THE 2-LIEN of a 2-GERBE by PRABHU VENKATARAMAN A

THE 2-LIEN OF A 2-GERBE

By PRABHU VENKATARAMAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

1 °c 2008 Prabhu Venkataraman

2 To my parents.

3 ACKNOWLEDGMENTS I would like to thank my advisor, Richard Crew, for guiding me through the literature, for suggesting this problem, and for patiently answering my questions over the years. I have learned a great deal from him. I am also grateful to David Groisser and Paul Robinson. Both of them helped me many times and served on my committee. Thanks also go to committee members Peter Sin and Bernard Whiting for their feedback on this project. The support of my family and friends has been invaluable to me. I thank all of them.

4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... 4

ABSTRACT ...... 7

CHAPTER

1 INTRODUCTION ...... 8

2 GERBES, LIENS AND 2-GERBES ...... 12 2.1 Torsors and H1 ...... 12 2.2 Fibered Categories and Stacks ...... 13 2.3 Gerbes and their Liens ...... 17 2.4 2-Categories, 2-Functors, 2-Natural Transformations ...... 21 2.5 Fibered 2-Categories, 2-Stacks and 2-Gerbes ...... 23 2.6 Representable Functors ...... 28 2.7 2-Representability ...... 32 2.8 Giraud’s approach to Liens of Gerbes ...... 34 3 EQUALIZERS AND COEQUALIZERS ...... 39

3.1 Equalizers ...... 39 3.2 Representability of Equalizers in CAT ...... 40 3.3 Coequalizers ...... 43 3.4 Representability of Coequalizers in CAT ...... 44 4 GROUP CATEGORIES ...... 51 4.1 Inner Equivalences of Group Categories ...... 51 4.2 Action of a Group Category on a Category ...... 54 5 COEQUALIZERS AND QUOTIENTS IN ST ACKS ...... 58 5.1 Action of a Group Stack on a Stack ...... 58 5.2 Coequalizers in ST ACKS ...... 59

6 THE 2-LIEN OF A 2-GERBE ...... 71 6.1 Deﬁnition of a 2-Lien ...... 71 6.2 The 2-Lien of a 2-Gerbe ...... 72 6.3 Cocycle Description of the 2-Lien of a 2-Gerbe ...... 79

7 2-LIENS, PICARD STACKS AND H3 ...... 81 7.1 2-Gerbes and 3-Cocycles ...... 81 7.2 2-Liens and Strict Picard Stacks ...... 83 7.3 Strict Picard Stacks and H3 ...... 86

5 8 CONCLUSION ...... 92 REFERENCES ...... 93

BIOGRAPHICAL SKETCH ...... 95

6 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy THE 2-LIEN OF A 2-GERBE By Prabhu Venkataraman May 2008 Chair: Richard Crew Major: Mathematics Principal bundles have a well-known description in terms of nonabelian cocycles

of degree 1 with values in a sheaf. A more general notion than that of a sheaf on a space X is that of a lien on X. A lien on X is an object that is locally defined by a sheaf of groups, with descent data given up to inner conjugation. Equivalences classes of gerbes with a given lien L are classified by nonabelian degree 2 cocycles. In his work, Lawrence Breen has given a similar classification of 2-gerbes using nonabelian degree 3 cocycles that take values in a family of group stacks. In our work, we defined the notion of a 2-lien on a space X. It is an object that is given locally by a group stack, with 2-descent given up to inner equivalence. We have proved some theorems about 2-liens

of 2-gerbes which correspond to well known results about liens of gerbes. Also, Deligne has shown that any strict Picard stack G corresponds to a 2-term complex of abelian

d sheaves K· = [ K0 / K1 ]. In this case we proved that Hˇ 3(X, G) is isomorphic to the hypercohomology group Hˇ 3(X,K·).

7 CHAPTER 1 INTRODUCTION

In geometry a fundamental concept is the notion of a principal G-bundle (also known as a G-torsor). If G is a sheaf on a space X, a G-torsor on X is a geometric realization of G-valued Cechˇ 1-cocycle. The problem of deﬁning non-abelian degree 2 sheaf cohomology leads to the concept of a gerbe. A gerbe on X may be thought of naively as a “sheaf of categories” with certain gluing axioms for objects and arrows. A gerbe on X is a geometric realization of a 2-cocycle on X with values in a non-abelian sheaf (or more generally a lien on X). This theory was worked out by Giraud [15].

Brylinski [7] developed a theory of differential geometry for gerbes. By his theory one can realize classes in H3(M, Z) as equivalence classes of abelian gerbes via the exponential map H2(M, C×) ∼ / H3(M, Z) . Then Murray invented the notion of a bundle gerbe [21]. The theory of bundle gerbes has proven very useful to physicists; for example bundle gerbes have been used to explore anomalies in quantum field theory [1], [2]. Hitchin has used the theory of gerbes for studying mirror symmetry [17]. Brylinski has used gerbes to give an interpretation of Beilinson’s regulator maps in algebraic K-theory [8]. One can then ask, what kinds of objects are classified by non-abelian degree 3 sheaf cohomology? The answer is a notion which is built up from 2-categories which is due to Breen. He calls such objects 2-gerbes on X [4]. Brylinski and McLaughlin have studied a certain class of 2-gerbes [9], [10], namely 2-gerbes bound by the abelian sheaf C×, which give rise to classes in H4(M, Z) via the exponential map H3(M, C×) ∼ / H4(M, Z) . They also prove that the class of the canonical 2-gerbe associated to a principal bundle P in H4(M, Z) is the same as the first Pontryagin class of P . This provides the motivation for our project. Given a 2-gerbe on X the associated

3-cocycle takes its values in a family of group stacks Gi. Breen then points out that one could introduce the appropriately deﬁned notion of a 2-lien L on a space X. Then the

8 3-cocycle could be viewed as deﬁning a class in a corresponding L-valued cohomology set H3(X, L).

In this thesis we deﬁne the notion of a 2-lien on a space X and show that each

2-gerbe gives rise to such a 2-lien. After proving some analogues of theorems about liens of gerbes, we establish some results about 2-liens that are strict Picard stacks, and an associated degree 3 hypercohomology group. We now provide a more detailed description of the contents of this thesis. In chapter 2, we review the necessary background material that forms the basis of this area, namely nonabelian cohomology. We begin with the definition of a G-torsor for a sheaf of groups G and explain their classification by H1(X,G). We then describe the objects that embody degree 2 cohomology: that is, gerbes. To do so, we first give the definition of a stack, a morphism between stacks and the definition of a 2-arrow between such morphisms. This data gives us a 2-category of stacks on a space X. After this setup, we give the definition of a gerbe and that of its associated lien. We then provide a description of the objects that embody degree 3 cohomology: that is, 2-gerbes. As with the case for gerbes, we first review the notion of 2-stacks prior to defining a 2-gerbe. The latter half of Chapter 2 is devoted to reviewing two notions from category theory that will prove essential to defining the notion of a quotient in the 2-category of stacks: these notions are those of the representability of a functor and of a 2-functor. We close by reviewing the original approach taken by Giraud [15] to introduce gerbes. It is this approach that we will adapt in order to define a 2-lien of a 2-gerbe in chapter 6. The key to defining a 2-lien is to figure out how to define a quotient in the 2-category of stacks. To answer this question locally, we first need to define a quotient in the 2-category of (small) categories CAT . This is what is accomplished in chapter 3: we prove the existence of a coequalizer in the 2-category of (small) categories CAT . Since the definition of an equalizer is used to define a coequalizer, we first prove the representability of an equalizer in CAT . We then construct a category Coeq corresponding to a (small)

9 pair of functors, and prove that this category represents the coequalizer of this pair in CAT .

In chapter 4, we begin by giving a deﬁnition of inner equivalence for group categories

and show that such inner equivalences for a fixed group category form a categorical group. We then define what it means for a group category to act on a category, and define the quotient by such an action using the concept of a coequalizer developed in chapter 3.

In chapter 5, we begin by deﬁning what it means for a group stack to act on a stack. Since the goal is to deﬁne the quotient by such an action, we do the work of proving that coequalizers are representable in the 2-category ST ACKS. This is done using the

coequalizer construction in chapter 3, and via the universal properties of the coequalizer. After proving the existence of a coequalizer in ST ACKS we are able to deﬁne a quotient

of an action by a group stack on a stack. In chapter 6, we deﬁne the notion of a 2-lien on a space X, following the approach of Giraud for an ordinary lien. We then do the necessary work of showing how to any 2-gerbe, one may asociate such a 2-lien, and that this association is given up to canonical 2-equivalence. After establishing this, we give a more down-to-earth description of the 2-lien of a 2-gerbe: it is an object that is given locally by a group stack, with descent data given up to inner equivalence.

In chapter 7, our goal is to prove some theorems about G-2-gerbes, whose 2-lien arises from a strict Picard stack G. For this purpose, in section 1 we review Breen’s treatment of the 3-cocycle description of a G-2-gerbe [4]. In section 2 of chapter 7, we prove results about G-2-gerbes which are analogues of theorems about for liens of gerbes. In section 3, we wish to give a nice cohomological description of connected G-2-gerbes where G is strict Picard. Deligne has shown that any strict Picard stack G corresponds to a 2-term complex

d of abelian sheaves K· = [ K0 / K1 ]. We begin by reviewing Deligne’s contruction, and then giving the deﬁnition of a degree 3 hypercohomology group. We then prove that the

10 set of equivalence classes Hˇ 3(X, G) of connected gerbes with 2-lien G is isomorphic to the hypercohomology group Hˇ 3(X,K·).

We conclude with chapter 8, where we summarize our results, and indicate some directions for future work.

11 CHAPTER 2 GERBES, LIENS AND 2-GERBES

2.1 Torsors and H1

We will rapidly review some elements of the theory of torsors/principal bundles [5],

[7].

Recall that given sets X, Y and Z with maps f : X → Z, g : Y → Z, the

ﬁbered product X ×Z Y is the subset of the product X × Y consisting of (x, y) such that f(x) = g(y). Let G be a sheaf of groups on a space X. Deﬁnition 2.1.1. A right G-torsor on X is a space π : P → X above X, together with a right group action P × G → P of G on P such that the induced morphism

P ×X G ' P ×X P (2–1)

(p, g) 7→ (p, pg)

is an isomorphism. In addition, we require that there exist a family of local sections

si : Ui → P of π, for some open cover U = (Ui)i∈I of X. The trivial G-torsor is X × G itself, with the action of G given by multiplication. Any

G-torsor is locally isomorphic to the trivial torsor TG.A G-torsor is isomorphic to TG if and only if it has a global section s : X → P . Now, given a space X and a sheaf G of groups ; we wish to consider isomorphism

classes of G-torsors q : P → X. Given an open covering (Ui)i∈I and a section si of q over

Ui, we have a function gij : Uij → G|Uij such that sj = si · gij (recall that G acts on the

right on P ). These transition functions satisfy the equality gik = gijgjk over Uijk. If the

0 −1 section si is replaced over Ui by si = si · hi, then gij is replaced by hi gijhj. This leads to the appropriate notion of 1-cocycles and coboundaries. For a sheaf G

of (possibly nonabelian) groups on a space X, and (Ui)i∈I an open covering, we deﬁne a ˇ Cech 1-cocyle with values in G to consist of a family aij ∈ Γ(Uij,G) such that the cocycle

12 relation

aik = aijajk (2–2)

ˇ 0 holds. Next, two Cech 1-cocycles aij and aij are said to be cohomologous if there exists a section hi of G over Ui such that

0 −1 aij = hi aijhj (2–3) thus deﬁning an equivalence relation on the set of Cechˇ 1-cocycles. Deﬁnition 2.1.2. 1. For a sheaf G of groups on the space X, and an open covering

ˇ 1 U = (Ui)i∈I of X, the ﬁrst cohomology set H (U,G) is deﬁned as the quotient of the set of 1-cocycles with values in G by the equivalence relation:“a is cohomologous to a¯”.

2. The first cohomology set H1(X,G) is defined as the direct limit limHˇ 1(U,G) where −→ U the limit is taken over the set of all open coverings of X, ordered by the relation of refinement.

Note that the set H1(X,G) has a distinguished element, the class of the trivial 1-cocycle 1. So it is a pointed set.

The deﬁnition of ﬁrst cohomology makes the following result immediate. Proposition 2.1.3. The set of isomorphism classes of G-torsors over X is in a natural bijection with the sheaf cohomology set H1(X,G). 2.2 Fibered Categories and Stacks

We will review the deﬁnitions from category theory [4], [5] relevant to our discussion. Recall that a groupoid is a category in which every morphism is invertible.

Deﬁnition 2.2.1. A category ﬁbered in groupoids above a space X consists of a family of groupoids CU , for each open set U in X, together with an inverse image functor

∗ / f : CU CU1 (2–4)

13 associated to every inclusion of open sets f : U1 ⊂ U (which is the identity whenever f = 1U ), and a natural transformation

∗ +3 ∗ ∗ φf,g :(fg) g f (2–5)

for every pair of composable inclusions

g / f / U2 U1 U. (2–6)

For each triple of composable inclusions

h / g / f / U3 U2 U1 U (2–7)

we require that the composite natural transformations

∗ +3 ∗ ∗ +3 ∗ ∗ ∗ ψf,g,h :(fgh) h (fg) h (g f ) (2–8)

and

∗ +3 ∗ ∗ +3 ∗ ∗ ∗ χf,g,h :(fgh) (gh) f (h g )f (2–9)

coincide.

We will frequently refer to the inverse image of an object x ∈ CU by an inclusion / i : V U as the restriction x|V of x above V . Deﬁnition 2.2.2. A Cartesian functor F : C / D between ﬁbered categories consists of / a family of functors FU : CU DU indexed by the open sets U, together with, for every / morphism f : U2 U1 , a natural transformation

∗ / ∗ ϕf : f ◦ FU1 FU2 ◦ f . (2–10)

This is required to be compatible, for any pair of composable inclusions f and g, with

the natural transformations (fg)∗ +3 g∗f ∗ for C and for D. Finally, a 2-arrow

Ψ: F +3 G between a pair of Cartesian functors F and G from C to D is deﬁned

14 +3 by a family of natural transformations ΨU : FU GU which are compatible with the restriction functors f ∗.

Deﬁnition 2.2.3. • A prestack in groupoids above a space X is a ﬁbered category in

groupoids above X such that “arrows glue” i.e. for every pair of objects x, y ∈ CU ,

the presheaf HomCU (x, y) is a sheaf on U. • If, in addition, “objects glue” i.e. descent is eﬀective for objects in C, then the prestack is called a stack in groupoids above X. • A morphism of stacks is just a morphism of the underlying ﬁbered categories.

By descent data we mean that we are given, for every open cover U = (Uα) of

an open set U ⊂ X, a family of objects xα ∈ CUα , and a family of isomorphisms / φαβ : xα|Uαβ xβ|Uαβ such that

φαβφβγ = φαγ (2–11)

above Uαβγ.

The descent condition (xi, φij, ψαβγ) is eﬀective if there exists an object x ∈ CU

together with isomorphisms fα : x|Uα ' xα compatible with the morphisms φαβ i.e. the following diagram commutes:

x|Uαβ (2–12) w GG fα ww GG fβ ww GG ww GG w{ w G# / xα|Uαβ xβ|Uαβ φαβ By a construction based on the corresponding construction for sheaves, given any prestack C on X, there corresponds an associated stack C, together with a cartesian functor

i : C → C (2–13)

which is universal for cartesian functors from C into stacks (see [18]).

15 Deﬁnition 2.2.4. A stack in groupoids C on X, endowed with a monoidal associative Cartesian functor · : C × C → C, with identity objects and with inverses will be called

a group stack (or just a gr-stack) on X. For any Vi ∈ Ob(CU ) (i=1, 2, 3, 4) the

associativity isomorphisms αViVj Vk :(Vi · Vj) · Vk 7→ Vi · (Vj · Vk) must sit within the following two commutative diagrams.

