If T is a subcanonical topology on C then ♦T is a subcanonical topology on ♦C. The following proof is inspired by the proof that if C is subcanonical site then C/S is also subcanonical. Here C/S is the category of objects over S with the usual induced topology. See e.g., Proposition 2.59 in [5] for the details.
Proof. We need to show that for any covering {fi : Ui → U}i in ♦C the sequence Y Y Hom(U, X) → Hom(Ui,X) ⇒ Hom(Ui ×U Uj,X) i i,j is an equalizer. We do this in two steps: 32
1. We begin by showing that the first arrow is injective. Take two arrows g, h : U → X that Q have the same image in i Hom(Ui,X). For an arrow a : α → β in SU we have subcovers Uα,
Uβ satisfying the SCP. This gives us at least one arrow ai : αi → βi in some SUi such that S fi (ai) = a. Therefore
S S S S g (a) = g fi (ai) = (g ◦ fi) (ai) S S S = (h ◦ fi) (ai) = h fi (ai) = hS(a).
Since this is true for all a : α → β we have that gS = hS, i.e., g and h have the same underlying ν ν S S functor. Now consider gα, hα : U(α) → X(g α) = X(h α) for some α in SU . These belong to S Hom(U(α),X(g α)) and agree when pulled back to the cover U|α (note that we do not need S ν ν the SCP here). Since Hom( · ,X(g α)) is a sheaf, this implies that gα = hα. This is true for
all α ∈ SU so that g = h. Q 2. Now suppose that we have an element (gi) in i Hom(Ui,X) with the property that for all ∗ ∗ pairs i, j the equality pr1gi = pr2gj holds in Hom(Ui ×U Uj,X). We need to construct an
arrow g in Hom(U, X) such that g ◦ fi = gi for all i. For an arrow a : α → β in SU we have
subcovers Uα, Uβ satisfying the SCP. This gives us a collection of arrows ai : αi → βi such S ∗ ∗ S that fi (ai) = a. The condition that pr1gi = pr gj implies that all these ai get sent via gi 0 0 0 S 0 to the same arrow a : α → β in SX and we can therefore define g a = a . It is now clear S S S that we have g fi = gi for all i, so we are half-way done. Now we have the covering Uα of ν S ν U(α) and arrows (gi )αi in Hom(Ui(αi),X(gi αi)) for every (fi )αi in Uα. These agree when S S S pulled back to any Hom(Ui(αi) ×U(gi αi) Uj(αj),X(gi αi)) and since Hom( · ,X(gi αi)) is a ν S ν sheaf this gives us an element gα in Hom(U(α),X(gi αi)) that pulls back to the (gi )αi . The last thing that we need to do is to show that this defines a natural transformation gν . We 0 ν ν 0 must show that the two elements X(a ) ◦ gα, gβ ◦ U(a) are equal in Hom(U(α),X(β )). But 0 ν 0 ν ν ν Hom( · ,X(β )) is a sheaf and (fi )αi ◦ X(a ) ◦ gα = (fi )αi ◦ gβ ◦ U(a) as can be seen from the diagram U(α) dJ gν t J (f ν ) α tt JJ i αi tt JJ tt U(a) JJ tt JJ 0 zt X(α ) (g ν ) Ui(αi) k i αi U(β) X(a0) ν dJ g t J (f ν ) β tt JJ i βi tt JJ tt JJ tt JJ 0 zt X(β ) (g ν ) Ui(βi) j i βi 0 ν ν so X(a ) ◦ gα = gβ ◦ U(a) as required. Chapter 5
Configuring stacks
From now on, whenever C is a site with a topology T we will consider ♦C with the configured topology ♦T . If F → C is a stack [2, SGA3] (see also Definition A.16) we can ask whether the fibered category
♦F → ♦C is also a stack. We proceed in two steps: first proving that if F → C is a prestack then so is ♦F → ♦C and then proving that if F → C is a stack then so is ♦F → ♦C.
Configuring prestacks
Assume that F → C is a prestack. Let U be an object of ♦C and U = {fi : Xi → X} a covering. To prove that ♦F → ♦C is a prestack we need to show that the functor
♦F(U) → ♦F({fi : Xi → X}) is fully faithful. To do that let ξ and η be objects of ♦F(U), ξi and ηi the pullbacks to Xi, ξij and ∗ ∗ ηij the pullbacks to Xij. Suppose there are arrows δi : ξi → ηi in ♦F(Xi), such that pr1δi = pr2δj : ∗ ξij → ηij for all i and j. We need to define an arrow δ : ξ → η in ♦F(X) such that fi δ = δi for all i. Such an arrow ξ → η is a natural transformation between the functors ξ and η whose components project to identities in C. Take α → β in SX and consider the coverings U|α, U|β of X(α) and X(β). ν For any (fi )αi : Xi(αi) → X(α) in U|α we can pull the objects ξ(α), η(α) back to Xi(αi) and there we have an arrow (δ ) :(f ν )∗ ξ(α) → (f ν )∗ η(α). Since F → C is a prestack and these arrows are i αi i αi i αi compatible when pulled further back to the fibered products they give us an arrow δα : ξ(α) → η(α).
Similarly we get an arrow δβ : ξ(β) → η(β). We need to show that δα and δβ are components of a natural transformation δ. To do that take coverings Uα of X(α) and Uβ of X(β) satisfying the SCP. ν ν Fix an arrow (fi )αi in Uα and take a complementary arrow (fi )βi in Uβ. We need to show that the
33 34 square in the middle (on the top) commutes.
(f ν )∗ ξ(α) (f ν )∗ ξ(β) i αi / ξ(α) / ξ(β) o i βi (δ ) δ (δi) h( NN i αi h( NNNδα h( NNN β h( NN βi ( NN (( NNN (( NNN ( NNN (( N' ( N' ( N' (( ' ( (f ν )∗ η(α) ( η(α) ( η(β) ( (f ν )∗ η(β) i αi ( / ( / o i βi (( S ( S ( S (( S ( (( (( ( (( ( ( (( ( ( ( ( (( (( (( (( ( ( ( ( (( (( (( ( ( ( ( (( ( ν ( ( ν ( Ù (fi )αi Ù s Ù (fi )βi Ù Xi(αi) / X(α) / X(β) o Xi(βi) 3 s0
This is equivalent to pulling that square back to F(X(α)) and asking that square to be commutative
ξ(α) s∗ξ(β) . (5.1) / ∗ N s δβ NN NNN NNN NN δα N' N' η(α) / s∗η(β)
ν Now pull this square back to Xi(αi) along some (fi )αi in Uα. But we can also pull ξ(β) → η(β) 0 ν back to Xi(αi) along s ◦ (fj )βj yielding the diagram
(s0)∗(f ν )∗ ξ(β) j βj t dJJ tt JJ tt JJ tt JJ ztt J (f ν )∗ s∗ξ(β)(f ν )∗ ξ(α) i αi o i αi 0 ∗ ν ∗ 0 ∗ (s ) (f ) δβ =(s ) (δ ) j βj j βj (s0)∗(f ν )∗ η(β) ν ∗ ∗ j βj ν ∗ (f ) s δβ (f ) δα=(δi)α i αi t dJJ i αi i tt JJ tt JJ tt JJ ztt J (f ν )∗ s∗η(β)(f ν )∗ η(α) i αi o i αi .
The square in the back on the left obviously commutes and square in the back on the right commutes because δi is a natural transformation. The triangles on the top and the bottom are easily seen to be commutative and therefore the whole diagram is commutative. ν Since this is true for any (fi )αi in Uα we get that the square (5.1) is commutative and this completes the proof of the following Theorem.
Theorem 5.1. If F → C is a prestack then the fibered category ♦F → ♦C is also a prestack.
Configuring stacks
The next thing to consider is what happens when F → C is a stack. The Theorem above tells us that ♦F → ♦C is at least a prestack and that will be useful. To prove that it is actually a stack wee need to show that for any covering U = {fi : Xi → X}, any object with descent data ({ξi}, {φij}) ∗ is effective, i.e., we can find an object ξ over X such that fi ξ is isomorphic to ξi, for all arrows fi 35
in U. For any α in SX we have a covering U|α of X(α) and an object with descent data, given to us as a restriction of ({ξi}, {φij}). Since F → C is a stack this gives us an object ξα over X(α).
This defines the object-part of a functor ξ : SX → C. We still need to construct the arrow-part, so let α → β be an arrow in SX and choose subcovers Uα and Uβ satisfying the SCP. Constructing an ∗ arrow ξα → ξβ over s : X(α) → X(β) is equivalent to constructing an arrow ξα → s ξβ in F(X(α)). But → is a prestack so it suffices to construct arrows (f ν )∗ ξ → (f ν )∗ s∗ξ for every (f ν ) ♦F ♦C i αi α i αi β i αi in U , that are compatible when pulled back to the fibered products. But (f ν )∗ ξ ∼ ξ (α ) and α i αi α = i i
(f ν )∗ s∗ξ ∼ t∗(f ν )∗ ξ ∼ t∗ξ (β ) i αi β = i βi β = i i
∗ and we have an arrow ξi(αi) → t ξi(βi) coming from the given arrow ξi(αi → βi) (which lies over t). We have proved the following theorem.
Theorem 5.2. If F → C is a stack then the fibered category ♦F → ♦C is also a stack. Part II
Serre Functors
36 Chapter 6
Configurations
In [13] Lunts defines and uses poset categories which we have generalized to pseudo-poset categories (Definition 3.9) and shown to be equivalent to fibers in certain fibered diagram categories (Theorem 3.13). We will use these fibers below because they are easier to generalize if we want to consider more general shapes of diagrams than posets. Below we will work with various different functors arising as pullbacks, pushforwards, etc. and will be using the notation of Illusie in [11] and [12] and of Grothendieck in his unpublished manuscript on derivateurs [8]. This notation will sometimes be inconsistent with that of Lunts in [13].
6.1 Poset schemes and configuration schemes
We fix a category S where all our shapes of diagrams will live. We will mostly consider the cases where S = Pos, the category of posets and order-preserving morphisms, and S = FinPos, the category of finite posets and order-preserving morphisms. It is reasonable to expect that our results extend to more general situations. We will treat a poset S as a category in the following way: The objects are elements α ∈ S and there is an arrow α → β if and only if α ≤ β. Note that in this type of category there is either a unique arrow between two objects or none at all. An order-preserving map between posets is equivalent to a functor between the associated categories. See further details in Section A.2. Recall the following definitions from [13]:
Definition 6.1. A poset of schemes X is a functor X : S → Sch, where S is a poset. If the poset S is finite we say that X is finite. If X is finite and for any α ≤ β in S the morphism X(α) → X(β) is a closed embedding then we call X a configuration scheme. If X(α) is smooth for all α in S then we say that X is smooth. We will sometimes use the notation X(S) to denote the configuration scheme.
