                                             Source file: src/stacks.tex

STACKS

Contents

Section 1. Introduction 1 Section 2. Definition 1 Subsection 2.1. Explanation 1 Subsection 2.2. Examples 4 References 4

Section 1. Introduction

Stacks are defined in this document. See [DM69].

Section 2. Definition

Let C be a site. The 2-category of stacks over C will be a full sub-2-category of the 2-category of categories over C, see Categories, Definition 3.1.5. Thus a stack will be given by a functor of categories p : S → C which satisfies the following conditions:

(1) p : S → C is a category fibred in groupoids, see Categories, Definition 3.1.1,

(2) descent for morphisms holds, and

(3) descent data for objects are effective.

Subsection 2.1. Explanation. To explain this, we choose a collection of pullback functors as in Categories, Lemma 3.1.3. Another approach is to use Categories, Lemma 3.3.2. First, suppose that x, y ∈ Ob(S_{U}) are objects in the fibre category over U. We are going to define a contravariant functor

Isom(x, y) : C/U ⟶ Sets.

In other words this will be a presheaf on C/U, see Sites, Definition 2.1.5. Namely, for f : V → U we set

Isom(x, y)(f : V → U) = Mor_{S_{V}} (f^{∗}x, f^{∗}y).

We also have to define the restriction map corresponding to a morphism (g, id_{U}) : (f′ : V′ → U) → (f : V → U) in C/U (in other words f′ = f ∘ g). This will be a map

Mor_{S_{V}} (f^{∗}x, f^{∗}y) ⟶ Mor_{S_{V ′}} (f′^{∗}x, f′^{∗}y).

This map will basically be g^{∗}, except that this transforms an element ϕ of the left hand side into an element g^{∗}ϕ of Mor_{S_{V ′}} (g^{∗}f^{∗}x, g^{∗}f^{∗}y). At this point we use the transformation t of Categories, Lemma 3.1.3. In a formula, the restriction map maps ϕ to

t_{y} ∘ g^{∗}ϕ ∘ (t_{x})^{−1}.

Lemma 2.1.1. This actually does give a presheaf.

Proof. Let (g′, id_{U}) : (f′ : V′ → U) → (f : V → U) and (g′′, id_{U}) : (f′′ : V′′ → U) → (f′ : V′ → U) be morphisms in C/U, and let ϕ ∈ Mor_{S_{V}} (f^{∗}x, f^{∗}y). It suffices to show that

[(Isom(x, y))(g′ ∘ g′′)](ϕ) = [(Isom(x, y))(g′′)]([(Isom(x, y))(g′)](ϕ))

By Lemma 3.1.3 there are pullback functors r : (g′ ∘ g′′)^{∗}f^{′∗} ⟶ f^{′′∗}, t′ : g^{′∗}f^{∗} ⟶ f^{′∗}, t′′ : g^{′′∗}f^{′∗} ⟶ f^{′′∗} and u : g^{′′∗}g^{′∗} ⟶ (g′ ∘ g′′)^{∗}. It follows from the uniqueness part of the lemma that t′′_{y} ∘ g^{′′∗}t′_{y} = r_{y} ∘ u_{f^{∗}y} and g^{′′∗}t_{x}^{′−1} ∘ t_{x}^{′′−1} = u_{f^{∗}x}^{−1} ∘ r_{x}^{−1}. We now have that

[(Isom(x, y))(g′ ∘ g′′)](ϕ) = r_{y} ∘ (g′ ∘ g′′)^{∗}ϕ ∘ r_{x}^{−1}
= r_{y} ∘ u_{f^{∗}y} ∘ g^{′′∗}g^{′∗}ϕ ∘ u_{f^{∗}x}^{−1} ∘ r_{x}^{−1}
= t′′_{y} ∘ g^{′′∗}t′_{y} ∘ g^{′′∗}g^{′∗}ϕ ∘ g^{′′∗}t_{x}^{′−1} ∘ t_{x}^{′′−1}
= t′′_{y} ∘ g^{′′∗}(t′_{y} ∘ g^{′∗}ϕ ∘ t_{x}^{′−1}) ∘ t_{x}^{′′−1}
= [(Isom(x, y))(g′′)]([(Isom(x, y))(g′)](ϕ)).