1. Pentagon Axiom.

((V1 · V2) · V3)R· V4 (2–14) lll RRR α1,2,3·id4ll RRα12,3,4 lll RRR lll RRR lu ll RR) (V1 · (V2 · V3)) · V4 (V1 · V2) · (V3 · V4)

α1,23,4 α1,2,34 V1 · ((V2 · V3) · V4) / V1 · (V2 · (V3 · V4)) id1·α2,3,4

2. Triangle Axiom. α (V · 1) · V / V · (1 · V ) 1 LL2 r1 2 (2–15) LLL rrr LLL rrr ρ·id LL% yrrr id·λ V1 · V2

where ρVi : Vi · 1 7→ Vi and λVi : 1 · Vi 7→ Vi. Deﬁnition 2.2.5. • A group category G is said to be braided when its group law is

endowed with a commutivity isomorphism SX,Y : XY → YX which is functorial in X and Y and sits in a commutative square

X1 S / 1X (2–16)

m S X X

and two hexagonal “associativity” diagrams.

• G is said to be Picard if in addition SY,X ◦ SX,Y = 1XY for all X, Y in G.

• G is said to be a strict Picard category if in addition to all the above, SX,X = 1X for all X.

Deﬁnition 2.2.6. A gr-stack is said to be a strict Picard stack if it is endowed with a commutativity natural transformation S for the group law SX,Y and satisﬁes the corresponding conditions.

16 2.3 Gerbes and their Liens

The following deﬁnitions are due to Giraud [15]. We follow the presentation in [4], [5] and [20].

To say that a stack G is locally non-empty means there exists a covering U = (Ui) of

X for which the set of objects of the category GUi is non-empty. The locally connectedness

condition on G is the requirement that for any pair of objects x and y in GU , there exists

an open cover V = (Vα) of U such that the set of arrows from x|Vα to y|Vα is non-empty for all α.

Deﬁnition 2.3.1. • A (1)-gerbe on a space X is a stack in groupoids G on X which is locally non-empty and locally connected.

• A morphism of gerbes is just a morphism of the underlying ﬁbered categories.

• A gerbe G on X is said to be neutral (or trivial) when the ﬁber category GX is non-empty.

A lien on a space X, as ﬁrst deﬁned by Giraud in his thesis [15], is a collection (Gi) of sheaves of groups corresponding to an open cover (Ui) of X with descent data up to inner conjugation.

In other words a lien on X is an object which is defined locally by a sheaf of groups, but in a category where morphisms between groups differing by inner conjugation are identified.

We give a cocycle description of a lien. Let U = (Ui)ı∈I be an open cover of X, and

suppose we have a family of sheaves Gi of groups, deﬁned on the open sets Ui. Denote

each sheaf by lien(Gi). The sheaves lien(Gi) and lien(Gj) are glued on the open set

Uij by a section ψij of the quotient sheaf Out(Gj,Gi) := Isom(Gj,Gi)/Int(Gi) on Uij.

Thus the lien L is determined by a family of sections of Out(Gj,Gi) on Uij satisfying the

1-cocycle condition ψij ◦ ψjk = ψik in Out(Gk,Gi) and the normalization condition ψii = 1 for all i.

17 0 0 Two liens L and L on X, each deﬁned locally by families of groups (Gα) and (Gβ ),

are isomorphic whenever there exists a common reﬁnement V = (Vi) of the deﬁning covers

0 0 U and U , and a family of isomorphisms χi : lien(Gi) → lien(Gi) on the open sets Vi, 0 which are compatible with the gluing data. The cocycles (ψij) and (ψij ) are therefore

0 −1 related by the coboundary conditions ψij = χiψijχj , the isomorphisms χi are viewed as

sections on the open sets Vi of the sheaf Out(Gi,Gi) of outer automorphisms of Gi.

Let G be a sheaf of groups, and let Gi = G|Ui . Then when a lien is locally of the form

lien(Gi), each ψij is a section on the set Uij of the sheaf Out(G) of outer automorphisms of G. Following [4] we will call such liens, which are locally isomorphic to the lien lien(G), G-liens. It follows from this description that they are classiﬁed by the non-abelian cohomology set H1(X, Out(G)).

Let G be a gerbe. It is locally non-empty so there exists an open cover (Uα) of X

such that G(Uα) is non-empty. So choose a family of objects xα ∈ G(Uα). Now Aut(xα)

is a sheaf on Uα. Now G is also locally connected so for each α, β there exists an open

ξ ξ ξ cover of (Uαβ) of Uαβ and for each ξ an arrow fαβ : xβ → xα in G(Uαβ). Conjugation by ξ this arrow defines an isomorphism of sheaves of groups λαβ : Aut(xβ)| ξ → Aut(xα)| ξ . Uαβ Uαβ ξ ξ This isomorphism λαβ depends on the choice of the fαβ but different choices define the ξ ζ ξ same outer isomorphism. In particular, on overlaps Uαβ ∩ Uαβ the two isomorphisms λαβ ζ ξ and λαβ define the same outer isomorphism. Thus for fixed α, β the family λαβ defines an

“outer isomorphism” of sheaves on Uαβ:

λαβ : Aut(xβ)|Uαβ → Aut(xα)|Uαβ (2–17)

ξ which does not depend on the choice of fαβ. This system of sheaves of groups Aut(xα) and

outer isomorphisms λαβ is called the lien of the gerbe G. To any gerbe with lien L, we can attach a 2-cocycle that takes its values in L, see [20] for details. Deﬁnition 2.3.2. Let K be a lien on X. A gerbe with lien K is a gerbe G on X together with an isomorphism of liens θ : lien(G) ' K. Two gerbes with lien K are said to be

18 equivalent if there is an equivalence between the underlying ﬁbered categories. We designate by H2(X,K) the set of equivalence classes of gerbes with lien K.

Example 2.3.3. Let G be a bundle of groups on X. The stack T ors(G) of right G-torsors

on X is a gerbe on X. It is (globally) non-empty since its fiber on X always contains the trivial G-torsor on X. It is also locally connected since any G-torsor is locally isomorphic to the trivial G-torsor. Thus T ors(G) is in fact a neutral gerbe. Its lien is the lien represented by the group G, denoted lien(G). Example 2.3.4. Let 1 → A → C → B → 1 be an exact sequence of (possibly infinite dimensional) Lie groups such that the projection C → B is a locally trivial A-bundle. Let p : P → X be a smooth B-bundle over a manifold X. Consider the problem of finding a C-bundle q : Q → X such that the associated B-bundle Q/A → X is isomorphic to

P → X. The ﬁbered category of local solutions to this is a gerbe on X. The lien associated

to this gerbe is isomorphic to the sheaf AX of smooth A-valued functions. Deﬁnition 2.3.5. Let P be a gerbe on X, let G be a sheaf of groups on X, and let

U = (Ui)i∈I be an open cover of X. We say that P is a G-gerbe on X if there exists for

each i ∈ I an object xi ∈ Ob(PUi ) and an isomorphism of sheaves of groups on Ui

/ Aut (x ) ηi : G|Ui PUi i (2–18)

where Aut (x ) is the sheaf of automorphisms of the object x above U . PUi i i i Deﬁnition 2.3.6. Let P be a gerbe on X, let G be a sheaf of groups on X. Suppose there exists, for every object x in a ﬁber category PU , an isomorphism of sheaves of groups

/ ηx : G|U AutPU (x) , (2–19)

and that, for any morphism f : x → y in PU , the corresponding diagram of sheaves on U

G|U (2–20) t JJ η tt JJ ηy xtt JJ tt JJ ztt J$ Aut (x) / Aut (y) PU λ PU

19 determined by the morphism λ associated to f commutes. The gerbe P is then called an abelian G-gerbe on X.

The map λ associated to f is the isomorphism

λ : AutPU (x) → AutPU (y) (2–21)

u 7→ fuf −1. (2–22)

An abelian G-gerbe is a G-gerbe by deﬁnition. In fact, we now show that in this situation the sheaf G must be abelian. In particular, the commutivity of the group law in

G can be veriﬁed locally, for sections of a sheaf G|U . Let g be a section of this sheaf and

consider the above triangle associated to the corresponding arrow u = ηx(g): x → x. This is the diagram

G|U (2–23) t JJ η tt JJ η xtt JJx tt JJ ztt J$ / AutPU (x) AutPU (x) iu

where iu denotes inner conjugation by u in the sheaf AutPU (x). Commutivity of this diagram implies that iu is the identity map whence the sheaf AutPU (x), and therefore G|U is abelian.

We state two propositions (see [4] for proofs) to illustrate how the notion of a lien is helpful in characterizing a G-gerbe.

Proposition 2.3.7. Let G be a sheaf of groups on a space X. A gerbe G on X is a G-gerbe if and only if its lien is locally isomorphic to lien(G). Proposition 2.3.8. Let G be a sheaf of abelian groups on a space X. A gerbe G on X is an abelian G-gerbe if and only if lien(G) ' lien(G). We now state a theorem of Giraud (see [15]) giving the connection between gerbes with G-liens where G is an abelian sheaf, and cohomology.

Theorem 2.3.9. Let G be a sheaf of abelian groups on X. Let L = lien(G). A gerbe G with lien L gives rise to a degree-2 Cechˇ cocycle with values in L as follows. Choose an

20 open covering (Ui) of X for which each G(Ui) is nonempty, and such that any two objects

of G(Uij) are isomorphic. Choose objects Pi of G(Ui), and let φij : Pj|Uij → Pi|Uij be an isomorphism between objects in the category G(Uij). We deﬁne a section hijk of L over

Uijk by

−1 φijk = φik ◦ φij ◦ φjk ∈ Aut(Pk) ' L (2–24)

ˇ 2 so that (h) = (hijk) is an L-valued Cech 2-cocycle. The corresponding class in H (X,L) is independent of all choices, and the assignment G 7→ (h) gives a one-to-one correspondence between equivalence classes of gerbes with lien L and the elements of H2(X,L).

We conclude this section with the following deﬁnition. Deﬁnition 2.3.10. Let G be a gr-stack on X. A right G-torsor on X is a stack Q on X together with a right action functor

Q × G → Q (2–25) which is coherently associative and satisﬁes the unit condition, and for which the induced functor Q × G → Q × Q (2–26)

(q, g) 7→ (q, qg) (2–27) is an equivalence. In addition we require that Q be locally non-empty. 2.4 2-Categories, 2-Functors, 2-Natural Transformations

We will review, in an informal spirit, some deﬁnitions from the theory of 2-categories. Deﬁnition 2.4.1. A 2-category A consists of the following data:

1. a collection of objects A, B, C,..., 2. for each ordered pair of objects (A, B), a small category A(A, B),

3. for each triple A, B, C of objects, a bifunctor

cA,B,C : A(A, B) × A(B,C) → A(A, C), (2–28)

21 4. for each object A, a functor

uA : 1 → A(A, A). (2–29)

Here 1 denotes the terminal category (with one object and one arrow) in the category of small categories. These elements of data are required to satisfy associativity laws for composition and the requirement that uA provides a left and right identity for this composition. Example 2.4.2. A basic example is the 2-category CAT , whose objects are small categories, arrows are functors, and 2-arrows are natural transformations.

Deﬁnition 2.4.3. Given two 2-categories A and B, a 2-functor F : A → B consists in giving

1. for each object A ∈ A, an object F (A) ∈ B, 2. for each pair of objects A, A0 in A, a functor

0 0 FA,A0 : A(A, A ) → B(F (A),F (A )) (2–30)

(For brevity’s sake we often write F instead of FA,A0 ). This data is required to satisfy the following axioms: 1. Compatibility with composition: given three objects A, A0, A00 in A, the following diagram commutes.

c A(A, A0) × A(A0,A00) AA0A00 / A(A, A00) . (2–31)

FAA0 ×FA0A00 FAA00

B(F (A),F (A0)) × B(F (A0),F (A00)) / B(F (A),F (A00)) cF (A)F (A0)F (A00)

22 2. Unit: for every object A ∈ A the following diagram commutes.

uA / 1 II A(A, A) (2–32) II II II II II II FAA uF (A) II II II I$ B(F (A),F (A))

F / Deﬁnition 2.4.4. Consider two 2-categories A, B, and two 2-functors A / B between G them. A 2-natural transformation θ : F ⇒ G consists in giving, for each object A ∈ A, an

arrow θA : F (A) → G(A) such that the following diagram commutes for each pair of objects A, A0. F A(A, A0) AA0 / B(F (A),F (A0)) . (2–33)

GAA0 B(1F (A),θA0 )

B(G(A),G(A0)) / B(F (A),F (A0)) B(θA,1G(A0))

2.5 Fibered 2-Categories, 2-Stacks and 2-Gerbes

Deﬁnition 2.5.1. A ﬁbered 2-category in 2-groupoids above a space X consists of a family of 2-groupoids CU , for each open set U in X, together with an inverse image 2-functor

∗ / f : CU CU1 (2–34)

associated to every inclusion of open sets f : U1 ⊂ U (which is the identity whenever f = 1U ), and a natural transformation

∗ +3 ∗ ∗ φf,g :(fg) g f (2–35)

for every pair of composable inclusions

g / f / U2 U1 U. (2–36)

23 For each triple of composable inclusions

h / g / f / U3 U2 U1 U (2–37) we require a modiﬁcation (which is a map between natural transformations)

ψf,g,h

& ∗ αf,g,h ∗ ∗ ∗ (fgh) h8@ g f (2–38)

χf,g,h between the composite natural transformations

∗ +3 ∗ ∗ +3 ∗ ∗ ∗ ψf,g,h :(fgh) h (fg) h (g f ) (2–39) and

∗ +3 ∗ ∗ +3 ∗ ∗ ∗ χf,g,h :(fgh) (gh) f (h g )f . (2–40)

Finally, for any U4 ,→ U3, the two methods by which the modiﬁcations α compare the composite two arrows

(fghk)∗ +3 (ghk)∗(f)∗ +3 ((hk)∗g∗)f ∗ +3 k∗h∗g∗f ∗ (2–41) and (fghk)∗ +3 k∗(fgh)∗ 3+ k∗(h∗(fg)∗) +3 k∗h∗g∗f ∗ (2–42) must coincide.

Deﬁnition 2.5.2. A Cartesian 2-functor F : C / D between ﬁbered 2-categories over / X consists of a family of 2-functors FU : CU DU indexed by the open sets U, together / with, for every morphism f : U2 U1 , a natural transformation of 2-functors

∗ +3 ∗ φf : f ◦ FU1 FU2 ◦ f . (2–43)

For any pair of composable inclusions f and g, a 3-arrow αf,g is also given, which com-

∗ ∗ ∗ pares the natural transformation from (fg) ◦ FU1 to FU3 ◦ g ◦ f deﬁned by φf and φg with

24 that constructed from φfg. This 3-arrow is required to satisfy a coherence condition relating the two induced natural transformations associated to a triplet (f, g, h) of composable inclusions. This condition will not be made explicit here.

Deﬁnition 2.5.3. • A 2-prestack in groupoids C above a space X is a ﬁbered 2-

category in 2-groupoids above X such that for every pair of objects x, y ∈ CU , the

ﬁbered category ArCU (x, y) is a stack on U. • If, in addition, 2-descent is eﬀective for objects in C, then the 2-prestack is called a 2-stack in 2-groupoids above X.

• A morphism of 2-stacks is just a morphism of the underlying ﬁbered 2-categories.