Definition 6.2. Posets of schemes (respectively finite poset schemes, configuration schemes) are the objects in a category PosSch (respectively FinPosSch, ConfSch) , whose arrows are defined as follows: An arrow f : X → Y between two objects X : SX → Sch and Y : SY → Sch consists of: 37 38
S 1. a functor f : SX → SY (in S), and
2. a natural transformation f ν : X → Y f S
SX f S
f ν X +3 SY
" Sch s Y .
Example 6.3. Let X2 be a scheme with a divisor X1. The diagram X1 → X2 is a configuration scheme associated to the poset S = {1 ≤ 2}.
Here is an example of a morphism between two poset schemes:
1 1 3 Example 6.4. Let SX = {•1 → •2} and X be the Segre embedding of P × P into P . Let 1 2 5 S SY = {•3 → •4} and Y be the Segre embedding of P × P into P . We define f to be the functor shown in the diagram below.
1 × 1 /o /o /o /o / 1 × 2 P P f ν P P Segre o 1 o ooo ooo ooo oooSegre 3 wo/o /o /o /o /o /o /o 5 wo P O ν / P O f2
X Y
•1 /o /o /o /o /o /o / •3 _ oo f S _ oo ooo ooo oo S oo ooo f ooo •2 w /o /o /o /o /o /o /o / •4 w
We construct the natural transformation part of the arrow f as follows:
ν f1 : ([x : y], [a : b]) 7→ ([x : y], [a : b : 0]) ν f2 :[α : β : γ : δ] 7→ [α : β : 0 : γ : δ : 0], and we leave it to the reader to check that the square on top commutes.
Example 6.5. Consider the three coordinate planes inside A3
2 2 2 Ax=0, Ay=0, Az=0, and the three axes 1 1 1 Ax=y=0, Ax=z=0, Ay=z=0, and finally the origin 0 • = Ax=y=z=0. 39
Using the inclusion maps between these schemes we can build a configuration scheme X,
2 Ax=0 jjj4 ? jjj jjjj 1 j Ax=y=0 t: JJ tt JJ tt JJ tt JJ tt JJ tt JJ tt JJ tt $ • t 1 2 J / Ax=z=0 Ay=0 JJ ? : JJ ?? tt JJ ? tt JJ ?? tt JJ ? tt JJ tt? JJ tt ?? J$ tt ? 1 ?? Ay=z=0 ?? TTT ?? TTTT ? TTT ? TT* 2 Az=0
6.2 The order topology on a poset
There is a topology called the order topology on any poset S defined by letting the sets
Uα = {β ∈ S | β ≤ α} be a basis. Note that this implies that the subset
[ 0 0 Zα = {γ ∈ S | α ≤ γ} = S − {β ∈ S | β ≤ β } β6≥α is a closed subset. An order-preserving map of posets is equivalent to a continuous function in this topology. This is an Alexandroff topology [1] (the open sets are the upper sets). See further details in Section A.2.
Example 6.6. If α ≤ β then the map j : {α},→{α, β} is an open embedding and the map i : {β},→{α, β} is a closed embedding.
Using this topology we can define open and closed embeddings of poset schemes as follows: We S say that i : X → Z is a closed embedding if i : SX → SZ is a closed embedding and every component ν S of i is an isomorphism. Similarly we say that j : Y → Z is an open embedding if j : SY → SZ is an open embedding and every component of jν is an isomorphism. It might be interesting to relax these definitions of embeddings by allowing the components of the natural transformations to be closed (resp. open) embeddings themselves.
Example 6.7. This definition implies in Example 6.3 that X1 is open in the configuration scheme and X2 is closed. 40
Example 6.8. In Example 6.5 we see that U 2 , shown below, Ax=0
2 Ax=0 jjj4 ? jjj jjjj 1 j Ax=y=0 : tt tt tt tt tt tt tt tt 1 • / Ax=z=0
is open inside the configuration scheme X and that U 2 , U 2 , U 2 can be used to cover X in Ax=0 Ay=0 Az=0 the sense of Definition-Proposition 4.7.
6.3 Poset categories and configuration categories
We now turn our attention to operations on poset categories, viewed as categories of sections.
Definition 6.9. Let S be a poset. Given a fibered category p : F → S we define the category of sections Γ(p : F → S) = Γ(p),
0 whose objects are sections x : S → F (i.e., px = idS) and whose arrows η : x → x are natural transformations with the property that p(ηα) = idα.
Given a fibered category p : F → S and a subposet T ⊂ S we form the fibered product T ×S F, which will be a fibered category over T and gives rise to the section category
Γ(T |p) := Γ(π1 : T ×S F → T ) = M(T, ΦF |T ).
Similarly, given a fibered category p : F → Sch and a poset scheme X : S → Sch we get for any subposet T ⊂ S a new poset scheme X|T : T → Sch, given by restriction, which gives rise to the category F(X|T ) = M(T, ΦF◦ (X|T )).
We use this to define coherent and quasi-coherent sheaves on a configuration:
Definition 6.10. 1. Consider the fibered category Coh → Sch.A coherent sheaf on a configura-
tion scheme X : S → Sch is an object in M(S, ΦCoh◦ X). We denote the category of all coherent sheaves on X by Coh(X).
2. Consider the fibered category QCoh → Sch.A quasi-coherent sheaf on a configuration scheme
X : S → Sch is an object in M(S, ΦQCoh◦ X). We denote the category of all quasi-coherent sheaves on X by QCoh(X). 41
Remark 6.11. By Theorem (3.13) both Coh(X) and QCoh(X), for any poset scheme X, can be viewed as either pseudo-poset categories or as fibers. They are poset categories if Coh and QCoh are replaced with their equivalent split categories.
For any scheme the category of (quasi-)coherent sheaves on it is abelian and furthermore for any morphism of schemes f the functor f ∗ is always right-exact, so the categories Coh(X), QCoh(X) are abelian poset categories, as defined in Defintion 2.1 in [13]. By Lemma 2.2 in [13] it follows that these categories are abelian themselves. For a general scheme morphism f the functor f ∗ always has a right adjoint given by the direct image f∗ which in general is only left-exact. If we assume that X is a configuration scheme then
X(a)∗ is exact for any a : α → β in S, so Coh(X) and QCoh(X) are configuration categories, as defined in Definition 2.3 in [13]. Note that we could relax the requirement that all the X(a) be closed embeddings by taking them to be affine, in which case X(a)∗ is still exact.
6.4 A note on Serre duality
Recall the following definition of [3]:
Definition 6.12. Let A be a triangulated C-linear category with finite-dimensional Hom’s. A covariant additive functor S : A → A that commutes with shifts will be called a Serre functor if it is a category equivalence and there are given bi-functorial isomorphisms
∼ ∨ φE,G : HomA(E,G) → HomA(G, S(E)) for E,G ∈ ObA, with the following property: the composite
−1 ∗ ∨ (φG,S(E)) ◦ φE,G : HomA(E,G) → HomA(G, S(E)) → HomA(S(E),S(G)) coincides with the isomorphism induced by S. The set of isomorphisms {φE,G} with this property will be called a Serre structure.
We will need Proposition 3.3 from [3]:
Proposition 6.13. Every Serre functor is exact, i.e., takes distinguished triangles into distinguished.
By Yoneda’s lemma we can phrase Serre duality as follows:
Lemma 6.14. A functor S : A → A as in Definition 6.12 above is a Serre functor if and only if there exists a map tr : Hom(F,S(F )) → k, called the trace map, that induces a perfect pairing
Hom(F,G) × Hom(G, S(F )) → Hom(F,S(F )) → k, which induces the Serre structure. 42
Proof. If the object S(F ) represents the functor Hom(F, ·)∨ then an isomorphism from that functor to Hom(G, S(F )) is given by a universal object. For G = S(F ) this is an element of Hom(F,S(F ))∨, i.e., a map Hom(F,S(F )) → k.
6.5 Building adjunctions
We use the notation F G to mean that the functor F : C ← D is the left adjoint of G : C → D and G is the right adjoint of F . When this is the case there are natural transformations
ε : FG → idC, η : idD → GF, called the counit and the unit of the adjunction, with the properties
idFY = εFY ◦ F (ηY ), idGX = G(εX ) ◦ ηGX for any objects X in C and Y in D.
The inclusion of a vertex
Let M = Γ(p) be any abelian poset category, given by a fibered category p : F → S. Consider the inclusion k : {α},→S of a vertex (which in general is neither an open nor a closed embedding).
There is a full subcategory M{α} of objects that satisfy x(β) = 0 for any β 6= α. By restriction we get another abelian poset category Γ({α}|p) which is just F(α). We have a pullback functor
∗ k : Γ(p) → Γ({α}|p), which is exact. This functor has a left adjoint
k! : Γ({α}|p) → Γ(p), defined by as follows: Let y be an object in Γ({α}|p) = F(α).
• For β ∈ S we define a∗y if there exists an arrow a : β → α, k!(y)β = 0 otherwise.
Note that here we use that there is either a unique arrow β → α or none at all. (This functor ∗ can also be defined with a colimit, as k!(y)β = colima:β→α a y.)
∗ • For an arrow b : γ → β we define an arrow k!(y)b = b k!(y)β → k!(y)γ using the the inverse of ∗ ∗ ∗ the component of the canonical isomorphism νb,a :(ab) → b a at y.
When pushforwards exist (as they will for the cases we consider) the functor k∗ also has a right adjoint
k∗ : Γ({α}|p) → Γ(p) defined as follows: Let y be an object in Γ({α}|p) = F(α). 43
• For β ∈ S we define a∗y if there exists an arrow a : α → β, k∗(y)β = 0 otherwise.
(This functor can also be defined with a limit, as k∗(y)β = lima:α→β a∗y.)
• For an arrow b : β → γ we define an arrow
∗ ∗ k∗(y)b : b k∗(y)γ → k∗(y)β = b (ba)∗y → a∗y
∗ using the (unique) isomorphism (ba)∗y ' b∗a∗y and the counit b b∗ → id.
∗ We now have a sequence of adjoint functors k! k k∗ for any inclusion k : {α} → S of a vertex. If this is actually an open or a closed embedding of a vertex then this sequence can be extended. Before we do that consider general open and closed embeddings.
For any object x in M define its support Supp(x) = {α ∈ S | x(α) 6= 0}. Let U ⊂ S be an open subset and let Z := S \ U be the complementary closed subset. Define two full subcategories of M;
MU consisting of objects with support contained in U and MZ consisting of objects with support contained in Z. By restriction we get the abelian poset categories Γ(U |p) and Γ(Z |p). Let j : U → S and i : Z → S be the inclusions. We get exact pullback functors
∗ ∗ j : Γ(p) → Γ(U |p), i : Γ(p) → Γ(Z |p).