Alternatively we can argue with Lemma 3.3.2 which says that S → C is equivalent to the category assoicated to a contravariant function F: C → Groupoids. Then t_{y} is the identity transformation for all y so the restriction map is g ↦ g^{∗}ϕ = F (g)ϕ which is clearly functorial making Isom(x, y) a presheaf. □

OK, so the second condition listed in Section 2 is simply the condition that Isom(x, y) is a sheaf on the site C/U! In order to explain the meaning of the third condition, we must define a descent datum. First, we introduce some notation. Suppose that {f_{i} : U_{i} → U}_{i∈I} is a covering in the site C. We will be looking at fibre products U_{ij} = U_{i} ×_{U} U_{j} and U_{ijk} = U_{i}×_{U} U_{j} ×_{U} U_{k}. It is very important to allow i = j and even i = j = k in the following. The projection maps from U_{ij} to U_{i}, resp. U_{j} are denoted pr_{1}, resp. pr_{2}. The projection maps from U_{ijk} to the twofold fibre products are pr_{12} : U_{ijk} → U_{ij}, pr_{13} : U_{ijk} → U_{ik}, and pr_{23} : U_{ijk} → U_{jk}. The projection maps from U_{ijk} to U_{i}, U_{j}, and U_{k} are pr_{1}, pr_{2}, and pr_{3}, respectively. This notation is potentially ambiguous, but it is standard in the literature. For an example of the ambiguity: note the relations pr_{1} = pr_{1} ∘ pr_{12} = pr_{1} ∘ pr_{13}, pr_{2} = pr_{2} ∘ pr_{12} = pr_{1} ∘ pr_{23}, and pr_{3} = pr_{2} ∘ pr_{13} = pr_{3} ∘ pr_{23}. Let x_{i} ∈ Ob(S_{U_{i}}), i ∈ I be a collection of objects, and let

ϕ_{ij} : pr_{1}^{∗}x_{i} ⟶ pr_{2}^{∗}x_{j} (i, j ∈ I)

be a collection of morphisms in the fibre categories S_{U_{ij}}. In the definition that follows we will identify, for instance, pr_{1}^{∗}x_{i} on U_{ijk} with both pr_{12}^{∗}pr_{1}^{∗}x_{i} and pr_{13}^{∗}pr_{1}^{∗}x_{i}.

Definition 2.1.2. The collection {x_{i}, ϕ_{ij}} is a descent datum if the following cocycle condition is satisfied: For every triple (i, j, k) ∈ I^{3} the diagram

\xymatrix{pr_{1}^{∗}x_{i} \ar[rr]^{pr_{13}^{∗}ϕ_{ik}} \ar[dr]_{pr_{12}^{∗}ϕ_{ij}} &  & pr_{3}^{∗}x_{k} \\  & pr_{2}^{∗}x_{j} \ar[ur]_{pr_{23}^{∗}ϕ_{jk}}}

in the fibre category S_{U_{ijk}} is commutative.

Remarks 2.1.3. Two remarks about this definition are in order.

(1) There is a diagonal morphism Δ_{i} : U_{i} → U_{ii}. We can pull back ϕ_{ii} via this morphism to get an automorphism Δ_{i}^{∗}ϕ_{ii} ∈ Aut_{U_{i}}(x_{i}). On pulling back the cocycle condition for the triple (i, i, i) by Δ_{123} : U_{i} → U_{iii} we deduce that Δ_{i}^{∗}ϕ_{ii} ∘Δ_{i}^{∗}ϕ_{ii} = Δ_{i}^{∗}ϕ_{ii}; thus Δ_{i}^{∗}ϕ_{ii} = id_{x_{i}}.