By 2-descent data we mean that we are given, for an open cover U = (Uα) of an open / set U ⊂ X, a family of objects xi ∈ CUi , of 1-arrows φαβ : xα|Uαβ xβ|Uαβ and a family

of 2-arrows ψαβγ : φαγ ◦ φβγ ⇒ φαβ

x β B (2–44) φ || BB βγ || B8@Bφαβ || ψ BB ~|| B! xγ / xα φαγ

for which the tetrahedral diagram of 2-arrows induced by ψ in CUαβγδ commutes:

xδ C (2–45) || CC || CC || CC |} | C! x ______/ x γ B > α BB || BB || BB || B || xβ

The 2-descent condition (xi, φij, ψαβγ) is eﬀective if there exists an object x ∈ CU

together with isomorphisms fα : x|Uα ' xα compatible with the morphisms φij and ψαβγ. For any 2-prestack C on X, one defines by the same sheafification method as for

sheaves and 1-stacks, an “associated 2-stack” 2-functor

a : C → C (2–46)

25 which is universal for Cartesian 2-functors b : C → D from C into 2-stacks. The latter property characterizes C upto 2-equivalence. For, suppose that a Cartesian 2-functor b : C → D satisﬁes:

1. b is ﬁberwise fully faithful .i.e. for any pair of objects x and y in a ﬁber category CU , the induced map of stacks on U

Ar(x, y) → Ar(b(x), b(y)) (2–47)

is an equivalence.

2. every object in D is locally isomorphic to one in the image of C.

Then D is an associated 2-stack of C. Deﬁnition 2.5.4. • A 2-gerbe P on a space X is a 2-stack in 2-groupoids on X which is locally non-empty, locally connected, in which 1-arrows are weakly invertible, and 2-arrows are invertible.

• A morphism of 2-gerbes is just a morphism of the underlying 2-stacks.

To say that P is locally non-empty means there exists a covering U = (Ui) of X for

which the set of objects of the 2-category PUi is non-empty. The connectedness condition on P is the requirement that for any pair of objects x and y in the ﬁbered 2-category

GU , there exists an open cover V = (Vα) of U such that the set of 1-arrows from x|Vα to

y|Vα is non-empty for all α. To say that 1-arrows are weakly invertible means that for any

1-arrow f : x → y in a ﬁbered 2-category PU , there exists an arrow g : y → x in PU

which is both a left and right inverse of f (upto a pair of 2-arrows λ : g ◦ f ⇒ 1x and

ρ : f ◦ g ⇒ 1y). Example 2.5.5. (Breen) Let L be a lien on X. We can ask when is L isomorphic to a lien of the form lien(G) for some gerbe G on X? Locally the answer is always, since the lien L is locally isomorphic to a lien of the form lien(G) for some sheaf of groups G,

whence it is realized by the neutral gerbe T ors(G) corresponding to G. Globally this gives a

2-gerbe on X.

26 For a 2-gerbe P, we can associate to each object x ∈ PU a gr-stack Px := ArU (x, x) above U.

Deﬁnition 2.5.6. Let P be a 2-gerbe on X, and let G be a gr-stack on X. We say that P is a G-2-gerbe on X if there exists an open cover U = (Ui)i∈I of X, for each i ∈ I an object

xi ∈ Ob(PUi ) and an equivalence GUi 'Gxi over Ui. Deﬁnition 2.5.7. Let P be a 2-gerbe on X, let G be a gr-stack on X. Suppose there exists, for every object x in a ﬁber 2-category PU , an isomorphism of sheaves of gr-stacks

/ ηx : G|U EqPU (x) , (2–48)

and for any morphism f : x → y in PU , a 2-arrow ηf : λf ◦ ηx ⇒ ηy

G|U (2–49) u II ηx uu II ηy uu 5=II uu ηf II zuu I$ Eq (x) / Eq (y) PU λ PU where λf is the morphism of gr-stacks

λ : EqPU (x) → EqPU (y) (2–50)

u 7→ fuf −1. (2–51) deﬁned by f. The natural transformations ηf are required to respect the group structures, and satisfy the following transitivity and normalization conditions:

1. For any pair of composable morphisms f : x → y and g : y → z in PU , the composite

2-arrow obtained by pasting ηf and ηg is equal to ηgf .

2. For any 2-arrow φ : f ⇒ g between a pair of morphisms f, g : x → y in PU ,

ηf = ηg ◦ λφ, where λφ : λf ⇒ λg is conjugation by φ.

3. For every x in PU , η1x = 1 A 2-gerbe P satisfying these conditions is called an abelian G-2-gerbe on X.

27 2.6 Representable Functors

We will rapidly review the notion of a representable functor. The primary source for

this summary is [12]. Let C be a category and let SET denote the category of sets. The category of presheaves of sets on C is the functor category Hom(Cop,SET ). Now choose an object X of C and deﬁne a presheaf hX by:

hX (Y ) = HomC(Y,X), hX (f) = HomC(f, X) := ◦f (2–52)

for any object Y of C and arrow f in C. This functor is called the functor represented by X. This construction is functorial in X, i.e. gives a functor

h : C → Hom(Cop,SET ) (2–53)

deﬁned by h(X) = hX , and for f ∈ Hom(X,Y ),

h(f)(Z): Hom(Z,X) → Hom(Z,Y ) (2–54)

g 7→ f ◦ g. (2–55)

Deﬁnition 2.6.1. A functor F ∈ Hom(Cop,SET ) is representable if it is in the essential

image of h, i.e. if it is isomorphic to some hX . We say that F is represented by (the object) X. Note that the isomorphism is part of the data.

If (X, α : hX ' F ) represents F then we obtain a universal element ξ = αX (idX ) ∈ F (X). The isomorphism can be recovered from ξ. To see this, if f : X → Y is any arrow in C, then the diagram α Hom(X,X) X / F (X) . (2–56)

Hom(f,X) F (f)

α Hom(Y,X) Y / F (Y )

28 commutes, since α is a natural transformation. But idX ∈ Hom(X,X) maps under Hom(f, X) to f ∈ Hom(Y,X), whence by the above diagram we have

αY (f) = F (f)(αX (idX )) = F (f)(ξ) (2–57)

which shows how to compute α from ξ. Thus we can also say F is represented by the pair (X, ξ). Yoneda’s lemma states that:

Theorem 2.6.2. The functor h : C → Hom(Cop,SET ) is fully faithful. Corollary 2.6.3. If a functor F is representable then the representing object in C is deﬁned uniquely up to a unique isomorphism.

In the category SET we have definitions for products, fibered products, equalizers, coequalizers, etc. We can use representable functors to import these notions from SET to any arbitrary category. To see how this is done we will examine carefully the definition of product.

Products. The product of two sets X and Y , together with the projection maps

p1 : X × Y → X, p2 : X × Y → Y , is characterized up to isomorphism by a universal

property: given any sets Z with maps q1 : Z → X, q2 : Z → Y , there is a unique map

0 q : Z → X × Y such that q1 = p1 ◦ q and q2 = p2 ◦ q. In fact, if P and P are two sets with this universal property, then there exist maps f : P → Q, g : Q → P and uniqueness implies g ◦ f = idP and f ◦ g = idQ. Thus the product of two sets can be defined as the object with this universal property. This description of the product of the sets X and Y mentions only objects and morphisms; it says nothing about ordered pairs or any thing specifically about the objects under consideration. So we can import this definition to any arbitrary category. In particular, if C is any random category and X, Y are objects of C, we can define the product of X and Y as being an object P of C with morphisms p1 : P → X, p2 : P → Y which is universal, i.e. if Z is any other object of C with maps q1 : Z → X, q2 : Z → Y , there is a unique map q : Z → P such that q1 = p1 ◦ q and

29 q2 = p2 ◦ q. In general such a universal object may not exist. In the category SET it does: it is the set consisting of the ordered pairs (x, y) for x ∈ X, y ∈ Y .

Universal properties (such as products) can be expressed by saying that a certain

presheaf is representable. Deﬁnition 2.6.4. If for every pair of objects X and Y of a category C, the presheaf

hX × hY : Z 7→ Hom(Z,X) × Hom(Z,Y ) (2–58)

is representable, then we say that products are representable in C. The object that repre-

sents this functor is unique up to unique isomorphism whence we say that the product of X and Y is the object that represents this functor.

To see why this deﬁnition makes sense, say an object P represents the product of

X and Y . By deﬁnition this means we have an isomorphism hP ' hX × hY , i.e. an isomorphism

αZ : Hom(Z,P ) → Hom(Z,X) × Hom(Z,Y ) (2–59)

for each Z ∈ Ob(C). Then plugging P in for Z above, we get that the universal element

αZ (idP ) is a pair of morphisms p1 : P → X, p2 : P → Y with the property that given

any pair of morphisms q1 : Z → X, q2 : Z → Y , both factor through a unique morphism Z → P , see diagram below.

q ∈ Hom(Z,P ) ∼ / Hom(Z,X) × Hom(Z,Y ) (2–60) O O

Hom(q,P ) (q1=p1q,q2=p2q)

∼ / idP ∈ Hom(P,P ) Hom(P,X) × Hom(P,Y ) The deﬁnition for arbitrary products is an obvious extension of the above deﬁnition.

We can make similar deﬁnitions for ﬁbered products, equalizers, and coequalizers.

First we list these deﬁnitions for the category SET . Given sets X, Y and Z with maps

30 f : X → Z, g : Y → Z, the ﬁbered product X ×Z Y is the subset of the product X × Y consisting of (x, y) such that f(x) = g(y). Given two maps f, g : X → Y of sets, the equalizer Eq(f, g) of f and g is the set {x ∈ X|f(x) = g(x)}. If i : Eq(f, g) → X is the inclusion, then f ◦ i = g ◦ i, and Eq(f, g) is universal for this property. The deﬁnition of the coequalizer is just dual to that of the equalizer.

We now make this definitions for any arbitrary category C. Definition 2.6.5. 1. Given arrows f : X → Z, g : Y → Z in C, we define the fibered

product X ×Z Y of X and Y to be the object P which represents the functor

Q 7→ Hom(Q, X) ×Hom(Q,Z) Hom(Q, Y ). (2–61)

If the object P exists then we have a functorial isomorphism

αQ : Hom(Q, P ) ' Hom(Q, X) ×Hom(Q,Z) Hom(Q, Y ). (2–62)

The universal object αP (idP ) is a pair of morphisms p1 : P → X, p2 : P → Y .

Since it is universal, for any pair of morphisms q1 : Q → X, q2 : Q → Y such that

f ◦q1 = g ◦q2, there is a unique arrow h : Q → P such that q1 = p1 ◦f and q2 = p2 ◦f. 2. Given arrows f, g : X → Y , we deﬁne their equalizer to be the object K which represents the functor

f / Z 7→ Eq( Z / X / Y ). (2–63) g

i.e., Hom(Z,f) / Z 7→ Eq( Hom(Z,X) / Hom(Z,Y )). (2–64) Hom(Z,g) If K exists then there is a functorial isomorphism

Hom(Z,K) ' Eq(Hom(Z, f), Hom(Z, g)). (2–65)

31 Thus there is a canonical morphism i : K → X such that f ◦ i = g ◦ i, and any morphism h : Z → X such that f ◦ h = g ◦ h factors through a unique morphism Z → K.

3. Given arrows f, g : X → Y , we deﬁne their coequalizer to be the object K which represents the functor

Hom(f,Z) / Z 7→ Eq( Hom(X,Z) / Hom(Y,Z)). (2–66) Hom(g,Z)

If K exists then there is a functorial isomorphism

Hom(Z,K) ' Eq(Hom(f, Z), Hom(g, Z)). (2–67)

Thus there is a canonical map p : Y → K such that p ◦ f = p ◦ g, and any morphism h : Y → Z such that h ◦ f = h ◦ g factors through a unique morphism K → Z. 2.7 2-Representability

We will now briefly discuss the notion of 2-representability. These definitions can be found in [16]. Let C be a 2-category. Recall that Cop is obtained from C by “inverting the direction” of the 1-cells of C .i.e. if [X,Y ] is the collection of 1-cells in C from X to Y then [X,Y ] in Cop is [Y,X]. We will define 2-representability for 2-functors f : Cop → CAT . Let F (C) be the 2-category of 2-functors from Cop to CAT . We define the following strict 2-functor:

h : C → CAT (2–68)

hX ( ) = HomC( ,X) (2–69)

and which is deﬁned similarly on 1-cells and 2-cells. Let f be any 2-functor .i.e. f ∈

Ob(F (C)). Then for all X ∈ Ob(C), we can deﬁne the functor

hX, : HomF (C)(hX , f) → f(X). (2–70)

32 0 For u ∈ Ob(HomF (C)(hX , f)) we associate hX,f (u) = u(idX ) ∈ Ob(f(X)) and to φ : u → u , where φ ∈ Mor(HomF (C)(hX , f)) we associate

0 hX,f (φ) = φ(idX ): u(idX ) → u (idX ). (2–71)

The functor hX,f is an equivalence of categories with the quasi-inverse given by kX,f : f(X) → HomF (C)(hX , f) where for all ξ ∈ Ob(f(X)), kX,f (ξ) = uξ is the morphism of

0 2-functors uξ : hX → f, uξ( ) = f( )(ξ) and for φ : ξ → ξ ∈ Mor(f(X)),

kX,f (φ) = f( )(φ): uξ → uξ0 . (2–72)

In particular, letting f = hX0 we get the following proposition. Proposition 2.7.1. The 2-functor h : C → F (C) is 2-faithful. Deﬁnition 2.7.2. We say that a 2-functor f ∈ Ob(F (C) is 2-representable if f is equivalent to a 2-functor of the form hX . We say that the solution to the universal 2- problem determined by f is the pair (X, u) where X ∈ Ob(C) and u : hX ' f is an

equivalence of 2-functors. We say that the pair (X, ξ), where ξ = u(idX ) ∈ Ob(f(X)) gives a representation of f.

See [16] for the proofs of the following propositions. Proposition 2.7.3. Let f be a representable 2-functor in F (C) and let (X, ξ) and (X0, ξ0) be two representations of f. Then there exists a pair (λ, ²) which gives an equivalence in C:

0 0 λ : X ' X and a 2-isomorphism ² : f(λ)(ξ ) ' ξ. Further, for all other such pairs (λ1, ²1)

0 there exists a unique 2-isomorphism α : λ ' λ1 such that ²1 ◦ (f(λ)(ξ )) = ². Proposition 2.7.4. Let C be a 2-category and let F(C) denote the 2-category of 2- functors from Cop to CAT . Let F be a 2-functor which is representable in F(C), and suppose (X, ξ) and (X0, ξ0) are two given representations of F . Then there exists a

∼ ∼ pair (λ, ²) such that λ : X / X0 is an equivalence in C and ² : F (λ)(ξ0) / ξ0

is a 2-isomorphism. Further, for every other such pair (λ1, ²1) there exists a unique

∼ / 0 0 2-isomorphism α : λ λ such that ²1 ◦ (F (λ)(ξ )) = ².

33 2.8 Giraud’s approach to Liens of Gerbes

We now describe Giraud’s original definition of a lien given in [15]. It is this approach that we will generalize to define the 2-lien of a 2-gerbe. Let X be a topological space, and let F and G be sheaves of groups on X. Consider the sheaf Isom(F,G) (the sheaf of isomorphisms of sheaves of groups from F to G). Let Int(G) denote the sheaf of inner automorphisms of the sheaf G. Define the quotient sheaf

Out(F,G) := Isom(F,G)/Int(G) (2–73) where Int(G) acts on Isom(F,G) from the right via composition. Now, for every open U ⊂ X, let LI(X)(U) denote the category whose objects are the sheaves of groups on U, and whose arrows between two objects A and B are the global sections of the sheaf

Out(A, B)(U) i.e. HomLI(X)(U)(A, B) = Γ(U, Out(A, B)). The composition law is deﬁned after passing to the quotient sheaf Out, and is well-deﬁned (see the discussion of a lien of a gerbe in Section 2.3).