∗ The functor j has an exact left adjoint, j!, which is defined in the same way as the left adjoint of k∗, but is just extension by zero, since there is no arrow β → α such that β∈ / U, α ∈ U. The functor ∗ ∗ i has an exact right adjoint, i∗, which is defined in the same way as the right adjoint of k , but is just extension by zero, since there is no arrow α → β such that β∈ / Z, α ∈ Z. ∗ Since j and i∗ are right adjoints of exact functors it follows that they preserve injectives. All these functors above were defined in [13]. We define two new ones below.
Open and closed embeddings of vertices
When j is an open embedding of a vertex we can extend our sequence of adjunctions to the left:
? ∗ j j! j j∗, where j?(E) = coker(σE). When i is a closed embedding of a vertex we can extend our sequence of adjunctions to the right:
∗ ! i! i i∗ i ,
! E where i (E) = ker(σe ). We fix the following notations for the counits and units of these adjunctions: First for the open embedding j:
? ∗ j ? j! ∗ j ∗ ε : j j → id, ε ∗ : j j → id, ε : j j → id, j! ! j ! j∗ ∗ ∗ j! ? j ∗ j∗ ∗ η : id → j j , η : id → j j , η ∗ : id → j j . j? ! j! ! j ∗ 44
Similarly for the closed embedding i:
∗ i! ∗ i ∗ i∗ ! ε ∗ : i i → id, ε : i i → id, ε : i i → id, i ! i∗ ∗ i! ∗ ∗ ! i ∗ i∗ ∗ i ! η : id → i i , η ∗ : id → i i , η : id → i i . i! ! i ∗ i∗ ∗
Below we will be working in the derived categories of the abelian categories above, so we need to derive our sequences of adjunctions: When j is an open embedding of a vertex we have a sequence of adjunctions of derived functors:
? ∗ Lj j! j j∗.
Where
• j?(E) = coker(σE) and Lj?(E) = C(σE).
• j! is extension by pullback (i.e., extension by zero) and is exact.
• j∗ is restriction to U and is exact
• j∗ is extension by pushforward and is exact.
The last two of these preserve injectives. When i is a closed embedding of a vertex we have a sequence of adjunctions of derived functors:
∗ ! Li! i i∗ Ri .
Where
• i! is extension by pullback. Li! is extension by derived pullback.
• i∗ is restriction to Z and is exact.
• i∗ is extension by pushforward (i.e., extension by zero) and is exact.
! E ! E • i (E) = ker(σe ) and Ri (E) = C(σe )[−1]. The third one of these preserve injectives. We use the same notation for the counits and units of these adjunctions as for the un-derived ones. Using these adjunctions we get the following semi-orthogonal decompositions
∗ ? Li!i → id → j!Lj , (6.1) ∗ ∗ j!j → id → i∗i , (6.2) ! ∗ i∗Ri → id → j∗j . (6.3)
We will use the first two to describe the Serre functor for a configuration scheme which is the inclusion of a divisor in an ambient scheme. Chapter 7
Constructing the Serre functor for the inclusion of a divisor
Let us consider the configuration scheme X = f : D,→Y , the closed embedding of a divisor in an ambient scheme. Assume that both D and Y have dualizing sheaves ωD, ωY and Serre functors
SD(·) = · ⊗ ωD[dim D],SY (·) = · ⊗ ωY [dim Y ].
Let i denote the closed embedding of Y into X and j denote the open embedding of D into X.
In Section 7.1 we will give a description of the Serre functor S = SX of the configuration scheme which only depends on the Serre functors SD and SY , and not their description in terms of the dualizing sheaves, i.e., this description can be generalized to any triangulated category which has similar adjunction sequences as the derived category of the configuration X. We should note that this description is in terms of a cone so will only be up to a non-unique isomorphism. In Section 7.2 we will give a description of S using the two dualizing sheaves, again using a cone. Finally, in Section 7.3 we conjecture how the trace map for S can be defined in terms of the trace maps for the Serre functors of the individual schemes. This would give a complete description of the Serre functor.
7.1 Describing the Serre functor in terms of the individual Serre functors
Here we give a description of the Serre functor S = SX of the configuration scheme which only depends on the Serre functors SD and SY , and not their description in terms of the dualizing sheaves, i.e., this description can be generalized to any triangulated category which has similar adjunction sequences as the derived category of the configuration X. We should note that this description is in terms of a cone so will only be up to a non-unique isomorphism.
45 46
Theorem 7.1. For any sheaf E the Serre functor applied to E equals (up to a non-unique isomor- phism) the cone on a morphism
? θ ∗ j∗SD(Lj E)[−1] → i∗SY (i E).
The morphism is described in the proof.
We first consider two special cases:
Lemma 7.2. For any sheaf E on X we have
∗ ∼ ∗ S(Li!i E) = i∗SY (i E).
∗ ! Proof. Using the adjunction sequence Li! i i∗ Ri and Serre duality on Y we get a chain of isomorphisms, functorial in F , for any sheaf F on X:
∗ ∼ ∗ ∨ Hom(F,S(Li!i E)) = Hom(Li!i E,F ) =∼ Hom(i∗E, i∗F )∨ ∼ ∗ ∗ = Hom(i F,SY (i E)) ∼ ∗ = Hom(F, i∗SY (i E)).
This completes the proof by Yoneda’s lemma.
Lemma 7.3. For any sheaf E on X we have
? ∼ ? S(j!Lj E) = j∗SD(Lj E).
? ∗ Proof. Using the adjunction sequence Lj j! j j∗ and Serre duality on D we get a chain of isomorphisms, functorial in F , for any sheaf F on X:
? ∼ ? ∨ Hom(F,S(j!Lj E)) = Hom(j!Lj E,F ) ? ∗ ∨ =∼ Hom(Lj E, j F ) ∼ ∗ ? = Hom(j F,SD(Lj E))) ∼ ? = Hom(F, j∗SD(Lj E))).
This completes the proof by Yoneda’s lemma.
Proof of Theorem 7.1. Using the semi-orthogonal decomposition 6.1 above we get an exact triangle for any sheaf E on X: ∗ ? Li!i E → E → j!Lj E. When we apply the Serre functor S to this triangle and use Lemmas 7.2 and 7.3 above we get another exact triangle (by Proposition 6.13)
∗ ? i∗SY (i E) → S(E) → j∗SD(Lj E). 47
The terms on the left and right are known in the sense that they only depend on the Serre functors of the individual schemes. We want to use these terms to give a description of the middle term. We start by shifting to get the exact triangle
? θ ∗ j∗SD(Lj E)[−1] → i∗SY (i E) → S(E), so S(E) equals (up to a non-unique isomorphism) the cone on the morphism θ. In order to understand the cone we need to understand the morphism θ. To that end let trY be the trace map associated to the Serre functor SY and let trD be the trace map associated with the Serre functor SD. Now consider the semi-orthogonal decomposition 6.1 of E again:
∗ ? Li!i E → E → j!Lj E.
? ∗ By shifting we get a morphism v : j!Lj E[−1] → Li!i E which we pull back along j and precompose ∗ with the unit, ηj , of the adjunction j j∗ (which happens to be the identity): j! !
j∗ η ∗ ? j! ∗ ? j v ∗ ∗ Lj E[−1] → j j!Lj E[−1] → j Li!i E.
By Serre duality on D and Lemma 6.14 this composition is equivalent to a unique element
∗ w(·) = tr (· ◦ j∗v ◦ ηj ) ∈ Hom(j∗ i i∗E,S ( j?E)[−1])∨. D j! L ! D L
∗ By pulling back along j and post-composing with the counit, εj , of the adjunction j∗ j (which j∗ ∗ again is the identity) we get an element
∗ w0(·) = w(εj ◦ j∗(·)) ∈ Hom( i i∗E, j S ( j?E)[−1])∨. j∗ L ! ∗ D L
i! ∗ Applying Li! and post-composing with the counit, εi∗ , of the adjunction Li! i we get an element
00 0 i! ∗ ∗ ? ∨ w (·) = w (εi∗ ◦ Li!(·)) ∈ Hom(i E, i j∗SD(Lj E)[−1]) .
By Serre duality on Y and Lemma 6.14 this defines a unique element
0 ∗ ? θ ∗ i j∗SD(Lj E)[−1] → SY (i E) which finally gives us the morphism we want:
ηi∗ 0 ? i∗ ∗ ? i∗θ ∗ θ : j∗SD(Lj E)[−1] → i∗i j∗SD(Lj E)[−1] → i∗SY (i E),
i∗ ∗ where the first map is the unit, ηi∗ , of the adjuction i i∗.
7.2 Describing the Serre functor in terms of the dualizing sheaves
Let us consider the configuration scheme X = f : D,→Y , the closed embedding of a divisor in an ambient scheme. Assume that both D and Y have dualizing sheaves ωD, ωY . Let i denote the closed embedding of Y into X and j denote the open embedding of D into X. 48
Here we will give a description of S using the two dualizing sheaves, again using a cone, so this description will also only be up to a non-unique isomorphism.
Theorem 7.4. For any sheaf E the Serre functor applied to E equals (up to a non-unique isomor- phism) the cone on a morphism
∗ λ ∗ Li!(i E ⊗ ωY (D)) → j∗(j E ⊗ ωD). The morphism is described in the proof.
We will use two well-known results here:
∗ Lemma 7.5. Let f be the inclusion D,→Y above. Then f ωY (D) = ωD.
∗ Lemma 7.6 (Projection Fomula). Let f be the inclusion D,→Y above. Then f∗(A⊗f B) = f∗A⊗B. Proof Theorem 7.4. Let E be a sheaf on X and define two new sheaves on X as follows: The first one is ∗ T−1E = Li!(i E ⊗ ωY (D)), with the identity as the structure map. If we write out the components of this sheaf we have
∗ ∗ ∗ Lf (i E ⊗ ωY (D)) i E ⊗ ωY (D) _ _
f D / Y The second one is ∗ T0E = j∗(j E ⊗ ωD), with the counit (which happen to be the identity) as the structure map. If we write out the components of this sheaf we have
∗ ∗ j E ⊗ ωD f∗(j E ⊗ ωD) . _ _
f D / Y
We claim there exists a morphism λ : T−1E → T0E. The part of the morphism over Y is given E ∗ ∗ E as follows: We start with the map σe : i E → f∗j E induced from the structure map σ by the ∗ adjunction f f∗. Now tensor this map with ωY (D), giving a map
∗ ∗ i E ⊗ ωY (D) → f∗j E ⊗ ωY (D).