(2) There is a morphism Δ_{13} : U_{ij} → U_{iji} and we can pull back the cocycle condition for the triple (i, j, i) to get the identity (σ^{∗}ϕ_{ji})∘ϕ_{ij} = id_{pr_{1/2}^{∗}x_{i}}, where σ : U_{ij} → U_{ji} is the switching morphism.

A morphism of descent data α : {x_{i}, ϕ_{ij}} → {y_{i}, ψ_{ij}} is given by a collection of morphisms α_{i} : x_{i} → y_{i} in the fibre category over U_{i} such that the following diagrams

\xymatrix{pr_{1}^{∗}x_{i} \ar[r]^{pr_{1}^{∗}α_{i}} \ar[d]_{ϕ_{ij}} & pr_{1}^{∗}y_{i} \ar[d]^{ψ_{ij}} \\ pr_{2}^{∗}x_{j} \ar[r]_{pr_{2}^{∗}α_{j}} & pr_{2}^{∗}y_{j}}

commute. Note that every morphism of descent data is an isomorphism. An object x ∈ Ob(S_{U}) gives rise to a descent datum in the following manner. First we set x_{i} = f_{i}^{∗}x. Then we set ϕ_{ij} = t_{j}^{−1} ∘ t_{i}, where t_{i} : pr_{1}^{∗}f_{i}^{∗}x → (f_{i} ∘ pr_{1})^{∗}x and t_{j} : pr_{2}^{∗}f_{j}^{∗}x → (f_{j} ∘pr_{2})^{∗}x are the canonical isomorphisms guaranteed by Categories, Lemma 3.1.3. The lemma below shows this is a descent datum; we will call this the canonical descent datum associated to x.

Lemma 2.1.4. The cocycle condition holds for the datum described above.

Proof. First, note that f_{i}∘pr_{1} = f_{j} ∘pr_{2} = f_{k} ∘pr_{3}. Then note that pr_{13}^{∗}ϕ_{ik}, pr_{23}^{∗}ϕ_{jk} and pr_{12}^{∗}ϕ_{ij} factor uniquely through (f_{i} ∘ pr_{1})^{∗}x = (f_{j} ∘ pr_{2})^{∗}x = (f_{k} ∘ pr_{3})^{∗}x by Lemma 3.1.3. □

Definition 2.1.5. A descent datum {x_{i}, ϕ_{ij}} is said to be effective when there exists an x ∈ Ob(S_{U}) such that {x_{i}, ϕ_{ij}} is isomorphic to the canonical descent datum associated to x.

At this point we are ready to give the definition of a stack.

Definition 2.1.6. A stack (in groupoids) over a site C is a category p : S → C over C such that

(1) p : S → C is a category fibred in groupoids over C,

(2) for all U ∈ Ob(C) and all x, y ∈ Ob(S_{U}) the presheaf Isom(x, y) is a sheaf on C/U, and

(3) for all coverings U = {U_{i} → U} in C, all descent data {x_{i}, ϕ_{ij}} for U are effective.

Usually the hardest part to check is the third condition.

Lemma 2.1.7. Suppose that f : X → S and g : Y → S are morphisms of stacks over C. Let X ×_{S} Y, p, q, ψ be the explicit 2-fibre product of f and g in the 2category of categories over C described in Lemma 3.1.7. Then X ×_{S} Y is a stack. In particular the 2-category of stacks over C has 2-fibre products (and they are as described in Lemma 3.1.7).

Proof. FIXME. □

Subsection 2.2. Examples. FIXME: Here we need lots of examples. FIXME: To be continued. To continue reading,

(1) visit the next section: Stacks and Groupoids, Section 1, or

(2) go back to the table of contents: index.html#contents.

References

[DM69] P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Publ. Math. IHES, 36:75–110, 1969.