Now if V,→ U is an inclusion of open sets in X, then the formation of the quotient Out(A, B) commutes with the restriction of U to V . So we can define a fibered category LI(X). Let Grp(X) denote the fibered category of sheaves of groups on X. Then by construction we have a morphism of fibered categories

Grp(X) → LI(X). (2–74)

Now if A and B are any two objects of the category LI(X)(U), then the presheaf

HomLI(X)(U)(A, B) of arrows from A to B is identified with the sheaf Out(A, B). Thus LI(X) is a prestack. Thus upon applying the stackification functor, we obtain the associated stack LIEN(X). Composing the stackification functor [15] with the morphism (1) yields a morphism of stacks

lien(X): Grp(X) → LIEN(X). (2–75)

34 Deﬁnition 2.8.1. 1. The stack LIEN(X) is called the stack of liens on X. 2. We call a global section of this stack a lien over X.

For every U ⊂ X, we say that a lien of U is an object of the category LIEN(X|U )

which is the fiber over U of the stack of liens. We call LIEN(X|U ) the category of liens over U. To generalize this definition of a lien from topological spaces to topoi, we just replace “global section” in Definition 2.8.1.2 with “a cartesian section” of the stack

LIEN(X), as done by Giraud in [15]. Now let G be a stack on X. Recall that Grp(X) denotes the stack of sheaves of groups on X. We have a cartesian functor

AUT : G → Grp(X) (2–76)

AUT (G)(x) = AutU (x) (2–77)

where x ∈ Ob(G(U)). Composing this morphism with the morphism lien(X): Grp(X) → LIEN(X) gives a morphism of stacks

liau(G): G → LIEN(X) (2–78)

which to every object x ∈ G(U) associates the lien

liau(G)(x) = lien(AutU (x)) (2–79)

(called the lien represented by the sheaf of U-automorphisms of x) and to every U-isomorphism α : x → y of G associates the morphism

liau(G)(α) = lien(Int(α)) (2–80)

−1 (where Int(α): AutU (x) → AutU (y) deﬁned by Int(α)(a) = αaα is a morphism of sheaves of groups over X).

35 By the deﬁnition of the stack of liens, if m, n : x → y are two U-isomorphisms of G, we have

liau(G)(m) = liau(G)(n) (2–81)

since the functor lien transforms every inner automorphism to an identity arrow.

Now let m : F → G be a morphism of stacks. Then we have a morphism of morphisms of stacks (i.e a 2-arrow):

liau(m): liau(F) ⇒ liau(G) · m (2–82)

where, liau(m) = lien(X) ◦ AUT (m). (2–83)

For every x ∈ Ob(F(U)), the functor m induces a morphism of sheaves of groups

µx : AutU (x) → AutU (m(x)) (2–84)

and by deﬁnition of the map liau(m), the morphism liau(m(x)) is the morphism of liens represented by µx:

mx = lien(µx), mx : lien(AutU (x)) → lien(AutU (m(x))). (2–85)

Deﬁnition 2.8.2. Let F be a stack over X and let L be a lien over X i.e. it is a global section L : X → LIEN(X) of the stack of liens. An action of L on F is a morphism of morphisms of stacks: a : L ◦ f ⇒ liau(F) (2–86)

where f : F → X is the projection onto the Zariski site of X

X J (2–87) ? JJ f JJL JJ a JJ J% F ÑÙ / LIEN(X) liau(F)

36 The Zariski site of X is the category whose objects are the open sets of X, and the morphisms are the inclusion maps. By deﬁnition a is a morphism of (cartesian) functors

i.e. a family a(x): L(U) → lien(AutU (x)), x ∈ Ob(F(U)), U ⊂ X, of morphisms of liens on U that satisﬁes the conditions of compatibility with the restriction to open sets, and with the composition of morphisms.

Deﬁnition 2.8.3. Let (L, a) be a lien operating on a stack F and let u : L0 → L be a morphism of liens over X. Then the action induced by a and u is the morphism b deﬁned by b : L0 · f → liau(F ), (2–88)

b = a · (u ◦ f ◦ L) (2–89)

(where f : F → X is the projection onto the Zariski site of X). So b is the action such that, for every x ∈ Ob(F(U)), b(x) is the composition

u(U) a(x) L0(U) / L(U) / liau(F)(x) . (2–90)

Proposition 2.8.4. Let F be a gerbe on X.

1. Let (L, a) be a lien operating on F. The following are equivalent: (a) a is an isomorphism. (b) for every lien L0 on X and every action b of L0 on F there exists a unique morphism of liens u : L0 → L such that b is the action induced by a and u.

2. There exists a lien (L, a) operating on F that satisfies the conditions of (1). Definition 2.8.5. We say that the lien of a gerbe F is a lien operating on F and satisfying the conditions of the above proposition. Condition 1(b) characterizes the lien of a gerbe F up to canonical equivalence. So we are justified in saying “the” lien of a gerbe F. By abuse of notation, the map a is often not mentioned. If L is the lien of a gerbe F, we say F is bound by L. Conversely, if L is a lien on X, then an L-gerbe is a pair (F, a) where F is a gerbe and (L, a) is a lien of F.

37 The lien of a gerbe F was in fact made explicit before: it is a lien L and a family of isomorphisms of liens over U,

∼ / a(x): L(U) lien(AutU (x)) (2–91)

where x ∈ F(U), U ⊂ X. This family is required to be compatible with the restriction of open sets and also satisfy the condition that if i : x → y is a U-isomorphism of F, then

the morphism of sheaves of groups Int(i): AutU (x) → AutU (y) represents the identity morphism of L(U).

Proof of Proposition 2.8.4. We follow Giraud’s proof in [15]. Since F is a gerbe, the

0 projection f : F → Xzar to the Zariski site is fully faithful, hence the map Hom(L ,L) → Hom(L0f, Lf) given by u 7→ u ◦ f is bijective. This forces b = a(u ◦ f) whence we have that 1(a) implies 1(b). Next, we show that for every gerbe F, there exists a lien L and an isomorphism a : Lf ∼ / liau(F) . Then (2) follows trivially, and so does 1(b) ⇒ 1(a)

since 1(b) determines L upto unique isomorphism. Let I be the image category of liau(F). Clearly this is a ﬁbered category of LIEN(X) and the functor induced by liau(F),

L0 : F → I (2–92)

is cartesian. Moreover, the projection I → Xzar is both fully faithful and essentially surjective (this follows trivially from the fact that F → X is fully faithful and because liau(m) = liau(n) for any two U-isomorphisms m, n : x → y of F). By the universal property of the associated stack, it follows that X is the stack associated to I, and that

∼ / 0 there exists a cartesian section L0 : X → I of I and an isomorphism a0 : L0f L .

Then setting L = iL0 and a = i ◦ a0 (where i : I → LIEN(X) is the inclusion) the conclusion follows. Corollary 2.8.6. Let m : F → G be a morphism of gerbes and let (L, a) and (M, b) denote the liens of F and G respectively. Then there exists a unique morphism of liens u : L → M such that m is a u-morphism i.e. liau(m) · a = (b ◦ m)(u ◦ f).

38 CHAPTER 3 EQUALIZERS AND COEQUALIZERS

Recall that a lien is deﬁned by assigning a family of sheaves of groups (Gi) to the

open sets (Ui) that cover X, and these sheaves are glued on the overlaps Uij by a section

of the quotient sheaf Out(Gj,Gi) = Isom(Gj,Gi)/Inn(Gi). Our project involves appropriately defining the action of a gr-stack on a stack, and the quotient by such an action. To do this we first need to define these notions for group categories. 3.1 Equalizers

We begin by recalling the deﬁnition of equalizer for the category SET .

Given two maps f, g : X → Y of sets, the equalizer Eq(f, g) of f and g is the set {x ∈ X|f(x) = g(x)}.

For any arbitrary category C, given arrows f, g : X → Y , we deﬁne their equalizer to be the object K which represents the functor

f / Z 7→ Eq( Z / X / Y ). (3–1) g

i.e., Hom(Z,f) / Z 7→ Eq( Hom(Z,X) / Hom(Z,Y )). Hom(Z,g) If K exists then there is a functorial isomorphism

Hom(Z,K) ' Eq(Hom(Z, f), Hom(Z, g)). (3–2)

Thus there is a canonical morphism i : K → X such that f ◦ i = g ◦ i, and any morphism h : Z → X such that f ◦ h = g ◦ h factors through a unique morphism Z → K.

Equalizers are representable in the category SET by the object Eq(f, g). In fact we have a functorial isomorphism

Hom(Z, Eq(f, g)) ' Eq(Hom(Z, f), Hom(Z, g)). (3–3)

39 To see why this is true, let h : Z → Eq(f, g) be any arrow in Hom(Z, Eq(f, g)). Now Eq(f, g) = {x ∈ X|f(x) = g(x)} so Eq(f, g) is a subset of X. Thus h : Z → Eq(f, g) is an arrow from Z to X. Further for z ∈ Z we have (f ◦ h)(z) = f(h(z)) = g(h(z)) = (g ◦ h)(z) since h(z) ∈ Eq(f, g). Thus h is an element of Eq(Hom(Z, f), Hom(Z, g)). Thus we can deﬁne ϕ : Hom(Z, Eq(f, g)) → Eq(Hom(Z, f), Hom(Z, g)) to be the identity map, which is clearly an isomorphism.

3.2 Representability of Equalizers in CAT

Let X, Y ∈ Ob(CAT ) and let f,g : X → Y denote two functors.

Deﬁnition 3.2.1. The category Eq(f, g) of f and g is the category deﬁned as follows:

1. The objects of Eq(f, g) are pairs (x, α) where each x is an object of the category X and α : f(x) ∼ / g(x) .

2. Morphisms (x, α) → (y, β) are those arrows h : x → y in X such that the diagram below commutes.

∼ / f(x) α g(x) (3–4)

f(h) g(h) f(y) ∼ / g(y) β Observe that we have an arrow ψ : Eq(f, g) → X that sends each object (x, α) to x ∈ X, and is the inclusion on morphisms. We now deﬁne equalizers in an arbitrary 2-category.

Deﬁnition 3.2.2. Let C be a 2-category. We say that equalizers are representable in C if for every pair of objects X, Y in C and every pair of 1-cells F , G : X → Y in C the 2-functor

Eq(F,G): C → CAT, (3–5)

Hom(F,Z) / Z 7→ Eq( Hom(Z,X) / Hom(Z,Y )) Hom(G,Z) is 2-representable.

Proposition 3.2.3. Equalizers are representable in CAT .

40 In fact for every pair of objects X, Y in C and every pair of 1-cells f, g : X → Y in C, there is a 2-functorial equivalence of categories:

ϕ : Hom(Z, Eq(f, g)) ∼ / Eq(Hom(Z, f), Hom(Z, g)) (3–6)

where,

Hom(Z,f) / Eq(Hom(Z, f), Hom(Z, g)) := Eq[ Hom(Z,X) / Hom(Z,Y )]. Hom(Z,g)

Proof. Let f∗ and g∗ denote Hom(Z, f) and Hom(Z, g) respectively. Deﬁne ϕ : f∗ / Hom(Z, Eq(f, g)) → Eq( Hom(Z,X) / Hom(Z,Y ) ) as follows: g∗ • On objects: Given F : Z → Eq(f, g), denote the functor F as follows: F (Z) =

∼ / ∼ / (F0(Z), f(F0(Z)) g(F0(Z)) ). Thus F0 : Z → X and F1 : f ◦ F0 g ◦ F0 F1(Z)

are functors. Then deﬁne ϕ(F ) := ψ ◦ F0 where ψ : Eq(f, g) → X is the map deﬁned ∼ / above. Then ϕ(F ) is in Hom(Z,X). Also f∗(ϕ(F )) = f(ψ(F0)) g(ψ(F0)) = F1 f∗ / g∗(ϕ(F )). Thus (ϕ(F ),F1) is an object of Eq( Hom(Z,X) / Hom(Z,Y ) ). g∗ • On arrows: Suppose F and G are two functors in Hom(Z, Eq(f, g)), and let k : F →

G be a morphism of functors. Then ϕ(F ) := ψ ◦ F0 and ϕ(G) := ψ ◦ G0 are in f∗ / Eq( Hom(Z,X) / Hom(Z,Y ) ). By deﬁnition of k, if x and y are objects of Z g∗ and λ : x → y is an arrow in Z then the diagram:

F (x) kx / G(x) (3–7)

F (λ) G(λ) F (y) / G(y) ky

41 commutes. So deﬁne ϕ(K) := ψ ◦ k (recall that ψ : Eq(f, g) → X is the inclusion on morphisms). Thus the diagram

ϕk ϕF (x) x / ϕG(x) (3–8)

ϕF (λ) ϕG(λ) ϕF (y) / ϕG(y) ϕky

commutes commutes in Hom(Z,X). But ϕ(F ) and ϕ(G) are in f∗ / Eq( Hom(Z,X) / Hom(Z,Y ) ) so given a morphism of functors in Hom(Z,X) we g∗ f∗ / have in the category Eq( Hom(Z,X) / Hom(Z,Y ) ) a commutative cube: g∗

f∗(ϕk) / f∗(ϕF (x)) f∗(ϕG(x)) (3–9) o o × f∗(ϕF (λ)) × f∗(ϕG(λ))) / g∗(ϕF (x)) g∗(ϕG(x)) g∗(ϕk)

g∗(ϕG(λ))

f∗(ϕk) / g∗(ϕ(F (λ)) f∗(ϕF (y)) f∗(ϕG(y))

o o

Ô Ô / g∗(ϕF (y)) g∗(ϕG(y)) g∗(ϕk)

f∗ / Thus ϕ : Hom(Z, Eq(f, g)) → Eq( Hom(Z,X) / Hom(Z,Y ) ) is a well-deﬁned g∗ morphism.

We now show that ϕ is in fact an equivalence of categories.

• ϕ is faithful: Consider HomHom(Z,Eq(f,g))(F,G) → HomEq(f∗,g∗)(ϕ(F ), ϕ(G)) where f∗ / Eq(f∗, g∗) := Eq( Hom(Z,X) / Hom(Z,Y ) ). If k : F → G is in Hom(F,G) g∗

42 then recall that ϕ(k) := ψ ◦ k. Further recall the deﬁnition of ψ : Eq(f, g) → X. It takes an object (x, α) to x and it is the inclusion on arrows. So if ϕ(k) = ϕ(k0) then ψ ◦ k = ψ ◦ k0 which implies that k = k0 since ψ is the inclusion on arrows. So ϕ is

faithful.

• ϕ is full: Now suppose k : ϕ(F ) → ϕ(G) is an arrow. Then kx :(ψ ◦ F )(x) → (ψ ◦ G)(x) is an arrow for each x. But ψ is a functor from Eq(f, g) to X whence

αx / (ψ ◦ F0)(x) → (ψ ◦ G0)(x) = ψ( F0(x) G0(x) ) for some αx. Let α : F → G be

deﬁned by α = αx for each x in Ob(Z). Then ϕ(αx) = ψ ◦ αx = kx so ϕ(α) = k. Thus ϕ is full.

• ϕ is essentially surjective: Let (G, α) be an object of Eq(f∗, g∗). Then G is an arrow ∼ / from Z to X and α is a morphism of functors i.e. α : f∗(G) g∗(G) i.e. for k : G → G0 in Hom(Z,X) have a commutative diagram

αG / f∗(G) g∗(G) . (3–10)

f∗(k) g∗(k) 0 / 0 f∗(G ) g∗(G ) αG0

Consider F : Z → Eq(f, g) deﬁned as follows:

1. On objects: for x in Ob(Z), F (x) := (G(x), αG,x), 2. On arrows: for λ : x → y in Z, F (λ) := G(λ).

Then for any x in Ob(Z), ϕ(F (x)) = ψ◦F0(x) = ψ(F0(x)) = ψ((G0(x), αG,x)) = G(x). ∼ / Further ϕ(F ) = (G, β) where β is an isomorphism of functors f∗ g∗ as is α so α ∼ / β . Thus ϕ(F ) ∼ / (G, α) whence ϕ is essentially surjective.