∗ ∗ The right hand side above equals f∗(j E ⊗ f ωY (D)) by the Projection Formula (Lemma (7.6)). Finally use Lemma (7.5) to get a map
∗ ∗ i E ⊗ ωY (D) → f∗(j E ⊗ ωD),
∗ which is exactly what we need. To get the part of the map over D we use the adjunction Lf f∗. The claim is that when we apply the Serre functor for the configuration scheme to E we get the cone over the morphism we just constructed. We break the proof into several steps: 49
Step 1
Denote
S0(E) = cone(T−1E → T0E), and let’s look at RHom(F,S0E): Using the commutativity of RHom with the cone we get
RHom(F,S0E) = cone(RHom(F,T−1E) → RHom(F,T0E)). (7.1)
We can calculate the individual terms inside this cone; starting with the one on the left:
Step 2
Decompose of F as follows ∗ ∗ j!j F → F → i∗i F.
Begin by looking at the rightmost part of the exact triangle:
∗ ∗ ! RHom(i∗i F,T−1E) = RHom(i F, Ri T−1E) ∗ ∗ = RHom(i F, i E ⊗ ωY ) ∗ ∗ ∨ = RHom(i E, i F [n]) ,
! where the second equality comes from the fact that Ri T−1E) is a cone, shifted by −1, over the unit ∗ ∗ ∗ i E ⊗ ωY (D)) → f∗f (i E ⊗ ωY (D))), and
∗ ∗ ∗ ∗ ∗ ∗ f∗f (i E ⊗ ωY (D))) = f∗(f i E ⊗ ωD) = i E ⊗ f∗ωD = i E ⊗ (ωY ⊗ f∗OD(D)), and we have the exact triangle
ωY → ωY (D) → ωY ⊗ f∗OD(D).
The third equality follows from Serre duality on Y .
Now look at the leftmost part of the exact triangle:
∗ ∗ ∗ RHom(j!j F,T−1E) = RHom(j F, j T−1E) ∗ ∗ ∗ = RHom(j F, f (i E ⊗ ωY (D))) ∗ ∗ ∗ ∗ = RHom(j F, f i E ⊗ f ωY (D)) ∗ ∗ ∗ = RHom(j F, f i E ⊗ ωD) ∗ ∗ ∗ ∨ = RHom(f i E, j F [n − 1]) .
These calculations give an exact triangle
∗ ∗ RHom(i∗i F,T−1E) → RHom(F,T−1E) → RHom(j!j F,T−1E), 50 whose dualized version is
∗ ∗ ∗ ∨ ∗ ∗ RHom(f i E, j F [n − 1]) → RHom(F,T−1E) → RHom(i E, i F [n]), which gives us
∨ RHom(F,T−1E) = ∗ ∗ ∗ ∗ ∗ cone(RHom(i E, i F [n])[−1] → RHom(f i E, j F [n − 1])).
Step 3
Now look at the term on the right in the cone (7.1): Again using the decomposition of F
∗ ∗ j!j F → F → i∗i F, we get for the rightmost part
∗ ∗ ! RHom(i∗i F,T0E) = RHom(i F, Ri T0E) = 0, (7.2) since a cone on an identity map is zero. The leftmost part we can’t calculate directly using adjunction ∗ methods so we decompose j!j F further into the exact triangle
∗ ∗ ∗ j!j F → j∗j F → i∗f∗j F.
For the rightmost part we get
∗ ∗ ∗ RHom(i∗f∗j F,T0E) = RHom(i∗f∗j F, j∗(j E ⊗ ωD)) (7.3) ∗ ∗ ∗ = RHom(j i∗f∗j F, j E ⊗ ωD) = 0.
For the middle part we get
∗ ∗ ∗ RHom(j∗j F,T0E) = RHom(j∗j F, j∗(j E ⊗ ωD)) (7.4) ∗ ∗ ∗ = RHom(j j∗j F, j E ⊗ ωD) ∗ ∗ ∨ = RHom(j E, j F [n − 1]) .
∗ since j j∗ = id.
Now this implies that
(7.2) ∗ RHom(F,T0E) = RHom(j!j F,T0E)
(7.3) ∗ = RHom(j∗j F,T0E)
(7.4) ∗ ∗ ∨ = RHom(j E, j F [n − 1]) . 51
Step 4
Putting all this together, we get
(7.1) RHom(F,S0E) = cone(RHom(F,T−1E) → RHom(F,T0E)) ∗ ∗ =RHom(i E,f∗j F [n−1]) ∨ ∗ ∗ z ∗ ∗ }| ∗ { = cone cone (RHom(i E, i F [n − 1]) → RHom(f i E, j F [n − 1])) ∨ ∗ ∗ → RHom(j E, j F [n − 1])
∨ ∗ ∗ =RHom (E,i∗ cone(i F [n−1]→f∗j F [n−1])) z ∨ }| { ∗ ∗ ∗ = cone RHom(i E, cone(i F [n − 1] → f∗j F [n − 1]))
∨ ∗ =RHom (E,j∗j F [n−1]) z ∨ }| { ∗ ∗ → RHom(j E, j F [n − 1])
∨ ∗ = cone RHom(E, j∗j F [n − 1])
∗ ∗ → RHom(E, i∗ cone(i F [n − 1] → f∗j F [n − 1])) ∨ ∗ ∗ ∗ = RHom E, cone(j∗j F [n − 1] → i∗ cone(i F [n − 1] → f∗j F [n − 1])) .
Step 5
The last step is to prove that
∗ ∗ ∗ cone(j∗j F [n − 1] → i∗ cone(i F [n − 1] → f∗j F [n − 1])) = F [n]. (7.5)
0 ∗ 0 ∗ 0 ∗ ∗ 0 ∗ 0 Let F = F [n − 1], and σe : i F → f∗j F be the map coming from σ : f i F → j F via the ∗ adjunction f f∗. The inner cone in (7.5) is equal to cone(σe). Now
0 ∗ 0 i∗ cone(σe) = cone((id, σe): F → j∗j F ), so we have an exact triangle
0 ∗ 0 0 F → j∗j F → i∗ cone(σe) → F [1]. This shows that ∗ 0 0 cone(j∗j F → i∗ cone(σe)) = F [1] = F [n]. This completes the proof.
7.3 Conjectural description of the trace map
In this section we conjecture how the trace map for S can be defined in terms of the trace maps for the Serre functors of the individual schemes. This would give a complete description of the Serre functor. We will need some special cases as in Lemmas 7.2, 7.3 above: 52
Lemma 7.7. For any sheaf E on X we have
∗ ∼ ∗ S(j!j E) = j∗SD(j E).
? ∗ Proof. Using the adjunction sequence Lj j! j j∗ and Serre duality on D we get a chain of isomorphisms, functorial in F , for any sheaf F on X:
∗ ∼ ∗ ∨ Hom(F,S(j!j E)) = Hom(j!j E,F ) =∼ Hom(j∗E, j∗F )∨ ∼ ∗ ∗ = Hom(j F,SD(j E))) ∼ ? = Hom(F, j∗SD(Lj E))).
This completes the proof by Yoneda’s lemma.
Lemma 7.8. For any sheaf E on X we have
! ∼ ∗ Ri S(E) = SY (i E).
∗ ! Proof. Using the adjunction sequence Li! i i∗ Ri and Serre duality on Y we get a chain of isomorphisms, functorial in F , for any sheaf F on Y :
! ∼ Hom(F, Ri S(E)) = Hom(i∗F,S(E)) ∼ ∨ = Hom(E, i∗F ) =∼ Hom(i∗E,F )∨ ∼ ∗ = Hom(F,SY (i E)).
This completes the proof by Yoneda’s lemma.
We will need also Corollary 5, p. 243 of [6]:
Lemma 7.9. Given two exact triangles (top row and bottom row in the diagram) and a morphism between their middle parts (solid vertical arrow in the diagram)
A / B / C f A0 / B0 / C0 such that Hom(A, C0) = 0, there exist morphisms A → A0 and C → C0, making the diagram commutative. These morphisms are unique provided that Hom(A, C0[−1]) = 0.
Let E be any sheaf and consider the semi-orthogonal decomposition from 6.2:
∗ ∗ j!j E → E → i∗i E
Applying the Serre functor to this decomposition gives a decomposition of S(E):
∗ ∗ S(j!j E) → S(E) → S(i∗i E). 53
Notice that by Serre duality (on X) we have
∗ ∗ ∼ ∗ ∗ Hom(j!j E,S(i∗i E)) = Hom(i∗i E, j!j E) = 0, since the two terms in the Hom-set are supported at different vertices. If shifts are added to either term the Hom-set will still be trivial. By Lemma 7.9 this implies that any morphism ε : E → S(E)
∗ ∗ j!j E / E / i∗i E λ ε µ ∗ f ∗ S(j!j E) / S(E) / S(i∗i E) can be extended to a morphism of triangles in the diagram above. We conjecture:
Conjecture 7.10. The value of the trace map on the given morphism ε : E → S(E) equals the sum of the traces of λ and µ, which can be calculated with the trace maps trD and trY of the individual schemes, as described below.
∗ ∗ More precisely: By Lemma 7.7 the morphism λ : j!j E → S(j!j ) corresponds to a morphism
0 ∗ ∗ λ : j!j E → j∗SD(j E).
∗ Pulling back along j and pre-composing with the unit ηj of the adjunction j j∗ we get a morphism j! !
∗ λ00 = ηj ◦ j∗λ0 : j∗E → S (j∗E). j! D
00 00 Applying the trace map trD to λ produces an element trD(λ ) in k. ! ∗ ∗ Similarly, applying the functor Ri to the morphism µi∗i E → S(i∗i E) gives us
0 ∗ ! ∗ µ i E → Ri S(i∗i E), which by Lemma 7.8 corresponds to a morphism
00 ∗ ∗ ∗ µ : i E → SY (i i∗i E).
∗ Using the counit (inside S ), εi , of the adjunction i∗ i we get a morphism Y i∗ ∗
∗ 00 S (εi ) 000 ∗ µ ∗ ∗ Y i∗ ∗ µ = i E → SY (i i∗i E) −→ SY (i E).
000 000 Applying the trace map trY to µ produces an element trY (µ ) in k. We conjecture that 00 000 tr(ε) = trD(λ ) + trY (µ ). Appendix A
Review of some preliminaries
Here we collect some well-known definitions and results mostly for the convenience of the reader and to fix the notation we will be using.
A.1 Some category theory
Most of the material in this section can be found in the notes by Vistoli [5]. We go through the main definitions but point the reader to [5] for proofs and more detail. We will not make a distinction between large and small categories and will not worry about set-theoretic difficulties. Note that these can be overcome by arguments with universes.
Definition A.1. A 2-category is category enriched over the category of all categories, Cat. More concretely a 2-category C consists of
• A class of 0-arrows (or objects) A, B, . . . .
• For all objects A and B a category Hom(A, B). The objects, f : A → B, of this category are called 1-arrows and its morphisms α : f → g are called 2-arrows; the composition in this category is written α ◦ β and called vertical composition.
• For all objects A, B and C there is a functor
∗ : Hom(B,C) × Hom(A, B) → Hom(A, C),
called horizontal composition, which is associative and admits the identity 2-arrows of idA as identities.
• For any object A there is a functor from the terminal category (with a unique object and a unique arrow) to Hom(A, A).
Sometimes n-arrows are called n-cells for n = 0, 1, 2, and the notion C (A, B) is often used for Hom(A, B). 54 55
The notion of a 2-category differs from the more general notion of a bicategory in that composition of 1-arrows is required to be strictly associative, whereas in a bicategory it need only be associative up to a 2-isomorphism. Note that Cat, the category of all categories is the prototypical example of a 2-category.
Fibered categories and pseudo-functors
Here we will be interested in categories over C, i.e., categories F with a functor p : F → C. We will use the notation x 7→ X to mean that px = X and the commutativity of the diagram
x / y _ φ _
f XY/ will mean that pφ = f.
Definition A.2. An arrow φ : x → y in F is cartesian if given an object z 7→ Z and arrows ψ : z → y and h : Z → Y so the diagram of solid arrows below is commutative z _ ? ψ ? ? ? θ ? ? ? x /# y _ φ _
Z ?? ?? h ?? ?? g ?? ?? ? # XY/ f then there is a unique arrow θ : z → x which makes everything commutative and in particular has the property pθ = g. When x → y is cartesian arrow in F mapping to an arrow X → Y in C we also say that x is a pullback of y to X.
Remark A.3. This definition is taken from [5] where it is also noted that it is more restrictive than the one found in [9, SGA I, VI.6.1]; these cartesian arrows would be called strongly cartesian in [7]. However the resulting notions of a fibered category (see Definition A.6) coincide.
Remark A.4. It is easy to see that a pullback is unique up to a unique isomorphism.
We also leave the proof of the following facts to the reader. 56
Proposition A.5. 1. If F is a category over C, then the composition of cartesian arrows is cartesian.
2. If x → y and y → z are arrows in F and y → z is cartesian, then x → y is cartesian if and only if the composition x → z is cartesian.
3. An arrow F whose image in C is an isomorphism, is cartesian if and only if it is an isomor- phism.
4. Let p : F → C and q : G → F be functors and Φ: ξ → η an arrow in G. If Φ: ξ → η is cartesian over its image qΦ = φ : x → y in F and φ : x → y is cartesian over its image pφ = f : X → Y in C then Φ: ξ → η is cartesian over its image pqΦ = f : X → Y in C.
Definition A.6. A category p : F → C is a fibered category if given an arrow f : X → Y in C and an object y over Y , we can find an object x over X and a cartesian arrow φ : x → y such that pφ = f.
Example A.7. For an arbitrary category C, the arrow category Arr C with the codomain functor cod : Arr C → C is a fibered category precisely when the category C has all fibered products.
Definition A.8. A cleavage of a fibered category F → C consists of a class K of cartesian arrows in F such that for each arrow f : U → V in C and each object y over V there exists a unique arrow φ : x → y in K with image f.
Definition A.9. A pseudo-functor Φ on C consists of the following data.
1. For each object U of C a category ΦU.
2. For each arrow f : U → V a functor f ∗ :ΦV → ΦU.
∗ 3. For each object U of C an isomorphism εU : idU ' idΦU of functors ΦU → ΦU.
f g 4. For each pair of arrows U → V → W an isomorphism
∗ ∗ ∗ νf,g : f g ' (gf)
of functors ΦW → ΦU.
These data are required to satisfy the following compatibility conditions
a. If f : U → V is an arrow in C and η is an object of ΦV , we have
∗ ∗ ∗ ∗ νidU ,f (η) = εU (f η) : idU f η → f η
and ∗ ∗ ∗ ∗ νf,idV (η) = f εV (η): f idV η → f η. 57
f g b. Whenever we have arrows U → V → W →h T and an object θ of ΦT , the diagram
∗ νf,g (h θ) f ∗g∗h∗θ / (gf)∗h∗θ
∗ f νg,h(θ) νgf,h(θ)
νf,hg (θ) f ∗(hg)∗θ / (hgf)∗θ
commutes.
The identification below between fibered categories and pseudo-functors is due to Grothendieck.
Proposition A.10. A fibered category over C with a chosen cleavage is equivalent (see subsections 3.1.2 and 3.1.3 in [5] for the details) to a pseudo-functor on C.
Proof. See [5].
Grothendieck topologies
We will need the to talk about a generalization of a topology on a space which allows us to talk about sheaves over categories:
Definition A.11. Let C be a category. A Grothendieck topology on C is an assignment to each
object U of C of a collection of sets of arrows {Ui → U}i∈I , called coverings of U, so that the following properties are satisfied.
GT1. If V → U is an isomorphisms then the set {V → U} is a covering of U.
GT2. If {Ui → U}i∈I is a covering of U and V → U is any arrow then the fibered products Ui ×U V
exist and the collection of projections {Ui ×U V → V }i∈I is a covering of V .
GT3. If {Ui → U}i∈I is a covering of U, and for each index i we have a covering {Vij → Ui}j∈Ji ,
the collection of composites {Vij → Ui → U}i∈I,j∈Ji is a covering of U.
We will sometimes drop the subscript and write {Ui → U} instead of {Ui → U}i∈I . A category C with a Grothendieck topology is called a site.
Notice that (GT2) and (GT3) imply that if {Ui → U} and {Vj → U} are two coverings of U
then {Ui ×U Vj → U} is also a covering.
Example A.12 (The site of a topological space). Let X be a topological space and denote by Xcl the category whose objects are the open subsets of X and whose arrows are inclusions (the subscript stands for “classical”). Then we put a Grothendieck topology on this category by associating to
each open subset U of X the set of open coverings of U. Here, when we have arrows U1 → U and
U2 → U the fibered product U1 ×U U2 is the intersection U1 ∩ U2. 58
See [5] for more examples.
Definition A.13. A functor F : Cop → Set is called representable if it isomorphic to a functor of the form hX = Hom( · ,X) for some object X in C.
Definition A.14. A Grothendieck topology T on a category C is called subcanonical if every representible functor F : Cop → Set is a sheaf in T .A subcanonical site is a category with a subcanonical topology.
Prestacks and stacks
Let F → C be a fibered category over C with a fixed cleavage. For a covering {Ui → U} we abbreviate
Ui ×U Uj = Uij and Ui ×U Uj ×U Uk = Uijk for each triple of indices i, j and k.
Definition A.15. Let U = {Ui → U} be a covering. An object with descent data ξ = ({ξi}, {φij}) ∗ ∗ on U consists of a collection of objects ξi ∈ F(Ui) and isomorphisms φij : pr2ξj → pr1ξi in F(Uij) satisfying a cocycle condition:
∗ ∗ ∗ ∗ ∗ pr13φik = pr12φij ◦ pr23φjk : pr3ξk → pr1ξi,
th th th where prab and pra are projections on the a and b factor, and the a factor respectively. The isomorphisms φij are called the transition morphisms of the object.
An arrow between two objects with descent data ξ and η is a collection of arrows ξi → ηi in F(Ui) satisfying an obvious compatibility condition. We denote the category of all objects with descent data on U by F(U).
For any covering U = {fi : Ui → U} there is a functor F(U) → F(U) which sends an object ∗ ∗ ∗ ξ ∈ F(U) to ({fi ξi}, {φij}) where the transition morphisms are the unique isomorphisms pr2fj ξ → ∗ ∗ pr1fi ξ. The action on the arrows is defined similarly.
Definition A.16. Let F → C be a fibered category over a site C.
1. F is a prestack over C if for each covering U = {Ui → U} in C, the functor F(U) → F(U) is fully faithful.
2. F is a stack over C if for each covering U = {Ui → U} in C, the functor F(U) → F(U) is an equivalence of categories.
Definition A.17. An object with descent data is effective if it is in the image of F(U) → F(U).
We see that a fibered category is a stack if and only if it is a prestack and all objects with descent data are effective. 59
A.2 Posets
We briefly talk about posets because they are always the underlying type of diagram in [13]. We talk about how they can be treated as either topological spaces or as categories. A partially ordered set or poset is a set with a partial order, i.e., a binary relation ≤ which is reflexive, antisymmetric and transitive. In other words, a partial order is an antisymmetric preorder.
The order topology on a poset
We can put a topology on any poset S by letting the subsets of the form Uα = {β ∈ S | β ≤ α} be a basis. Notice that in this topology the set Zα = {γ ∈ S | γ ≥ α} is a closed subset of S. We will call this the order topology on S. This is an Alexandroff topology [1] (the open sets are the upper sets). When we take the product of two posets S and T we define the product order on S ×T as follows:
(α1, β1) ≤ (α2, β2) if and only if α1 ≤ α2 and β1 ≤ β2. It is easily verified that this is in fact a partial order on S × T . When we take products of posets we are faced with what could be two different choices for the topology on the product: The product of the order topologies and the order topology on the product. The next result shows that the two choices are in fact the same:
Proposition A.18. Let S1 and S2 be two posets and consider the product S1 × S2, with the product order. The order topology on S1 × S2 is equivalent to the product of the order topologies on S1 and
S2.
We will use the following simple result:
Lemma A.19. Let B and B0 be bases for topologies T and T 0, respectively, on X. Then the following are equivalent:
1. T 0 is finer than T .
2. For each x ∈ X and each basis element B ∈ B containing x, there is a basis element B0 ∈ B0 such that x ∈ B0 ⊆ B.
Proof of Proposition A.18. A subbasis for the product topology on S1 × S2 consists of sets of the −1 form πi (U), for open subsets U in Si; i = 1, 2. A basis then consists of finite intersections from the subbasis.
Now take a basis element U(α,β) for the order topology on S1 × S2, and an element (γ1, γ2) in
U(α,β), i.e., γ1 ≤ α and γ2 ≤ β. Therefore γ1 ∈ Uα and γ2 ∈ Uβ. This then implies that
−1 −1 (γ1, γ2) ∈ π1 (Uα) ∩ π2 (Uβ) ⊆ U(α,β).
Now take a basis element V = V1 ∩ · · · ∩ Vn for the product topology on S1 × S2, and an element −1 (γ1, γ2) in V . Assume that the Vi are ordered in such a way that Vi = π1 (Wi) with Wi open in S1 −1 for 1 ≤ i < m and Vi = π2 (Wi) with Wi open in S2 for m ≤ i ≤ n. Note that it is possible that 60
m = n + 1 or m = 1. For 1 ≤ i < m we have γ1 = π1(γ1, γ2) ∈ Wi and since Wi is open we have that Uγ1 ⊆ Wi. Similarly for m ≤ i ≤ n we have γ2 = π2(γ1, γ2) ∈ Wi and since Wi is open we have m−1 n that Uγ2 ⊆ Wi. Therefore Uγ1 ⊆ ∩i=1 Wi and Uγ2 ⊆ ∩i=mWi. Now
−1 −1 (γ1, γ2) ∈ U(γ1,γ2) ⊆ π1 (Uγ1 ) ∩ π2 (Uγ2 ) ⊆ V.