Thus ϕ is an equivalence of categories. 3.3 Coequalizers

The concept of coequalizer is a useful way of formulating quotients.

Let C be a category.

43 Deﬁnition 3.3.1. Given arrows f, g : X → Y , we deﬁne their coequalizer to be the object K which represents the functor

Hom(f,Z) / Z 7→ Eq( Hom(X,Z) / Hom(Y,Z)). (3–11) Hom(g,Z)

If K exists then there is a functorial isomorphism

Hom(Z,K) ' Eq(Hom(f, Z), Hom(g, Z)). (3–12)

Thus there is a canonical map p : Y → K such that p ◦ f = p ◦ g, and any morphism h : Y → Z such that h ◦ f = h ◦ g factors through a unique morphism K → Z.

Deﬁnition 3.3.2. Given two maps f, g : X → Y of sets, let Coeq(f, g) of f and g be the projection p : Y → Y/E on the quotient of Y by the smallest equivalence relation E ⊂ Y × Y which contains all pairs (f(x), g(x)) for x ∈ X. Theorem 3.3.3. Coequalizers are representable in the category SET by the object Coeq(f, g).

The proof is similar to that given for equalizers above, and we do not include it here.

If F : G × A → A is the action of a group G on a set A then the quotient by p2 / the action is just the coequalizer of the arrows G × A / A . It is this formulation of F quotients which lends itself to generalization i.e. we use this approach to deﬁne a quotient when a group category acts on a category.

3.4 Representability of Coequalizers in CAT

We have the deﬁnition of an equalizer in the 2-category CAT . We will now use this deﬁnition to establish that coequalizers exist in CAT .

First we deﬁne coequalizers in an arbitrary 2-category.

Deﬁnition 3.4.1. Let C be a 2-category. We say coequalizers are representable in C if for every pair of objects X, Y of C and every pair of 1-cells F , G : X → Y in C the 2-functor

Ceq(F,G): C → CAT,

44 Hom(f,Z) / Z 7→ Eq( Hom(Y,Z) / Hom(X,Z) ) (3–13) Hom(g,Z) is 2-representable. This definition is dual to the one given for the equalizer Eq(F,G). F / Definition 3.4.2. Given two functors C / D , there is an induced functor H : C → G D × D, x 7→ (F (x),G(x)). We say that F and G are a small pair of functors if the fibers of H are small categories i.e. for all (a, b) ∈ Ob(D × D), H−1(a, b) is a set. We will devote the rest of this section to proving that the coequalizer of a small pair

of functors is representable in CAT . Let C and D be categories, and let F and G denote a small pair of functors from C to D. Deﬁne a category Coeq(F,G) as follows. The objects

of Coeq(F,G) will be objects of D. To define the arrows in Coeq(F,G) we will first define a set of “prearrows”. We will then define the arrows in Coeq(F,G) to be the quotient of this set of prearrows by a certain equivalence relation.

We deﬁne the set of prearrows P reAr(F,G) to be the disjoint union of the following two sets:

1. Ar(D)

∼ / 2. The set S consisting of, for every x ∈ C, a formal isomorphism F (x) α G(x) and its inverse: that is, an arrow of the form G(x) ∼ / F (x). α−1 The source and target of an element of Ar(D) is just its usual source and target in D.

∼ / A member of S of the form F (x) α G(x) has source F (x) ∈ Ob(D) and target G(x) ∈ Ob(D), and an element of the form G(x) ∼ / F (x) has source G(x) ∈ Ob(D) and α−1 target F (x) ∈ Ob(D). So within S, after one ﬁxes a source s and a target t, the arrows between s and t form a set.

Deﬁnition 3.4.3. A prearrow is a ﬁnite sequence (s1, s2, ..., sn) where si ∈ Ar(D) ∪ S and target(si)=source(si−1) for all i. Let P reAr(F,G) be the set of prearrows on Ar(D) ∪ S. Next we introduce a binary operation on P reAr(F,G). Given two prearrows

(r1, r2, ··· , rm) and (s1, s2, ··· , sn), if the target(s1)= source(rm) then their product is

45 deﬁned as:

(r1, r2, ··· , rm) ∗ (s1, s2, ··· , sn) = (r1, r2, ··· , rm, s1, s2, ··· , sn) (3–14)

Observe that by this deﬁnition (s1, s2, ··· , sn) = s1 ∗ s2 ∗ · · · ∗ sn. We will henceforth refer to the above product as composition. Then there is a smallest equivalence relation ∼ on P reAr(F,G) such that 1. For any s ∈ S, if x =source(s) and y =target(s), then

s ∗ idx ∼ idy ∗ s ∼ s. (3–15)

2. For any composable α, β ∈ Ar(D), (α ∗ β, α ◦ β) is in the relation i.e. α ∗ β ∼ α ◦ β. 3. for any a, b ∈ Ob(C) and t : a → b in Ar(C)

F (t) G(t) F (a) / F (b) ∼ / G(b) ∼ F (a) ∼ / G(a) / G(b) . (3–16)

4. If α, β ∈ Ar(D) ∪ S and source(α)=target(β) then (s1, α) ∗ (β, s2) ∼ (s1, (α ◦ β), s2)

for si ∈ P reAr(F,G). Further, if α1, α2 and β1, β2 are elements of P reAr(F,G)

such that α1 ∼ β1 and α2 ∼ β2 and suppose α2 ◦ α1 and β2 ◦ β1 is deﬁned, then

α2 ◦ α1 ∼ β2 ◦ β1. Deﬁne Coeq(F,G) as follows: Ob(Coeq(F,G)) = Ob(D) and Ar(Coeq(F,G)) = P reAr(F,G)/ ∼. Proposition 3.4.4. Coeq(F,G) satisﬁes the axioms of a category.

Proof. Let [f] ∈ HomCoeq(F,G)(A, B) and [g] ∈ HomCoeq(F,G)(B,C). Deﬁne [g] ◦ [f] := [g ◦ f].

To see that this is well deﬁned, suppose f1, f2 ∈ [f] and g1, g2 ∈ [g]. Then f1 ∼ f2 and g1 ∼ g2, which implies that g1 ◦ f1 ∼ g2 ◦ f2 whence [g1 ◦ f1] = [g2 ◦ f2]. So we have a composition law. Similarly, the associative law holds since it holds on the level

of prearrows. For suppose [f] ∈ HomCoeq(F,G)(A, B), [g] ∈ HomCoeq(F,G)(B,C), and

[h] ∈ HomCoeq(F,G)(C,D). Then [h] ◦ ([g] ◦ [f]) = [h] ◦ ([g ◦ f]) = [h ◦ (g ◦ f)] = [(h ◦ g) ◦ f] = ([h ◦ g]) ◦ [f] = ([h] ◦ [g]) ◦ [f] where the second equality follows since h ◦ (g ◦ f) is in

46 P reAr(F,G). Thus we have associativity. Finally, consider 1A ∈ HomCoeq(F,G)(A, A).

If α ∼ 1A in Coeq(F,G) then 1A = 1A ◦ α = α which implies that [1A] = 1A in

Coeq(F,G). So for [f] ∈ HomCoeq(F,G)(A, B) and [g] ∈ HomCoeq(F,G)(B,C) we have that

[1B] ◦ [f] = [1B ◦ f] = [f] and [g] ◦ [1B] = [g ◦ 1B] = [g] so the identity axiom is true.

By construction of Coeq(F,G) we have an arrow D λ / Coeq(F,G) which is the identity on objects and sends arrows in D to their corresponding class in Coeq(F,G).

Theorem 3.4.5. The coequalizer of a small pair of functors is representable in CAT . In

F / λ fact the diagram C / D / Coeq(F,G) gives an equivalence G

∗ ∼ F / Hom(Coeq(F,G), E) / Eq(Hom(D, E) / Hom(C, E)) (3–17) G∗

i.e. the functor category Hom(Coeq(F,G), E) is equivalent to a category B whose objects consist of functors f : D → E together with isomorphisms f ◦ F ∼ / f ◦ G and whose arrows are morphisms f → g of functors such that for each object x in the category C we have a commutative diagram:

fF (x) ∼ / fG(x) (3–18)

gF (x) ∼ / gG(x) Proof. Deﬁne ϕ : Hom(Coeq(F,G), E) → B as follows: if J : Coeq(F,G) → E is a functor, deﬁne ϕ(J) = J ◦ λ. If f : D → E denotes this composition (i.e if f := J ◦ λ) then an explicit description of f on objects and arrows is as follows:

• Objects(D): for x ∈ Ob(D), f(x) = J(x). J([α]) • Arrows(D): if α : x → y is an arrow in D, send this arrow to J(x) / J(y).

We show that ϕ is fully faithful. Let α, β : J → K where J, K : Coeq(F,G) → E are functors. Let f = ϕ(J) and g = ϕ(K). Then for α ∈ HomHom(Coeq(F,G),E)(J, K), its image in HomB(f, g) is a map γ : f → g which is a morphism of functors. So by deﬁnition, γ is

determined when we give maps γx : f(x) → g(x) for each x ∈ Ob(D), compatible with the morphisms in D. But Ob(D) = Ob(Coeq(F,G)) and since f(x) = J(x) and g(x) = K(x)

47 for all x ∈ Ob(D), γx is deﬁned to be αx. Thus if α, β ∈ HomHom(Coeq(F,G),E)(J, K) have

the same image γ in HomB(f, g) then αx = βx for all x ∈ Ob(Coeq(F,G)) whence α = β.

Now let γ : f → g be a morphism in B .i.e. γ ∈ HomB(f, g). Deﬁne α : J → K by

αx = γx. This makes sense since Ob(D) = Ob(Coeq(F,G)). For x, y ∈ Ob(Coeq(F,G)) suppose [u]: x → y is an arrow in Coeq(F,G). Let u represent the class of [u]. Then u is a composition of arrows in D and isomorphisms F (x) ∼ / G(x) for x ∈ Ob(C).

Case 1: Suppose u ∈ Ar(D). Then since γ is a morphism of functors, the diagram

γ f(x) x / g(x) (3–19)

f(u) g(u) γ f(y) y / g(y)

commutes, whence the diagram

f(x) = J(x) / K(x) = g(x) (3–20)

f(u)=J([u]) K([u])=g(u) f(y) = J(y) / K(y) = g(y)

commutes.

Case 2: Suppose u : x → y where x = F (a) and y = G(a) for a ∈ C. Then by deﬁnition of γ the diagram fF (a) ∼ / fG(a) (3–21)

γF (a) γG(a) gF (a) ∼ / gG(a) commutes so

JF (a) ∼ / KF (a) (3–22)

αF (a)=γF (a) αG(a)=γG(a)

JG(a) ∼ / KG(a) commutes since the vertical arrows in the above two diagrams are the same by deﬁnition

of the map ϕ.

48 Since for any x, y ∈ Ob(Coeq(F,G)), u : x → y is a composition of the types of arrows in Cases 1 and 2, we get that the diagram

J([u]) J(x) / J(y) (3–23)

αx αy K(x) / K(y) K([u])

commutes for all [u] ∈ Ar(Coeq(F,G)). Thus we have a bijection between HomB(f, g) and

HomHom(Coeq(F,G),E)(J, K): that is, ϕ is fully faithful.

We now show that ϕ is essentially surjective. Let f : D → E with isomorphisms

f ◦ F ∼ / f ◦ G be an object of B. We wish to define J : Coeq(F,G) → E such that ϕ(J) ∼ / f . Define J : Coeq(F,G) → E by J(x) = f(x) on objects. Let [u] be an arrow in Coeq(F,G). Let u represent the class of [u]. Then u is just a composition of arrows in D and formal isomorphisms. We define J on morphisms by sending arrows f x i / y in D to f(x)(i) / f(y) , and sending formal isomorphisms F (a) ∼ / G(a) to fF (a) ∼ / fG(a) . To see that this is well defined, consider for any arrow t : x → y in

Ar(C) the arrows u1 and u2 deﬁned by:

∼ / G(t) / u1 : F (x) G(x) G(y) (3–24)

and

F (t) / ∼ / u2 : F (x) F (y) G(y) (3–25)

Then u1 ∼ u2. Then we have

∼ / fG(t) / J(u1): fF (x) fG(x) fG(y) (3–26)

and

fF (t) / ∼ / J(u2): fF (x) fF (y) fG(y) (3–27)

49 are the same arrows in D because in Coeq(F,G) the diagram

F (x) ∼ / G(x) (3–28)

F (t) G(t) F (y) ∼ / G(y) commutes whence the diagram

fF (x) ∼ / fG(x) (3–29)

fF (t) fG(t) fF (y) ∼ / fG(y) also commutes. Thus J(u1) = J(u2). Since u is a composition of arrows of the above type get that F (u) is well deﬁned. We claim ϕ(J) ∼ / f . Clearly ϕ(J) = f on objects. If J([u]) u : x → y is in Ar(D) then ϕ(J) sends this arrow to J(x) / J(y) which is isomorphic f(u) to f(x) / f(y) . Thus ϕ is essentially surjective.

50 CHAPTER 4 GROUP CATEGORIES

4.1 Inner Equivalences of Group Categories

We deﬁne the notion of inner equivalences for group categories. We mention that L.

Breen [6] has also given a similar description of this notion.

Let C be a group category. Let Y , Z ∈ Ob(C) such that ZY ' I ' YZ. Deﬁne

FYZ : C → C as follows:

• On objects: for A ∈ Ob(C) let FYZ (A) := (ZA)Y . • On arrows: given an arrow f : A → B in C, functoriality of multiplication gives an arrow F (f): F (A) → F (B) such that the diagram

f A / B . (4–1)

FYZ FYZ

F (f) F (A) YZ / F (B) commutes. So FYZ is a functor. Further, for A, B ∈ Ob(C) we have:

FYZ (AB) = (Z(AB))Y (4–2)

assoc. (ZA)(BY )

MC−axioms ((ZA)I)(BY )

I'YZ ((ZA)(YZ))(BY )

assoc. ((ZA)Y )((ZB)Y )

FYZ (A)FYZ (B)

So FYZ respects the group structure of C. For A, B, C in Ob(C) we have:

51 FYZ [A(BC)] (4–3)

(Z[A(BC)])Y

c F (assoc) (Z[(AB)C])Y i UU iiii UUUU iiii UUUU iiii UUUU it iii i1 j1 UUU* (Z[[(AB)I]C])Y / (Z[([AI]B)C])Y

i2 j2 (Z[[AB](YZ)]C])Y / (Z[([A(YZ)]B)C])Y

i3 j3 ((Z[AB])Y )((ZC)Y ) / ((ZA)Y )((Z(BC))Y )

i4 j4 ((Z[(AI)B])Y )((ZC)) / ((ZA)Y )((Z([BI]C))Y )

i5 j5 ((Z[(A(YZ))B])Y )((ZC)Y ) / ((ZA)Y )((Z([B(YZ)]C))Y )

i6 j6 ([(ZA)Y ][(ZB)Y ])((ZC)Y ) / ((ZA)Y )([(ZB)Y ][(ZC)Y ])

(F (A)F (B))F (C) F (A)(F (B)F (C))

• The horizontal arrows are the associative and identity laws.

• Arrows i1 and i2 are the axioms for a monoidal category.

• The source of the arrows i1 and i2 is the deﬁnition of FYZ [(AB)C]. • The triangle commutes because of the associative axiom.

• The second and ﬁfth square commute because of the general associative law. • The ﬁrst and third squares are obtained by repeated application of the functoriality

of the associative isomorphism, where each application gives a commutative square.

• The source of arrow i4 (and the target of i3) is the deﬁnition of F (AB)F (C).

• The source of arrow j4 (and the target of j3) is the deﬁnition of F (A)F (BC).