This shows that the two bases generate equivalent topologies.
Continuous map between posets
Proposition A.20. A function f : S1 → S2 between to posets is continuous if and only if it respects the order, i.e., α ≤ β implies f(α) ≤ f(β).
Proof. Assume f respects the order and take a basis element Uα in S2. We have
−1 [ f (Uα) = {γ ∈ S1|f(γ) ∈ Uα} = {γ ∈ S1|f(γ) ≤ α} = Uγ . f(γ)≤α
Now assume that f is continuous and choose α ≤ β. We have that f(α) ≤ f(β) if and only if −1 −1 −1 Uf(α) ⊆ Uf(β) if and only if f (Uf(α)) ⊆ f (Uf(β)). But β is in the open subset f (Uf(β)) so this subset must also contain Uβ, the smallest open subset containing β. This implies that α lies in −1 f (Uf(β)), i.e., f(α) ≤ f(β). So f respects the order.
Posets viewed as categories
We can view a poset S as a category in the following way: The objects are just the elements of the poset and for two objects α, β in S there is exactly one arrow α → β if α ≤ β. It is easy to see that continuous maps between posets correspond to functors between the corresponding categories so we have the following result.
Proposition A.21. Let f : S → T be a function between two posets. The following are equivalent:
1. The function f is continuous when the posets are given the order topology.
2. The function f preserves order.
3. The function f is a functor when the posets are viewed as categories. Appendix B
Auxiliary material
We collect here some auxiliary material not necessary for the main text.
B.1 A simpler version of ♦C
This simpler version is mentioned here because for many applications it is sufficient.
Definition-Proposition B.1. For an arbitrary category C we can construct a new category C in the following way:
1. The objects of C are functors X : S → C where S is a category in S.
2. An arrow f : X → Y between two objects X : SX → C and Y : SY → C consists of:
S (a) a functor f : SX → SY (from S), and (b) a natural transformation f ν : X → Y f S, all of whose components are isomorphisms.
SX f S
f ν X +3 SY
# C r Y .
S If we need to work with more than one category of shapes we will use the notation C.
Again is worth while writing out the definition above in terms of elements: Let α → β be an
61 62
arrow in SX . Then we have a commutative diagram
f ν X(α) α Y f S(α) .
ν fβ X(β) Y f S(β)
Proof. Analogous to the proof of Definition-Proposition 2.2.
We can obviously extend items 2.6–2.8 above to the category C.
B.2 Some observations on the topologies on ♦C
We recall some properties of Grothendieck topologies and then inspect how they behave when we form the discrete configured topology and the configured topology.
Definition B.2. Let C be a category and U = {Ui → U}i∈I a set of arrows. A refinement
V = {Va → U}a∈A of U is a set of arrows such that for each index a in A there is some index i in I such that Va → U factors through Ui → U.
V1 V2
55 U1 Ù U2 55 ÙÙ 55 ÙÙ 5 ÔÙÙ U
Notice that the choice of factorization Va → Ui → U is not part of the data; we simply require their existence.
A refinement of a refinement is obviously a refinement. Also, any covering is a refinement of itself: thus the relation of being a refinement is a pre-order on the set of coverings of an object U.
Definition B.3. Let C be a category, T and T 0 two topologies on C. We say that T is subordinate to T 0, and write T T 0, if every covering in T has a refinement that is a covering in T 0. If T T 0 and T 0 T , we say that T and T 0 are equivalent and write T ≡ T 0.
Proposition B.4. Let C be a category, T and T 0 two topologies on C. If T is subordinate to T 0 then every sheaf in T 0 is also a sheaf in T .
Proof. Left to the reader.
Definition B.5. A topology T on a category C is called saturated if a set of arrows {Ui → U} which has a refinement that is in T is also in T . The saturation T of T is the set of sets of arrows which have a refinement in T . 63
Proposition B.6. Let T be a topology on a category C.
1. The saturation T of T is a saturated topology.
2. T ⊆ T .
3. T is equivalent to T .
4. The topology T is saturated if and only if T = T .
5. A topology T 0 on C is subordinate to T if and only if T 0 ⊆ T .
6. A topology T 0 on C is equivalent to T if and only if T 0 = T .
7. A topology on C is equivalent to a unique saturated topology.
Proof. We leave the easy proofs to the reader.
The discrete configured topology
0 d d Lemma B.7. If T is subordinate to T then ♦T is subordinate to ♦T 0 . In particular, if T is 0 d d equivalent to T then ♦T is equivalent to ♦T 0 .
d Proof. Let U = {Xi → X} be a covering in ♦T , i.e., for each α ∈ SX the incidence set U|α is a 0 covering in T and thus has a refinement Vα = {vα,a → X(α)} in T . We need to show that U has d a refinement in ♦T 0 . To do that, define Vi in the following way: The underlying category SVi is the collection of all (α, a) such that vα,a → X(α) factors through some arrow inside Xi → X. This category is discrete in the sense that there are no non-identity arrows in it. Then we define the functor Vi : SVi → C by sending (α, a) to vα,a. Now we have an obvious arrow fi : Vi → X for which S ν fi (α, a) = α and fi (α, a): Vi(α, a) → X(α) = vα,a → X(α). I claim that V := {Vi → X} is a refinement of U. But this is obvious since Vi → X factors through Xi → X (almost) by definition. d Furthermore, I claim that V is a covering in ♦T 0 . To show this we must check that V|α is a covering 0 in T . But V|α = Vα so that is immediate.
Proposition B.8. Given a site C with a topology T the saturation of the discrete configuration of T and the discrete configuration of the saturation of T are equivalent as topologies on ♦C, i.e., d ≡ d . ♦T ♦T Proof. We will use the fact that T and T are equivalent topologies and Lemma (B.7): T ≡ T implies
♦T ≡ ♦T and thus d d d ♦T ≡ ♦T ≡ ♦T .
In fact the two topologies in the proposition are more than equivalent, they are equal as the next proposition shows.
Proposition B.9. Given a site C with a topology T the saturation of the discrete configuration of T and the discrete configuration of the saturation of T are equal as topologies on , i.e., d = d . ♦C ♦T ♦T 64
Proof. To prove that d ⊆ d let U = {f : X → X} be a covering in d , i.e., U has a refinement ♦T ♦T i i ♦T d V = {ga : Va → X} in ♦T , i.e.,
1. for each index a there is some index ia such that ga : Va → X factors through fia : Xia → X; and
2. for each α ∈ SX the incidence set V|α is a covering in T .
Now let α in SX be fixed and consider the incidence set U|α. I claim that U|α has the refinement ν V|α, which implies that U|α is in T . To see this take g ∈ V|α so g = ga (β) for some β ∈ SVa . Since ga factors through fia we can write ga = fia ◦ ha for some ha : Va → Xi and we have
g = gν (β) = (f ◦ h )ν (β) = f ν (hSβ) ◦ hν (β), a ia a ia a a i.e., g factors through f ν (hSβ): X (hSβ) → X(f S hSβ) = X(α), which is an arrow in U| . ia a ia a ia a α To prove that d ⊆ d let U = {X → X} be a covering in d , i.e., for all α in S the ♦T ♦T i ♦T X incidence set U|α is a covering in T , i.e., U|α has a refinement Vα = {vαa → X(α)} which is in d T . We need to show that U has a refinement V in ♦T . To do that, define Vi in the following way:
The underlying poset SVi is the collection of all αa such that vαa → X(α) factors through some arrow inside Xi → X. This poset is discrete in the sense that there are no arrows in it. Then we define the functor Vi : SVi → C by sending αa to vαa. Now we have obvious arrows fi : Vi → X for S ν which fi (αa) = α and fi (αa): Vi(αa) → X(α) = vαa → X(α). I claim that V := {Vi → X} is a refinement of U. But this is obvious since Vi → X factors through Xi → X (almost) by definition. Furthermore, I claim that V is a covering in ♦T . To show this we must check that V|α is a covering in T . But V|α = Vα so that is immediate.
The configured topology
We want similar statements to Lemma B.7 and Propositions B.8 and B.9 for the configured topology
♦T .
0 Lemma B.10. If T is subordinate to T then ♦T is subordinate to ♦T 0 . In particular, if T is 0 equivalent to T then ♦T is equivalent to ♦T 0 .
Proof. Let U = {Xi → X} be a covering in ♦T , so
1. for any element α ∈ SX the incidence set U|α is a covering in T ,
2. for any arrow s : X(α) → X(β) inside X there are subcovers Uα,s ⊆ U|α and Uβ,s ⊆ U|β such ν that for any (fi )αi in Uα,s the diagram on the left can be completed to the diagram on the 65
ν S right with (fi )βi in Uβ,s, the arrow Xi(αi) → Xi(βi) inside Xi, and fi (αi → βi) = α → β.
ν ν (fi )αi (fi )αi Xi(αi) / X(α) Xi(αi) / X(α)
ν (fi )βi X(β) Xi(βi) / X(β) .
3. there exist refinements
0 0 0 0 0 0 Uα,s = {Xi(αi) → X(α)} and Uβ,s = {Xi(βi) → X(β)}
0 of Uα,s and Uβ,s, respectively, that are coverings in T .
0 0 There are no natural arrows between the objects in the two collections Uα,s and Uβ,s so there is no reason why they should satisfy the SCP. To fix that define
00 0 0 Uα,s = Uβ,s ×X(β) X(α) ×X(α) Uα,s.
0 This bigger set is still a covering in T and it still factors through Uα,s. More importantly the covers 00 0 Uα,s and Uβ,s satisfy the SCP. Now we need to construct some diagrams: Let Vi consist of objects and arrows of the form 0 0 0 0 0 0 π1 : Xi(βi) × X(α) × Xi(αi) → Xi(βi), that factor through Xi. It is clear that Vi has a morphism to X which factors through Xi and we define V = {Vi → X}. It is clear that V is a refinement of U and moreover it is clear that it is a cover in ♦T 0 .
Proposition B.11. Given a site C with a topology T the saturation of the configuration of T and the configuration of the saturation of T are equivalent as topologies on ♦C, i.e., ♦T ≡ ♦T .
Proof. We will use the fact that T and T are equivalent topologies and Lemma B.10 above: T ≡ T implies ♦T ≡ ♦T and thus
♦T ≡ ♦T ≡ ♦T .