52 Thus we have a diagram

F (A(BC)) a / F (A)F (BC) b / F (A)(F (B)F (C)) (4–4) O c f F ((AB)C) d / F (AB)F (C) e / (F (A)F (B))F (C)

Since

a = j3 ◦ j2 ◦ j1 ◦ c,

b = j6 ◦ j5 ◦ j4, c = F (assoc.),

d = i3 ◦ i2 ◦ i1,

e = i6 ◦ i5 ◦ i4, f =associative law,

the above commutative diagram shows that b ◦ a ' f ◦ e ◦ d ◦ c.

Next we deﬁne an isomorphism FYZ ◦ FZY ' IdC:

(FYZ ◦ FZY )(A)

' FYZ (FZY (A))

' FYZ ((YA)Z) ' (Z((YA)Z))Y

' ((ZY )A(ZY )) ' (IAI) ' A where the fourth arrow above is just the associativity isomorphism. Similarly we deﬁne an

isomorphism FZY ◦ FYZ ' IdC. Thus FYZ gives an equivalence of group categories. Let Inn(C) denote the collection of equivalences of C which are isomorphic to ones

of the form FYZ . Deﬁne arrows between them to be morphisms of functors, and let the composition law be the usual vertical composition of natural transformations, which is again a morphism in Inn(C). Then Inn(C) is a groupoid. For G, H ∈ Ob(Inn(C)), if

53 G ' FYZ and H ' FY 0Z0 then we deﬁne an isomorphism between H ◦ G and some FIJ as follows:

(H ◦ G)(A)

= H(G(A))

' H(FYZ (A)) = H((ZA)Y )

' FY 0Z0 ((ZA)Y ) = (Z0((ZA)Y ))Y 0

' ((Z0Z)A(YY 0))

' F(YY 0)(Z0Z)(A) where the second last arrow is just the associative law. Thus the composition of functors

and horizontal composition of natural transformations deﬁnes a tensor functor:

⊗ : Inn(C) × Inn(C) → Inn(C).

∼ / We let I = IInn(C) and an inverse for an object G FYZ is obtained by taking the

quasi-inverse FZY . Thus Inn(C) is a categorical group. We call Inn(C) the group category of inner equivalences of C. Note that it is a full subcategory of Eq(C).

4.2 Action of a Group Category on a Category

If F : G × A → A is the action of a group G on a set A then the quotient by the p2 / action is just the coequalizer of the arrows G × A / A . We use this approach to deﬁne F a quotient when a group category acts on a category.

We will use notation similar to that of [22] but our deﬁnitions are more general.

Deﬁnition 4.2.1. Let G be a group category and C be a category. Let m denote multiplication in G and I its unit object. An action of G on C is a triple (F, α, β) where

54 F : G × C → C is a bifunctor that sits in the following two pasting diagrams:

m×Id G × G × C C / G × C (4–5)

IdG ×F α +3 F G × C / F C

F / G × C = C (4–6) O {{ {{ β {{ {{ I×IdC {{# {{ IdC {{ {{ {{ C where α : F ◦ (IdG × F ) ⇒ F ◦ (m × IdC) and β : F ◦ (I × IdC) ⇒ IdC are 2-isomorphisms. Further α and β must be compatible with associativity in G: that is, they sit in the pasting diagrams shown below.

Id×Id×F G × G × G × C / G × G × C (4–7) m×Id×Id Id×m×Id Id×F Id×α × × m×Id G × G × C Ô / G × C Id×F

asc F α v~ Id×F m×Id G × G × C / G × C

asc

m×Id F

α α Ô Ô G × C Õ xÐ / F C

55 G × I × C (4–8) ×C 77 ×× 77 ×× 77 ×× 77 ×× 77 Id×I×Id ×× 77Id×F ×× 77 ×× 77 × Id×β 7 ×× 77 ×× 77 × G × C I×Id / I × C

m6>×Id

γ F G? ×? C F ?? ?? α ?? ? I×Id × ?? Id F ?? ?? ?? β ? qy C Id / C Note that “asc” above refers to the associative law. The 2-arrow γ requires explanation. Note that the existence of the 2-isomorphisms α and β means that we have isomorphisms in C, natural in (g, h, x),

α : g · (h · x) ∼ / (gh) · x (4–9) and β : 1 · x ∼ / x . (4–10)

In particular there exists an isomorphism in C natural in g, 1, and a, obtained from the following composition

(g · 1) · a ∼ / g · (1 · a) ∼ / g · a ∼ / 1 · (g · a) ∼ / (1 · g) · a (4–11) α−1 β β−1 α which yields a 2-isomorphism γ that sits in the diagram

56 Id×I×Id G × C / G × I × C . (4–12)

F m×Id γ }Õ / G × C C I×Id Definition 4.2.2. Let G be a group category acting via the maps (F, α, β) on a category C. We define the quotient of C by G to be the category that represents the coequalizer of the p2 / diagram G × C / C in the 2-category CAT . Denote this quotient by C/G. F Definition 4.2.3. Let G and H be group categories. Let Eq(H, G) denote the group category of equivalences of group categories from H to G. Recall that Inn(G) is the group category of inner equivalences of G. Inn(G) acts from the right on Eq(H, G) by composition of morphisms: that is, given an equivalence u : H → G and an inner equivalence F , have F · u = u ◦ F . We define a group category

Out(H, G) := Eq(H, G)/Inn(G). (4–13)

57 CHAPTER 5 COEQUALIZERS AND QUOTIENTS IN ST ACKS

5.1 Action of a Group Stack on a Stack

Deﬁnition 5.1.1. Let G be a gr-stack on X and C be a stack on X. An action of G on C is a triple (F, α, β) where F : G × C → C is a morphism of stacks that sits in the following

two pasting diagrams in the 2-category StackX of stacks on X:

m×Id G × G × C C / G × C (5–1)

IdG ×F α +3 F G × C / F C

F / G × C = C (5–2) O {{ {{ β {{ {{ I×IdC {{# {{ IdC {{ {{ {{ C where m is the monoidal functor associated with the gr-stack G, I its identity object, and

α : F ◦ (IdG × F ) ⇒ F ◦ (m × IdC) and β : F ◦ (I × IdC) ⇒ IdC are 2-isomorphisms.

Id×Id×F G × G × G × C / G × G × C (5–3) m×Id×Id Id×m×Id Id×F Id×α × × m×Id G × G × C Ô / G × C Id×F

asc F α v~ Id×F m×Id G × G × C / G × C

asc

m×Id F

α α Ô Ô G × C Õ xÐ / F C

58 G × I × C (5–4) ×C 77 ×× 77 ×× 77 ×× 77 ×× 77 Id×I×Id ×× 77Id×F ×× 77 ×× 77 × Id×β 7 ×× 77 ×× 77 × G × C I×Id / I × C

m6>×Id

γ F G? ×? C F ?? ?? α ?? ? I×Id × ?? Id F ?? ?? ?? β ? qy C Id / C Further α and β must be compatible with associativity in G: that is, they sit in the pasting diagrams shown, where I refers to the identity object in G. The 2-arrow γ exists for the same reason as in the case for group categories.

5.2 Coequalizers in ST ACKS

Let ST ACKS denote the 2-category of stacks on X. F / Deﬁnition 5.2.1. 1. Given two morphisms of ﬁbered categories C / D , there is an G

induced morphism H : C → D × D where for each open U ⊂ X and x ∈ Ob(CU ),

x 7→ (FU (x),GU (x)). We say that F and G are a small pair of morphisms of ﬁbered

categories if the ﬁbers of HU are small categories i.e. for all (a, b) ∈ Ob(DU × DU ),

−1 HU (a, b) is a set. 2. If C and D are prestacks (resp. stacks), and F and G are morphisms of prestacks (resp. stacks), we say that F and G are a small pair of morphisms of prestacks (resp. stacks) if they form a small pair of morphisms of the underlying ﬁbered categories.

59 Our goal is to prove that the coequalizer of a small pair of morphisms of stacks is representable in ST ACKS. We will begin with the case of FIBER – the 2-category of ﬁbered categories.

Let C and D be ﬁbered categories, and let F and G denote a small pair of morphisms F / from C to D i.e. we have a diagram C / D . G Let Coeq(F,G) denote the following ﬁbered category. To each open U ⊂ X,

Coeq(F,G)(U) := Coeq(FU ,GU ). Let V,→ U be an inclusion of open sets in the space X. To deﬁne the restriction functors f ∗ for Coeq(F,G) consider the following diagram:

FU / ϕU C(U) / D(U) / Coeq(F,G)(U) (5–5) GU ∗ ∗ ∗ c d f FV / ϕV C(V ) / D(V ) / Coeq(F,G)(V ) GV

∗ ∗ where c and d denote the restriction functors of the stacks C and D respectively, ϕU and ϕV are the canonical maps into the coequalizer. Also, as part of the data, we have

∗ ∗ ∗ ∗ ∗ 2-arrows α : FV ◦ c ⇒ d ◦ FU and β : GV ◦ c ⇒ d ◦ GU . The dotted arrow f is the map we need to construct. Now, from the data of a coequalizer, there is an isomorphism

∼ / ϕV ◦ FV ϕV ◦ GV . (5–6)

∗ ∼ / ∗ ϕV ◦ FV ◦ c ϕV ◦ GV ◦ c . (5–7)

But horizontal composition with α yields

∗ ∼ / ∗ ϕV ◦ d ◦ FU ϕV ◦ GV ◦ c (5–8) and horizontal composition with β yields

∗ ∼ / ∗ ϕV ◦ d ◦ FU ϕV ◦ d ◦ GU . (5–9)

60 Therefore there exists an isomorphism

∗ ∼ / ∗ ϕV ◦ d ◦ FU ϕV ◦ d ◦ GU . (5–10)

Then by the universal property of coequalizers, there exists an arrow

f ∗ : Coeq(F,G)(U) → Coeq(F,G)(V ). (5–11)

That f ∗ is unique up to unique isomorphism follows from the following result of Hakim in

[16]. Proposition 5.2.2. Let C be a 2-category and let F(C) denote the 2-category of 2- functors from Cop to CAT . Let F be a 2-functor which is representable in F(C), and suppose (X, ξ) and (X0, ξ0) are two given representations of F . Then there exists a

∼ ∼ pair (λ, ²) such that λ : X / X0 is an equivalence in C and ² : F (λ)(ξ0) / ξ0

is a 2-isomorphism. Further, for every other such pair (λ1, ²1) there exists a unique

∼ / 0 0 2-isomorphism α : λ λ such that ²1 ◦ (F (λ)(ξ )) = ². Thus the natural transformation corresponding to every pair of composable inclusions of open sets is the unique 2-arrow which exists from the above result, and thus ensures

that for each triple of composable inclusions of open sets the corresponding composite natural transformations coincide.

ϕ By construction of Coeq(F,G) we have an arrow D / Coeq(F,G) which, over each

open set U, is the identity on objects and sends arrows in DU to their corresponding class in Coeq(F,G)(U). Theorem 5.2.3. The coequalizer of a small pair of morphisms of ﬁbered categories is representable in FIBER. In fact, the diagram

F / ϕ C / D / Coeq(F,G) (5–12) G

61 gives an equivalence

∗ ∼ F / Hom(Coeq(F,G), E) / Eq(Hom(D, E) / Hom(C, E)) . (5–13) G∗

F ∗ / Proof. Deﬁne λ : Hom(Coeq(F,G), E) / Eq(Hom(D, E) / Hom(C, E)) as follows: G∗ if J : Coeq(F,G) → E is an arrow, let λ(J) = J ◦ ϕ. By Theorem 3.14, for each open set U ⊂ X this yields an equivalence

∗ ∼ F / Hom(Coeq(F,G)(U), E(U)) / Eq(Hom(D(U), E(U)) / Hom(C(U), E(U))) (5–14) G∗

of categories. Since J : Coeq(F,G) → E is a map of ﬁbered categories so it is compatible with the restriction functors of Coeq(F,G) and E. Similarly λ(J): D → E is a map of ﬁbered categories so it is compatible with the restriction functors of D and E. Thus

∗ λ F / Hom(Coeq(F,G), E) / Eq(Hom(D, E) / Hom(C, E)) (5–15) G∗

is an equivalence.

F / Now let C and D be prestacks, and let C / D be a small pair of morphisms G between them. We wish to show that a coequalizer of such a small pair is representable in P REST ACKS.

Let F ib(F,G) be the ﬁbered category coequalizer of F and G. Construct a prestack H as follows: for each open set U ⊂ X and x, y in Ob(F ib(F,G)(U)) let Ob(H(U)) =

Ob(F ib(F,G)(U)), and HomH(U)(x, y) be the sheafification of the presheaf HomF ib(F,G)(U)(x, y). Proposition 5.2.4. The fibered category H constructed above is a prestack.

Proof. We need to specify the restriction functors and check that they have the required compatibility on triple inclusions of open sets. So let V ⊂ U be an inclusion of open sets

62 in X. Consider the diagram

τU / HomF ib(F,G)(U)(x, y) HomH(U)(x, y) (5–16)

f ∗ τV / HomF ib(F,G)(V )(x, y) HomH(V )(x, y)

∗ where τU , τV denote the sheaﬁﬁcation maps and f is the restriction functor from

∗ F ib(F,G)(U) → F ib(F,G)(V ). Then τV ◦f is an arrow from the presheaf HomF ib(F,G)(U)(x, y) to the sheaf HomH(V )(x, y). So by the universal property of sheaﬁﬁcation there exists a

∗ unique arrow h : HomH(U)(x, y) → HomH(V )(x, y) such that the diagram

τU / HomF ib(F,G)(U)(x, y) HomH(U)(x, y) (5–17)

f ∗ h∗ τV / HomF ib(F,G)(V )(x, y) HomH(V )(x, y)

commutes. The uniqueness of h∗ gives the required compatibility for triple inclusions of open sets.

Let τ : F ib(F,G) → H denote the morphism of ﬁbered categories that, over each open set U, sends the objects of F ib(F,G)(U) to the objects of H(U), and each presheaf

HomF ib(F,G)(U)(x, y) to its associated sheaf. So there is a diagram:

F / ϕ τ C / D / F ib(F,G) / H . (5–18) G

Let ω : D → H be the arrow τ ◦ ϕ. Theorem 5.2.5. The coequalizer of a small pair of morphisms of prestacks is representable in P REST ACKS. In fact, the diagram

F / ω C / D / H (5–19) G

gives an equivalence

∗ ∼ F / Hom(H, J ) / Eq(Hom(D, J ) / Hom(C, J )) . (5–20) G∗

63 F ∗ / Proof. Deﬁne λ : Hom(H, J ) / Eq(Hom(D, J ) / Hom(C, J )) as follows: if G∗ j : H → J is an arrow, then let λ(j) := j ◦ ω. We ﬁrst show λ is essentially surjective.

Let γ : D → J be a given morphism of prestacks together with an isomorphism

γ ◦ F ∼ / γ ◦ G . We need to construct an arrow η : H → J , such that λ(η) ∼ / γ , i.e. such that the following diagram commutes:

F / ϕ / τ / C / D F ib(F,G) j j H . (5–21) G j j γ j j j j η j j J jt

Now, by the universal property of coequalizers in FIBER there exists an arrow δ :

F ib(F,G) → J , unique up to unique isomorphism, such that the diagram

F / ϕ τ C / D / F ib(F,G) / H (5–22) G uu uu γ uu uu uz u δ J

commutes. For each open set U ⊂ X there is a diagram:

τU / HomF ib(F,G)(U)(x, y) HomH(U)(x, y) (5–23)

δU HomJ (U)(x, y)

where the vertical arrow δU is a map from a presheaf to a sheaf. Then by the universal property of sheaﬁﬁcation, there exists a unique arrow ηU : HomH(U)(x, y) → HomJ (U)(x, y) such that the diagram:

τU / HomF ib(F,G)(U)(x, y) HomH(U)(x, y) (5–24) k kkkk δ kkk U kkkη ku kkk U HomJ (U)(x, y)

64 commutes. Thus we obtain a morphism η : H → J such that the diagram

F / ϕ / τ / C / D F ib(F,G) j j H (5–25) G uu j j uu j γ uu δj j uu j η uz uj j J jt commutes and λ(η) ∼ / γ . So λ is essentially surjective.