I have not been able to produce a proof of the equivalent of Proposition (B.9) for the configured topology. I do however believe that it holds. Appendix C
Homological algebra of general shapes
C.1 Introduction
This is an elaboration on Example 2.9, where we attempt to generalize constructions from homolog- ical algebra such as cohomology groups, chain homotopies and quasi-isomorphisms to more general shapes. We start by thinking about ordinary homological algebra as methods of looking at functors A : Z → A, where A is an abelian category, with the restriction that
A(n → n + 1 → n + 2) = 0.
Then we replace the integers Z with a category S and try to mimic the restriction above using auto-equivalences on S.
C.2 Replacing the source Z
First let us consider a functor is A : Z → A where we think about the integers Z as a category with an arrow n → m for every n ≤ m. This is a complex, the basic object of study in homological algebra—except the main ingredient is missing: the condition that two adjacent morphisms in the complex always compose to zero. So let’s first of all assume that A is abelian so we can actually talk about zero morphisms. Then our goal is to try to impose this condition in a way that easily generalizes when we replace the integers Z with an arbitrary category S. Here is how we go about doing this: There is a very natural shift functor, i.e., an auto-equivalence [1] on Z that shifts everything to the right and requiring that every two adjacent morphisms in a complex A : Z → A compose to zero is equivalent to requiring that the composition A(a[1]) ◦ A(a) is the zero morphism for every arrow a : n → m that can be composed with its shift a[1].
66 67
Now we can easily generalize this: Let S be a category with a collection of shift functors {τi}.
Then A : S → A is called a configured complex, or confplex, if for any shift functor τi and any arrow a that can be composed with τia we have
A(τia) ◦ A(a) = 0.
This gives us the objects of a category we write as C (A). The morphisms in this category are ♦ defined in Definition-Proposition (C.4). With this in hand we try to generalize familiar notions from homological algebra, such as coho- mology groups, chain homotopies and quasi-isomorphisms to this more general setting.
C.3 Generalizing the category of complexes
Basic definitions
We fix a category of categories S which we call the category of shapes and refer to categories S in S as shapes. We furthermore assume that every shape S comes equipped with a collection TS = {τi} of translation (or shift) functors (i.e., auto-equivalences S → S). We will sometimes write the action of a shift functor τ as τα or as α[τ].
Definition C.1. Let S be a category with a shift functor τ and let a : α → β be an arrow in S that is not the identity. We say that a is τ-minimal (or just minimal if the shift functor is understood) if τ(α) = β, i.e., if a can be composed with τ(a).
Remark C.2. 1. If a : α → β is a τ-minimal arrow, then so is τa, τ 2a, etc.
2. If a : α → β and a0 : α → β0 are two τ-minimal arrows with the same source then β0 = τα = β so the target of a τ-minimal arrow is uniquely determined by its source.
Example C.3. 1. The poset structure on Z gives us a category structure such that there is an arrow α → β if and only if α ≤ β. Let [1] : Z → Z be the functor
α 7→ α + 1.
Then an arrow a : α → β in (Z, [1]) is minimal if and only if there is no object γ 6= α, β such that a factors as α → γ → β.
The arrow 1 → 2 is minimal, while 1 → 3 is not minimal, because we can factor it through 2.
2. Here is an example of a category with an element that is not the target of any minimal arrow: Consider the poset Z ∪ {+∞} where we define α < +∞ for all α ∈ Z, and we extend the shift functor [1] by defining +∞[1] = +∞. Then +∞ is not the target of any minimal arrow.
By taking the opposite poset of Z ∪ {+∞} we get an example of a category with an object that is not the source of any minimal arrow. 68
3. Consider the poset (Q, [q]) where q is a rational number and the functor [q] shifts by q to the right. Here any arrow of length q is minimal. We can also consider (R, [r]), with r a real number. We could of course also look at (Q, {[q]| q ∈ Q}) and (R, {[r]| r ∈ R}).
Definition-Proposition C.4. For an arbitrary abelian category A we can construct a new category C (A), whose objects we call the configured complexes (or confplexes), in the following way: ♦ 1. The objects of C (A) are functors A : S → A, where S is a category in S, satisfying the ♦ condition: given any translation functor τ associated to S and any τ-minimal arrow α → β in S the composition A(τa) ◦ A(a): A(α) → A(τ(β))
is the zero morphism. We abbreviate this and say that A satisfies the d2 = 0 property.
2. An arrow f : X → Y between two objects A : SA → C and B : SB → C consists of
S (a) a functor f : SA → SB (in S) that is compatible with the translation functors: given
τ ∈ TSA there exists a functor ι ∈ TSB , which we say is paired with τ; and (b) a natural transformation f ν : X → Y f S
SA f S
f ν A +3 SB
# A r B .
If we need to work with more than one category of shapes we will use the notation CS (C). ♦ Proof. We leave it to the reader to show that this is actually a category.
Note that the requirement that f Sτ = ιf S implies that f S sends τ-minimal arrows to ι-minimal arrows.
Example C.5. Let S = {idZ : Z → Z} and TZ = {[1]} where [1] is defined above. Then
C (A) = C(A). ♦ We can similarly realize the category of double complexes as a category of configured complexes: 2 2 2 2 Set S = {idZ : Z → Z } and TZ = {rght, up}, where the functors rght, up are the usual horizontal and vertical shift functors, respectively. Then it is clear that C (A) is the usual category of double ♦ complexes. We can also mix the category of complexes and the category of double complexes together by letting
2 2 2 S = {idZ : Z → Z, idZ : Z → Z , 2 2 in : Z → Z , jn : Z → Z }. 69
2 2 With TZ, TZ as above. Here in is the functor that embeds Z as the nth line in Z , jn is the functor that embeds as the nth row in 2. Now C (A) contains both the category of complexes of objects Z Z ♦ from A and also the double complexes. There are some nice morphisms between these objects: Given a complex A : Z → A and a double complex B : Z2 → A we have an obvious morphisms of A with the nth line or nth row of B, with the underlying functor being in and jn in each case. We would also like to have a morphism in the other direction, but we run into trouble with the following: Say 2 we want to construct a morphism f : B → A with underlying functor pr1 : Z → Z (projection to the first factor). Then all the up-minimal arrows would be sent to identity arrows. This is remedied in the next example.
Example C.6. Let
0 0 2 2 0 2 S = {idZ : Z → Z , idZ : Z → Z , 2 0 2 0 pr1 : Z → Z , pr2 : Z → Z }
0 where Z is Z with an added arrow φn : n → n (not the identity) for every n in Z:
φ0 φ2 ··· / 0/ 1/ 2 / ··· L φ1
0 The functors pr1, pr2 are projections onto Z such that pr1 sends every vertical arrow of the form
(n, m) → (n, m + 1) to φn and similarly pr2 sends every horizontal arrow of the form (n, m) → 0 0 (n + 1, m) to φm. We define TZ = {[1], idZ}. Then the usual arrows n → n + 1 in Z are [1]-minimal
2 and φn for every n are the idZ-minimal arrows. We define TZ = {rght, up} as before. Note that the category of complexes lives in C (A) as the full subcategory of configured complexes of the form ♦ 0 0 2 A : Z → A with A(φn) = 0 for all n. Given A : Z → A and B : Z → A we can define a morphism S 0 f : B → A by setting f = pr1, pairing rght with [1] and up with idZ .
Of course we can’t stop here but forge on and try to generalize the cohomology groups of com- plexes, chain homotopies and quasi-isomorphisms.
Cohomology groups, contractions and chain homotopies
Cohomology groups
Definition C.7. Let A : S → A be a confplex in C (A) and fix a shift functor τ on S. For a ♦ τ-minimal arrow a : α → β we define the (τ, a)-cohomology group of A as
ker A(τ(a)) Ha(A) = . τ im A(a)
We will sometimes drop τ from the notation. 70
Let f : A → B be a morphism in C (A) and τ be a shift functor on S which is paired with ι ♦ A S on SB. Let a : α → β be a τ-minimal arrow. Then f a is ι-minimal and we get an induced arrow
a a f Sa Hτ (f): Hτ (A) → Hι (B).
This arrow arises from the diagram
A(β)
A(a) A(τa) A(α) / A(τα) / A(τβ)
ν ν ν fα fτα fτβ
B(f Sa) B(f Sτa) B(f Sα) / B(f Sτα) / B(f Sτβ)
B(ιf Sa) B(ιf Sα) / B(ιf Sβ) just like in the case of the usual morphism on cohomology groups of complexes. When talking about a • the arrows Hτ (f) for all τ-minimal arrows a in SA we use the usual notation Hτ (f), and sometimes • H• (f) if we want to talk about the induced arrows for all shift functors.
a Proposition C.8. 1. Let A be a confplex with shift functor τ. Then Hτ (idA) = id for any τ-minimal arrow a.
2. Let f : A → B and g : B → C be morphisms of confplexes and τ a shift functor on SA that is
paired with ι on SB which is paired with ω on SC . Then for any τ-minimal arrow a : α → β
in SA we have a gSf Sa f Sa Hτ (g ◦ f) = Hω (g) ◦ Hι (f).
Proof. Easy and left to the reader.
Contractions of shift functors and total complexes
Definition C.9. A shape S is said to be classical if for each shift functor τ associated to S and each object α in S there is a unique τ-minimal arrow with α as its source. If A : S → C is a confplex and S is classical then we also say that A is classical.
Remark C.10. All the confplexes in Examples (C.3) and (C.5) are classical while a confplex 0 0 A : Z → A (where Z is described in Example (C.6)) will fail to be classical, provided that A(φn) ◦
A(φn) 6= A(φn) for some n.
Here we attempt to generalize the construction of the total complex of a double complex with anti- commuting differentials. Recall that a double complex in our terminology is a functor A : Z2 → A 71 along with two shift functors rght and up. The anti-commutativity condition means that given a rght-minimal arrow a : α → β and an up-minimal arrow b : α → γ we have
A(b[rght]) ◦ A(a) + A(a[up]) ◦ A(b) = 0.
We start by generalizing this condition in the definition below.
Definition C.11. Let A : S → A be a classical shape and τ, σ ∈ TSA be two shift functors associated with S. We say that τ and σ anti-commute, or form an anti-commutative pair, if for any τ-minimal arrow a : α → β and σ-minimal arrow b : α → γ we have τγ = σβ and
A(τb) ◦ A(a) + A(σa) ◦ A(b) = 0, i.e., adding together the two compositions below gives the zero morphism
A(a) A(α) / A(β) A(τα)
A(b) A(τb) A(γ) A(τγ) t ttttt A(σa) tttt A(σα) / A(σβ)
Remark C.12. The relation of forming an anti-commutative pair is symmetric on TSA , but not reflexive or transitive in general.
Remark C.13. This definition generalizes the usual notion of anti-commutativity of the two dif- ferentials of a double complex.
Example C.14. Here is an example of anti-commuting shift functors on a shape different from a double complex. Consider the shape S below
··· / α0 / α1 / α2 / ··· < < <
| | | ··· / β0 / β1 / β2 / ··· with shift functors rght, which shifts everything to the right, and twst, which sends the top row to the bottom row and vice versa. If A : S → A is a confplex then rght and twst anti-commute if and only if in all squares
A(αn) / A(αn+1) A(αn) / A(αn+1) ? ?
A(βn) / A(βn+1) A(βn) / A(βn+1) the two compositions add up to the zero morphism. 72
The next definition is made to get rid of certain pathological complexes or complexes with objects that play no role when we are interested in calculating cohomology groups, see Remark C.16
Definition C.15. Let S be a shape in S.
1. We say that α in S is τ-accessible if there exists an arrow α → τα in S. If α is τ-accessible
for every τ ∈ TS then we say that it is accessible.
2. The shape S is said to be τ-cohesive if any α in S τ-accessible. If S is τ-cohesive for all
τ ∈ TS then we say that S is cohesive. If A : S → A is a confplex then we say it is τ-cohesive (resp. cohesive) if the shape S is τ-cohesive (resp. cohesive).
Remark C.16. It is easy to see that α is τ-accessible if and only if there exists a τ-minimal arrow a α → τα. So if α is not τ-accessible then there will be no cohomology group Hτ (A) with a : α → τα.
Lemma C.17. A classical shape is cohesive.
Proof. Obvious.
We can view the total complex of a double complex (with anti-commuting differentials) as a projection Z × Z → Z = {(n, n) | n ∈ Z} of the double complex onto its diagonal. Furthermore, we can imagine that the two differentials on the double complex get contracted onto the differential of the total complex. This leads to the following definition.
Definition-Proposition C.18. Let A : SA → A be a classical confplex with two anti-commuting shift functors, τ and σ, and let S be a shape with shift functor µ. Assume that p : SA → S is a functor that
1. is compatible with these shift functors such that both τ and σ are paired with µ; and
2. only sends minimal arrows to minimal arrows.
We define the contraction of A to S as the confplex TotA : S → A that on objects gives
M 0 TotA(α) = A(α ) pα0=α and on arrows a : α → β in S we set
TotA(a) : TotA(α) → TotA(β) to be the arrow given as follows: An arrow M M A(α0) → A(β0) pα0=α pβ0=β is equivalent an arrow M A(α0) → A(β0) pβ0=β for each α0 with pα0 = α. For a fixed α0 there are two possibilities: 73
a. there are arrows α0 → β0 that p sends to α → β and these give us arrows
A(α0) → A(β0).
(Note that α0 is fixed but β0 can vary.) We then add together all these arrows to produce a single arrow M A(α0) → A(β0); pβ0=β b. there is no arrow α0 → β0 that p sends to α → β and we let M A(α0) → A(β0) pβ0=β be the zero morphism.
Proof. We need to show that TotA : S → A is actually a confplex, i.e., if a : α → β is µ-minimal then the composition
TotA(α) → TotA(β) → TotA(µβ)
0 is zero. It suffices to restrict to one factor A(α ) of TotA(α). The arrow out of this factor is a sum of two arrows A(a0): A(α0) → A(β0) and A(a00): A(α0) → A(β00), where a0 is τ-minimal and a00 is σ-minimal, and
pa0 = a = pa00.
0 0 00 0 Now A(β ) = A(τα ) and A(β ) = A(σα ) are factors of TotA(β) and each of them is the source of a unique τ-minimal arrow and a unique σ-minimal arrow. It is easy to see that A(σa0), A(τa0) are τ-minimal and A(τa00), A(σa00) are σ-minimal. These arrows form the diagram below,
A(a0) A(τa0) A(α0) / A(β0) A(τα0) / A(τβ0)
A(a00) A(τa00) A(β00) A(τβ00) t ttttt A(σa0) tttt A(σα0) / A(σβ0)
A(σa00)
A(σβ00)
0 and the composition TotA(α) → TotA(β) → TotA(µβ) restricted to the factor A(α ) is exactly the sum
A(τa0) ◦ A(a0) + (A(τa00) ◦ A(a0) + A(σa0) ◦ A(a00)) + A(σa00) ◦ A(a00) = 0 + 0 + 0 = 0.
This completes the proof. 74
Chain homotopies and the homotopy category
Now we need to define a chain homotopy between two morphisms f, g : A → B. When f and g have the same underlying functor f S = gS the definition is a rather straight-forward generalization of the usual one and we do that case first.
Definition C.19 (Special case). Given two morphisms f, g : A → B in C (A), a chain homotopy ♦ between f and g consists of the following data: for every shift functor τ in TSA , paired with ι in τ S TSB , and every τ-minimal arrow a : α → β in S there is a morphism ha : A(β) → B(f α),
A(a) A(α) / A(β) f ν gν f ν gν α α τ β β ha B(f Sa) B(f Sα) / B(f Sβ) such that ν ν τ S τ fβ − gβ = hτaA(τa) + B(f a)ha,
A(a) A(τa) A(α) / A(β) / A(τβ) f ν gν f ν gν f ν gν α α τ β β τ τβ τβ ha hτa B(f Sa) B(f Sτa) B(f Sα) / B(f Sβ) / B(f Sτβ) If a chain homotopy exists between f and g we say they are chain homotopic and write f ∼ g.
Now let’s look at what happens when f and g have different underlying functors f S and gS:
Definition C.20. Given two morphisms f, g : A → B in C (A) a chain homotopy between f and ♦ g consists of the following data: for every shift functor τ in TSA , paired with ι in TSB there is first S S of all a natural isomorphism λ : Bf → Bg and for every τ-minimal arrow a : α → β in SA there τ S is a morphism ha : A(β) → B(f α),
A(α) A(a) ZZZZZZZ 55 ZZZZZ, 5 A(β) 55 5 5 gν 5 55 α 55 5 5 ν ν 5 5 gβ f τ 5 α h 5 5 a 5 5 5 f ν 5 β 55 S S 5 B(g α) ZZ B(g a) 5 λα ZZZZ nn6 ZZZZ, S nnn B(g β) Ó n λβ 6 B(f Sα) nnn ZZZZZ nnn S ZZZZ B(f a) Z, B(f Sβ) 75 such that ν ν τ S τ λβfβ − gβ = λβhτaA(τa) + B(g a)λαha. Or equivalently ν ν τ S τ λβfβ − gβ = λβhτaA(τa) + λβB(f a)ha. It can be helpful to look at the diagram below
A(α) ZZ A(a) 5 ZZZZZZZ 5 ZZZ, A(τa) 5 A(β) ZZZ 55 5 ZZZZZZZ 5 ν 5 ZZ, A(τβ) 5 gα 5 5 55 5 5 5 ν 5 ν 5 5 gβ 5 fα τ 5 5 ha 5 5 τ 5 5 5 hτa 5 ν 5 ν 5 gτβ fβ 5 S S 55 5 B(g α) B(g a) ν 5 ZZ 5 f 5 λα ZZZZ τβ 5 nn6 ZZZ, S S 5 nnn B(g β) ZZ B(g τa) 5 Ó n λβ ZZZZ S nn6 ZZZ, S B(f α) Z nn B(g τβ) ZZZZ Ó nn λτβ S ZZZZ 6 B(f a) Z, B(f Sβ) nnn ZZZZ nnn S ZZZZ B(f τa) Z, B(f Sτβ)
If a chain homotopy exists between f and g we say they are chain homotopic and write f ∼ g.
If f and g have the same underlying functor and we choose λ = idBf S then it is clear that this definition degenerates to the special case above.
Lemma C.21. If f, g : A → B are chain homotopic then they induce the same morphism on cohomology, i.e., • • Hτ (f) = Hτ (g), for any shift functor τ associated to A.
Proof. The proof is exactly the same as for usual one-complexes and therefore omitted.
Proposition C.22. Being chain homotopic is an equivalence relation.
Proof. We break the proof into three pieces.
1. Reflexive: This is clear; we choose λ = idBf S and h = 0.
0 −1 S S τ 0 τ 2. Symmetric: This is almost clear; we choose λ = λ : Bg → Bf and set (ha) = −λαha. The details are left to the reader.
3. Transitive: This is a bit more work. Assume we have three arrows f, g and h from A to B and f ∼ g (through λ and h) and g ∼ l (through µ and k), i.e.,
ν ν τ S τ λβfβ − gβ = λβhτaA(τa) + B(g a)λαha
and ν ν τ S τ µβgβ − lβ = µβkτaA(τa) + B(l a)µαka . 76
Consider the diagram
B(lSα) B(lSa) lν g3 ZZZZZ α ggg E ZZZZ S gggg Z, B(lSβ) B(l τa) gggg µ lν 3 ZZZ ggg α β ggg E ZZZZZZ A(α) gg A(a) gggg Z, S Z ggg ν B(l τβ) ZZZ gg µβ l 5 ZZZZZ ggg τα ggg3 E 5 ZZZ, A(β) gg A(τa) gggg 55 ZZZZZ ggggg 5 5 ZZZZZZZ gggg 5 gν 55 , A(τβ) 5 α τ 5 55 h 5 5 5 ν τa 5 ν 5 5 gβ 5 µτβ f τ τ 5 α h 5 ka 5 a 5 5 5 ν ν 5 5 g 5 f τ 5 τα β k 5 S S 55τa 5 B(g α) B(g a) ν 5 ZZ 5 f 5 λα ZZZZ τβ 5 nn6 ZZZ, S S 5 nnn B(g β) ZZ B(g τa) 5 Ó n λβ ZZZZ S nn6 ZZZ, S B(f α) Z nn B(g τβ) ZZZZ Ó nn λτβ S ZZZZ 6 B(f a) Z, B(f Sβ) nnn ZZZZ nnn S ZZZZ B(f τa) Z, B(f Sτβ)
I claim that f ∼ l through µλ and hτ + λ−1k and to check this we must calculate
τ −1 τ S τ −1 τ µβλβ(hτa + λβ kτa)A(τa) + B(l a)µαλα(ha + λα ka ) τ τ S τ S τ =µβλβhτaA(τa) + µβkτaA(τa) + B(l a)µαλαha + B(l a)µαka τ τ S τ S τ =µβλβhτaA(τa) + µβkτaA(τa) + µβB(g a)λαha + B(l a)µαka ν ν ν ν =µβ(λβfβ − gβ) + µβgβ − lβ ν ν =µβλβfβ − lβ,
which is exactly what we need!
Now that we have this equivalence relation at our disposal we define:
Definition C.23. We define the category K (A), the homotopy category of confplexes in A, as the ♦ category with the same objects as C (A) but where the morphisms are “up to chain homotopy”, ♦ i.e.,
HomK (A)(A, B) = HomC (A)(A, B)/ ∼ ♦ ♦ where ∼ is the equivalence relation defined above.
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