To show that λ is fully faithful, we need to establish a bijection between the sets HomHom(H,J )(a, b) and HomB(λ(a), λ(b)), where B is the equalizer category F ∗ / Eq(Hom(D, J ) / Hom(C, J )) . But the elements of both these sets are morphisms G∗ of prestacks. Recall that a morphism of prestacks is just a morphism of the underlying ﬁbered categories. A morphism of prestacks consists of a family of natural transformations

(each corresponding to each open subset U of X) which are compatible with the restriction functors of the source and target ﬁbered categories. Further, natural transformations are determined solely by their action on the objects of a category. Since the objects of F ib(F,G)(U) and H(U) are the same over each open set U, we have that

∼ / HomHom(H,J )(a, b) HomHom(F ib(F,G)),J )(a, b) . (5–26)

Since the map

F ∗ / / ϕ : HomHom(F ib(F,G)),J )(a, b) Eq(Hom(D, J ) / Hom(C, J )) (5–27) G∗ is fully faithful by the previous theorem, it follows that λ is fully faithful as well.

F / We now come to the case of stacks. Now let C and D be stacks, and let C / D be G a small pair of morphisms between them. Form the prestack coequalizer F / ϕ C / D / P re(F,G) of F and G. Let St(F,G) be the associated stack and G α : P re(F,G) → St(F,G) be the canonical cartesian functor. Thus there is a diagram:

F / α◦ϕ C / D / St(F,G) . (5–28) G

65 Let ω : D → St(F,G) be the arrow α ◦ ϕ. Theorem 5.2.6. The coequalizer of a small pair of morphisms of stacks is representable in ST ACKS. In fact, the diagram

F / ω C / D / St(F,G) (5–29) G

gives an equivalence

∗ ∼ F / Hom(St(F,G), J ) / Eq(Hom(D, J ) / Hom(C, J )) . (5–30) G∗

F ∗ / Proof. Deﬁne λ : Hom(St(F,G), J ) / Eq(Hom(D, J ) / Hom(C, J )) as follows: if G∗ j : St(F,G) → J is an arrow, then let λ(j) := j ◦ ω. We show λ is essentially surjective. Let J be any stack and β : D → J be a given morphism of stacks together with an

isomorphism β ◦ F ∼ / β ◦ G . Consider the diagram:

F / ϕ α C / D / P re(F,G) / St(F,G) . (5–31) G β J

By the universal property of coequalizers in P REST ACKS there exists an arrow γ : P re(F,G) → J such that the diagram

F / ϕ α C / D / P re(F,G) / St(F,G) (5–32) G tt tt β tt tt γ ztt J

commutes. Since the associated stack cartesian functor is fully faithful and universal for cartesian functors from P re(F,G) into ST ACKS ([18] Lemma 3.2), there is a morphism of stacks η : St(F,G) → J , which is unique up to unique isomorphism such that the diagram

F / ϕ α / / P re(F,G) / St(F,G) (5–33) C D h G tt h h tt h β tt h h tt γ h h η ztth h J th

66 commutes. Thus λ(η) ∼ / γ whence λ is essentially surjective.

We now show λ is faithful. Let ϕ1, ϕ2 : St(F,G) → J denote morphisms of stacks.

Then λ(ϕ1), λ(ϕ2): D → J . Suppose δ, β are in HomHom(St(F,G),J )(ϕ1, ϕ2) with

λ(δ) = λ(β), where λ(δ) and λ(β) are in HomEq(λ(δ), λ(β)) (here Eq denotes the category F ∗ / Eq(Hom(D, J ) / Hom(C, J )) . We need to show that δ = β. Now λ(δ) = λ(β) means G∗ that for each open set U and each object a in D(U), the arrow

λ(δ)U / λ(ϕ1(a)) λ(ϕ2)(a) (5–34)

is the same arrow as

λ(β)U / λ(ϕ1(a)) λ(ϕ2)(a) . (5–35)

But the deﬁnition of λ is λ(−) = − ◦ ω. So the arrow

λ(δ)U / (ϕ1 ◦ ω)(a) (ϕ2 ◦ ω)(a) (5–36)

is the same as the arrow

λ(β)U / (ϕ1 ◦ ω)(a) (ϕ2 ◦ ω)(a)

i.e. the arrow

δU / ϕ1(ω(a)) ϕ2(ω(a)) (5–37)

is the same as the arrow

βU / ϕ1(ω(a)) ϕ2(ω(a)) . (5–38)

Thus δU and βU agree over objects in St(F,G)(U) which are in the image of ω. Now, recall that ω = α ◦ ϕ where ϕ : D → P re(F,G) is the prestack coequalizer, and α : P re(F,G) → St(F,G) is the canonical stackiﬁcation functor. The maps ϕ and α both have the property that every object in their target is locally contained in their essential image, and so this property is carried by their composition ω. In other words, every object

in St(F,G)(U) is locally contained in the essential image of ω. So let b be an object in

∼ / St(F,G)(U) and suppose (Ui) is an open cover of U where ω(ai) b|Ui for some object ti

67 ai in D(Ui). (There is such an isomorphism for each i). Then over each Ui there exist commutative diagrams:

λ(δ) / ϕ1(ω(ai)) ϕ2(ω(ai)) (5–39)

ϕ1(ti) ϕ2(ti) ϕ (b) / ϕ (b) 1 δ 2 and

λ(β) / ϕ1(ω(ai)) ϕ2(ω(ai)) . (5–40)

ϕ1(t) ϕ2(t) ϕ (b) / ϕ (b) 1 β 2 By hypothesis, the two vertical and the top horizontal arrows are the same for both diagrams whence the lower horizontal arrows are the same in both diagrams i.e. δ and β

agree over each Ui. Since Ar(St(F,G)(U)) is a sheaf, δ = β over U. Since this is true for each open set U, get that δ = β. Thus λ is faithful.

Finally, we show that λ is full. Again, let ϕ1, ϕ2 : St(F,G) → J denote morphisms

of stacks. Then λ(ϕ1), λ(ϕ2): D → J . Suppose δ is an arrow in HomEq(λ(δ), λ(β)).

We need to show that there exists an arrow δ in HomHom(St(F,G),J )(ϕ1, ϕ2) such that λ(δ) = δ. By deﬁnition, δ consists of, for each open set U, a natural transformation

δU : λ(ϕ1)U → λ(ϕ2)U . Then for an object a in D(U),

δU : λ(ϕ1)U (a) → λ(ϕ2)U (a) (5–41)

δU / (ϕ1 ◦ ω)(a) (ϕ2 ◦ ω)(a) (5–42)

over U, so

δU / ϕ1(ω(a)) ϕ2(ω(a)) (5–43)

where ω(a) is an object in St(F,G)(U). Let b be any object in St(F,G)(U). Since every

object in St(F,G)(U) is locally contained in the essential image of ω (see explanation in

paragraph above), b is locally in the essential image of ω i.e. there exists an open cover

68 ∼ / (Ui) of U and objects bi in D(Ui) with arrows ω(bi)|Ui b|U . Then over each Ui we ti i have arrows δ Ui / ϕ1(ω(bi)) ϕ2(ω(bi)) . (5–44)

Further, on overlaps Uij,

ω(b )| ∼ / ∼ / ∼ / ω(b )| i Uij (b|Ui )|Uij (b|Uj )|Uij −1 j Uij . (5–45) ti tj

Consider the diagram δ Ui / ϕ1(ω(bi)) ϕ2(ω(bi)) (5–46)

ϕ1(ti) ϕ2(ti) ϕ1(b) ϕ2(b)

Let δUi : ϕ1(b) → ϕ2(b) be the unique arrow that makes the above diagram into a

commutative square. Since Ar(St(F,G)(U)) is a sheaf, the collection of arrows {δUi :

ϕ1(b) → ϕ2(b)} glue over the various Ui to give an arrow over U:

δ : ϕ1(b) → ϕ2(b) (5–47)

such that λ(δ) = δ. Thus λ is full. Deﬁnition 5.2.7. Let G be a group category acting on a category C via a map F . Con- sider the induced functor H : G × C → C × C which on objects is given by (g, c) 7→ (g · c, c).

We say that F is a small action if the action F : G × C → C and the projection p2 : G × C → C are a small pair i.e. for any pair of objects (c1, c2) in C × C, the subcategory of g in Ob(G) such that g · c1 = c2 is a small category. Proposition 5.2.8. Let G be a group category acting on a groupoid C. Then the action of

G on C is small if and only if the stabilizer StabG(c) is a set for each object c of C.

Proof. Let c1, c2 be any two objects of C and g, h be two objects of G such that g · c1 = c2 ∼ / and h · c1 = c2. Then if k is any object of G such that hk 1 then kg is in StabG(c1).

69 Deﬁnition 5.2.9. Let G be a gr-stack acting on a stack C via a morphism F . We say that F is a small action if F is a locally small action i.e. for any open subset U of X the action of G on C over U is small.

Proposition 5.2.10. Let G be a gr-stack acting on a stack C. Then the action of G on C is small if and only if for each open set U ⊂ X, and each object c of C(U), the stabilizer

StabG(U)(c) is a set. Proof. Since C is a stack, for any open set U ⊂ X, C(U) is a groupoid. The result now follows from the previous proposition.

Deﬁnition 5.2.11. Let G be a gr-stack acting via the maps (F, α, β) on a stack C. We deﬁne a quotient of C by G to be a coequalizer (C/G, α : C → C/G) of the pair p2 / H × G / G when it is representable in the 2-category ST ACKS. Note that by F the preceding theorem if the action is small then this coequalizer is always representable.

70 CHAPTER 6 THE 2-LIEN OF A 2-GERBE

6.1 Deﬁnition of a 2-Lien

We now deﬁne the notion of a 2-lien on a space X.

Deﬁnition 6.1.1. An inner equivalence ϕ of a gr-stack G is an equivalence of G which is locally isomorphic to an inner equivalence of categorical groups. In other words ϕ is an inner equivalence of G if for every point x ∈ X there exists a neighborhood U about

x such that the restriction of ϕ to U is an inner equivalence of categorical groups. This

is equivalent to saying that there exists an open cover (Ui) of the space X such that

ϕ|Ui : G(Ui) → G(Ui) is an inner equivalence of categorical groups. Thus ϕ is a cartesian

functor such that each ϕ|Ui is an object of the group category Inn(G(Ui)). Let X be a topological space, and let F and G be gr-stacks on X. Let Eq(F, G) denote the ﬁbered category of equivalences of gr-stacks from F to G. Thus for each open

U ⊂ X, Eq(F, G)(U) is deﬁned to be Eq(F|U , G|U ) i.e. the category of equivalences from the restricted gr-stack F|U to the restricted gr-stack G|U . By Corollary II.2.1.5 of [15] this is a stack on X. Let Inn(G) denote the ﬁbered category of inner equivalences of the stack G. Thus

for each open U ⊂ X, Inn(G)(U) is the group category Inn(G|U ) i.e. the group

category of inner equivalences of the restricted gr-stack G|U . The restriction functors and corresponding natural transformations are the same as the ones defined in the stack Eq(G, G). Recall that an equivalence is inner if and only if it is so locally. Thus since the conditions defining Inn(G) as a sub-fibered category of Eq(G, G) are local, Inn(G) is a stack on X. Let Out(F, G) denote a quotient stack Eq(F, G)/Inn(G).

We now deﬁne a ﬁbered 2-category LI2(X) as follows. For every open U ⊂ X,

let LI2(X)(U) denote the 2-category whose objects are the gr-stacks on U, and whose arrows between two objects A and B are the global sections of the stack Out(A, B)(U) i.e.

HomLI2(X)(U)(A, B) = Γ(U, Out(A, B)). The composition law is the same as the one in

71 the quotient stack Out. If V,→ U is an inclusion of open sets in X, then the formation of the quotient Out(A, B) commutes with the restriction of U to V (see the discussion after Deﬁnition 5.2).

Let Gr(X) denote the ﬁbered 2-category of gr-stacks on X. Then by construction we have a morphism of ﬁbered 2-categories

Gr(X) → LI2(X). (6–1)

Now if A and B are any two objects of the 2-category LI2(X)(U), then the prestack

HomLI2(X)(U)(A, B) of arrows from A to B is identiﬁed with the stack Out(A, B). Thus

LI2(X) is a pre-2-stack. Thus upon applying the 2-stackiﬁcation functor, we obtain

the associated 2-stack LIEN2(X). Since Gr(X) is already a 2-stack, composing the 2-stackiﬁcation functor with the morphism (3) yields a morphism of 2-stacks

lien2(X): Gr(X) → LIEN2(X). (6–2)

Deﬁnition 6.1.2. 1. The 2-stack LIEN2(X) is called the 2-stack of 2-liens on X.

2. We call a global section of the 2-stack LIEN2(X) a 2-lien over X.

0 Two 2-liens L and L on X, each deﬁned locally by families of gr-stacks (Gα) and

0 (Gβ), are equivalent whenever there exists a common reﬁnement V = (Vi) of the deﬁning

0 0 covers U and U , and a family of equivalences χi : lien(Gi) → lien(Gi) on the open sets Vi, which are compatible with the gluing data. To generalize this deﬁnition of a 2-lien from topological spaces to topoi, we just replace “global section” in the above deﬁnition with “a

2-cartesian section” of the 2-stack LIEN2(X), as done by Giraud for 1-gerbes in [15]. 6.2 The 2-Lien of a 2-Gerbe

Let G be a 2-stack on X. By [4], over any open set U ⊂ X, any 1-arrow f in G(U) has a quasi-inverse, deﬁned up to a unique 2-arrow. Further, Breen shows that local inverses for 1-arrows always descend to global ones, so for any object x ∈ G(U), the prestack in groupoids Eq(x) of self-equivalences of x is a gr-stack on X, after speciﬁc inverses for the

72 1-arrows have been chosen. In the following we will assume that this has been done, and denote by α−1 the chosen inverse of a 1-arrow α in G. Recall that Gr(X) denotes the 2-stack of gr-stacks on X. Then there is a Cartesian 2-functor

EQV : G → Gr(X) (6–3)

EQV (G)(x) = EqU (x) (6–4)

where x ∈ Ob(G(U)) and EqU (x) denotes the gr-stack of self-equivalences of x. Composing this morphism with the morphism

lien2(X): Gr(X) → LIEN2(X) (6–5)

gives a morphism of 2-stacks

liau2(G): G → LIEN2(X) (6–6)

which to every object x ∈ G(U) associates the 2-lien

liau2(G)(x) = lien2(EqU (x)) (6–7)

(called the 2-lien represented by the stack of equivalences of x over U) and to every equivalence α : x → y of G(U) associates the morphism

liau2(G)(α) = lien2(Inn(α)) (6–8)

in LIEN2(X)(U) where

Inn(α): EqU (x) → EqU (y) (6–9)

deﬁned by

Inn(α)(a) = αaα−1 (6–10)

is a morphism of gr-stacks over X, where

α ◦ α−1 ' Id ' α−1 ◦ α. (6–11)

73 We check that composition of morphisms is well deﬁned. Let α an β denote a pair of composable 1-arrows in G(U), and let γ denote the chosen inverse of β ◦ α in G(U). Then Inn(β ◦ α)(a) = (β ◦ α)aγ. Now (β ◦ α) ◦ (α−1 ◦ β−1) ' Id so there exists a unique invertible

2-arrow η : α−1 ◦ β−1 ⇒ γ. Thus, Inn(β) ◦ Inn(α)(a) = Inn(β)(αaα−1)

= β(αaα−1)β−1 = (β ◦ α)a(α−1 ◦ β−1) →˜ (β ◦ α)aγ

= Inn(β ◦ α)(a). We also need to check that diﬀerent choices of inverses for α : x → y where x, y ∈ G(U), give us canonically equivalent categories Inn(α). So suppose β, β0 : y → x are two inverses for α. By [4], page 62, there exists a unique 2-arrow η : β ⇒ β0. So for any arrow g : y → y, there is a unique commutative square

η β(y) / β0(y) . (6–12)

β(g) β0(g)

η β(y) / β0(y)

Then left composition by an equivalence a ∈ Eq(x) of x yields another unique commutative square I η aβ(y) a / aβ0(y) . (6–13)

aβ(g) aβ0(g)

I η aβ(y) a / aβ0(y)

74 Then left composition by the 1-arrow α : x → y yields unique commutative squares

I I η αaβ(y) α a / aβ0(y) . (6–14)

αaβ(g) αaβ0(g)

I I η αaβ(y) α a / αaβ0(y)

for every g : y → y. Thus ν := IαIaη deﬁnes a unique 2-isomorphism between αaβ and αaβ0, that is, we get a canonical equivalence of categories Inn(α) with choice of inverse β and Inn(α) with choice of inverse of inverse β0. By the deﬁnition of the 2-stack of 2-liens, if m, n : x → y are two equivalences of G(U), we have

∼ / liau2(G)(m) liau2(G)(n) (6–15)

since the 2-functor lien2 gives a natural isomorphism between every inner equivalence and the identity arrow.

If m : F → G is a morphism of 2-stacks, the composition

m / / F G liau2(G) (6–16)

factors through a morphism of morphisms of 2-stacks:

liau2(m): liau2(F) → liau2(G). (6–17)

For every x ∈ Ob(F(U)), the 2-functor m induces a morphism of gr-stacks

µx : EqU (x) → EqU (m(x)) (6–18)

and by deﬁnition of the map liau2(m), the morphism liau(m(x)) is the morphism of

2-liens represented by µx:

mx = lien2(µx), mx : lien2(EqU (x)) → lien2(EqU (m(x))). (6–19)

75 Deﬁnition 6.2.1. Let F be a 2-stack over X and let L be a 2-lien over X: that is, a global section L : X → LIEN(X) of the 2-stack of 2-liens. Let f : F → X denote the projection map onto the Zariski site of X. An action of L on F consists of a 2-arrow

a : L ◦ f ⇒ liau(F) (6–20)

that is,

X J (6–21) ? JJ f JJL JJ a JJ J% F ÑÙ / LIEN(X) liau(F)

and a 2-arrow af . The 2-arrow a is a morphism of cartesian 2-functors: that is, a family of arrows

a(x): L(U) → lien2(EqU (x)) (6–22)

where x is in Ob(F(U)), and U ⊂ X. For each inclusion f : V,→ U of open sets, the

2-arrow af sits in a diagram:

a(x) / L(U) lien(EqU (x)) . (6–23)

af ~Ö / L(V ) lien(EqV (y)) a(y)

In addition af satisﬁes the following conditions: g f 1. for a double inclusion W / V / U of open sets, there is a 3-arrow (i.e. a modiﬁcation)

αf,g : ag ◦ af _*4 ag◦f . (6–24)

h g f 2. for a triple inclusion Z / W / V / U of open sets, the two methods by which the modiﬁcations α compare the composite 2-arrows

ah◦g◦f ⇒ ah ◦ ag◦f ⇒ ah ◦ (ag ◦ af ) (6–25)

76 and

ah◦g◦f ⇒ ah◦g ◦ af ⇒ (ah ◦ ag) ◦ af (6–26)

are identical. Deﬁnition 6.2.2. Let (L, a) be a 2-lien operating on a 2-stack F and let u : L0 → L be a morphism of 2-liens over X. Then the action induced by a and u is the morphism b deﬁned by

0 b : L · f → liau2(F ), (6–27)

b = a · (u ◦ f ◦ L)

(where f : F → X is the projection onto the Zariski site of X). So b is the action such that, for every x ∈ Ob(F(U)), b(x) is the composition

u(U) a(x) L0(U) / L(U) / liau(F)(x) . (6–28)

Proposition 6.2.3. Let F be a 2-gerbe on X. 1. Let (L, a) be a 2-lien operating on F. The following are equivalent: (a) a is a 2-equivalence.

(b) for every 2-lien L0 on X and every action b of L0 on F there exists a unique

morphism of 2-liens u : L0 → L (up to 2-equivalence) such that b is equivalent to the action induced by a and u. 2. There exists a 2-lien (L, a) operating on F that satisﬁes the conditions of (1). The proof of this proposition is given below, following the proof of Proposition 6.2.5. Deﬁnition 6.2.4. A 2-lien operating on a 2-gerbe F and satisfying the conditions of the above proposition will be called a 2-lien associated to F

Condition 1(b) characterizes the 2-lien of a 2-gerbe F up to a canonical 2-equivalence. Thus we are justiﬁed in saying “the” 2-lien of a 2-gerbe F. By abuse of notation, the map a is often not mentioned. If L is the 2-lien of a 2-gerbe F, we say F is bound by L.

77 Conversely, if L is a 2-lien on X, then an L-2-gerbe is a pair (F, a) where F is a 2-gerbe and (L, a) is a 2-lien of F.

More explicitly, the 2-lien of a 2-gerbe F is a 2-lien L and a family of equivalences of

2-liens over U,

∼ / a(x): L(U) lien2(EqU (x)) (6–29)

where x ∈ F(U), U ⊂ X. This family is required to be compatible with the restriction of open sets and also satisfy the condition that if i : x → y is a U-equivalence of F , then the

morphism of gr-stacks Inn(i): EqU (x) → EqU (y) is isomorphic to the identity morphism of L(U).

Proposition 6.2.5. For every 2-gerbe F, there exists a 2-lien L and an equivalence

∼ / a : Lf liau2(F) .

Proof. Let I be the image category of liau2(F). Clearly this is a ﬁbered 2-category of

LIEN2(X) and the 2-functor induced by liau2(F),

L0 : F → I (6–30)

is 2-cartesian. Moreover, the projection I → Xzar is both fully 2-faithful and essentially surjective (this follows trivially from the fact that F → X is fully 2-faithful and because

∼ / liau2(m) liau2(n) for any two equivalences m, n : x → y of F over an open set U. By the universal property of the associated 2-stack, it follows that X is the 2-stack

associated to I, and that there exists a global section L0 : X → I of I and an equivalence

∼ / 0 a0 : L0f L .

Proof of Proposition 6.2.3. Since F is a 2-gerbe, the projection f : F → Xzar is fully 2-faithful. Hence, if (L, a) and (L0, b) are two 2-liens acting on F, the map Hom(L0,L) → Hom(L0f, Lf) given by u 7→ u ◦ f is an equivalence of group categories. This entails

that b ∼ / a(u ◦ f) whence we have that 1(a) implies 1(b). By the above proposition (2) follows trivially, and so does 1(b) ⇒ 1(a) since 1(b) determines L up to unique

78 equivalence. Then setting L = iL0 and a = i ◦ a0 (where i : I → LIEN2(X) is the inclusion) the conclusion follows.

Corollary 6.2.6. Let m : F → G be a morphism of 2-gerbes and let (L, a) and

(M, b) denote the 2-liens of F and G respectively. Then there exists a unique morphism of 2-liens u : L → M (up to 2-equivalence) such that m is a u-morphism .i.e.

∼ / liau2(m) · a (b ◦ m)(u ◦ f) . Following the convention for 1-gerbes, if a 2-lien is deﬁned by a family of gr-stacks

(Gi), we will call this family lien2(Gi). 6.3 Cocycle Description of the 2-Lien of a 2-Gerbe

Let G be a 2-gerbe on X and U an open subset of X. For an object a in G(U), the prestack in groupoids Eq(a) of self equivalences of a is endowed with a (strictly associative) monoidal structure determined by a composition of 1-arrows. Since G is a 2-gerbe, the 1-arrows are weakly invertible so this monoidal structure is group-like. In fact, local inverses for 1-arrows always descend to global ones (see [4], pg. 64-65) and so Eq(a) is a gr-stack on U once speciﬁc inverses for the 1-arrows have been chosen. In the following we will again assume that this has been done, and denote by α−1 the chosen

inverse of a 1-arrow α in G (see pages 72-73).

Thus given a 2-gerbe G on X ﬁx inverses for all the 1-arrows of G. Choose an open

cover X = (Uα), and for each α choose an object aα in G(Uα). For each α and β, choose

ξ ξ an open cover Uαβ = (Uαβ) and for each ξ a 1-arrow fαβ : aβ → aα in G(Uαβ). Then we can deﬁne an equivalence of gr-stacks:

ξ λαβ : Eq(aβ)| ξ → Eq(aα)| ξ (6–31) Uαβ Uαβ

ξ −1 ξ τ 7→ (fαβ) ◦ τ ◦ fαβ.

ξ ξ ξ1 This equivalence λαβ depends on the choice of fαβ. But consider two diﬀerent choices λαβ

ξ2 ξ1 ξ2 ξ1 ξ1 −1 ξ1 and λαβ corresponding to fαβ,fαβ : aβ → aα in G(Uαβ) i.e. suppose λαβ := (fαβ) ◦ τ ◦ fαβ

79 ξ2 ξ2 −1 ξ2 and λαβ := (fαβ) ◦ τ ◦ fαβ. Then there exists a 2-arrow:

ξ2 ξ2 −1 ξ1 ξ1 −1 ξ1 ξ1 −1 ξ2 λαβ ⇒ (fαβ) ◦ fαβ ◦ [(fαβ) ◦ τ ◦ fαβ] ◦ (fαβ) ◦ fαβ (6–32)

i.e.

ξ2 ξ2 −1 ξ1 ξ1 ξ2 −1 ξ1 −1 λαβ ⇒ [(fαβ) ◦ fαβ] ◦ λαβ ◦ [(fαβ) ◦ fαβ] (6–33)

where there exists an invertible 2-arrow

ξ2 −1 ξ1 ξ2 −1 ξ1 −1 [(fαβ) ◦ fαβ] ◦ [(fαβ) ◦ fαβ] ⇒ Id. (6–34)

ξ1 ξ2 Thus λαβ and λαβ diﬀer by an inner equivalence. In particular they deﬁne the same

ξ ξ ξ section of Out(Eq(aβ), Eq(aα)). So for ﬁxed α and β, the family {λαβ} deﬁnes an “outer”

equivalence of gr-stacks on Uαβ,

λαβ : Eq(aβ)|Uαβ → Eq(aα)|Uαβ (6–35)

ξ which does not depend on the choice of the fαβ. This system of gr-stacks Eq(aα) and outer equivalences λαβ will be called a cocycle representation of the 2-lien of the gerbe G.

80 CHAPTER 7 2-LIENS, PICARD STACKS AND H3

7.1 2-Gerbes and 3-Cocycles

The construction of a 3-cocycle associated to a given 2-gerbe P is lengthy and complicated. We will give a very brief outline of this construction, before specializing to the situation of strict Picard stacks. We follow the treatment in [4] and [5]. The reader may consult these sources for the details.

Let P be a G-2-gerbe on a given space X. We will assume that P is connected: that

is, for each pair of objects x, y in some ﬁber 2-category PU , the set of arrows in PU from x

to y is non-empty. Given an open cover (Ui) of X, we choose objects xi ∈ PUi and paths

φij : xj → xi in the 2-groupoid PUij , together with quasi-inverses ψij : xi → xj. To this we associate the following data:

1. An object λij in Eq(Gj|Uij , Gi|Uij ):

λij : Gj|Uij → Gi|Uij (7–1)

g 7→ φij ◦ g ◦ ψij, (7–2)

determined by φij and its inverse.

2. An arrow gijk (determined by the paths φij and their inverses) viewed as an object in

the ﬁber of the stack Gi over the open set Uijk:

φjk xk / xj (7–3) m φik ijk ,4 φij xi / xi gijk

3. An arrowm ˜ ijk (a natural transformation induced by the 2-arrow mijk in the diagram

above) in the category Eq(Gj, Gi)Uijk :

81 λjk / Gk Gj (7–4)

λik m˜ ijk -5 λij / Gi Gi igijk

where ig is the inner conjugation functor deﬁned by an object g in the gr-stack G,

and for γ ∈ Gk,

m˜ ijk(γ): igijk ◦ λij(γ) → λij ◦ λjk(γ) (7–5)

in Gi. By right multiplication by the inverse object of its source, this arrow corresponds to an arrow:

−1 {m˜ ijk, γ} : 1xi → λij ◦ λjk(γ) ◦ (igijk ◦ λik(γ)) (7–6)

in Gi sourced at the identity.

−1 −1 4. An arrow νijkl : 1xi ⇒ gijk ◦gikl ◦(gijl) ◦λij(gjkl) in the category Gi|Uijkl determined by the data listed above. It can also be viewed as a 2-arrow that sits in the diagram below.

gijl xi / xi (7–7)

gikl λij (gjkl) νijkl xi ÑÙ / xi gijk

Further, the following identity is valid for νijkl:

gijk λij (gikl) m˜ ijk( m˜ ikl)iνijkl = (λij(m ˜ jkl))( m˜ ijl) (7–8)

g where m denotes the left composition of m by g. In addition, νijkl|Uijklm satisﬁes the

following cocycle condition in Gi|Uijklm :

λij (gjkl) λij ◦λjk(gklm) gijk λij(νjklm)( νijlm)νijkl = ( νijkm){m˜ ijk, gklm}( νiklm) (7–9)

where gν denotes conjugation of an arrow ν by g.

82 By Breen ([4], [5]), such a quadruple of elements (νijkl, m˜ ijk, gijk, λij) is called a G-valued nonabelian 3-cocycle on the space X.

We now discuss the coboundary relations. We again review the relevant notions from

[4] and [5], before specializing to the strict Picard case.

0 0 Two nonabelian 3-cocycles (νijkl, m˜ ijk, gijk, λij) and (νijkl, µ˜ijk, γijk, λij) are cohomologous if there exist:

1. A family of objects (ρi)∗ in Eq(GUi ) and a family of objects hij in GUij .

2. A family of arrowsχ ˜ij in the category GUij :

0 χ˜ij :(ρi)∗ ◦ λij ⇒ ihij ◦ λij ◦ (ρi)∗ (7–10)

and a family of morphisms aijk:

aijk :(ρi)∗(γijk) ◦ hik ⇒ hij ◦ λij(hjk) ◦ gijk (7–11)

in GUijk , for which the following two identities are satisﬁed:

(ρi)∗(γijk) hij λij (hjk)m ˜ ijk)ia χ˜ik λij(˜χjk)˜χij(ρi)∗(˜µijk) = ( ijk

hij λij (hjk) (ρi)∗(γijk) 0 {mijk, hkl}aijk aikl(ρi)∗(νijkl) =

0 hij λij (hjk)(λij λjk(hkl)) hij (ρi)∗(λ (γjkl)) (νijkl) λij(ajkl){{χ˜ij, γjkl}} aijl

7.2 2-Liens and Strict Picard Stacks

In this section we prove some theorems which are analogs of the results for 1-gerbes stated in Chapter 2. Recall the following deﬁnitions from Chapter 2.

Deﬁnition 7.2.1. A group category G is said to be a strict Picard category when its group

law is endowed with a commutivity isomorphism SX,Y : XY → YX which is functorial in

83 X and Y and sits in a commutative square

X1 S / 1X (7–12)

m S X X

and two hexagonal “associativity” diagrams. In addition SY,X ◦ SX,Y = 1XY for all X,

Y in G and SX,X = 1X for all X.A gr-stack G is said to be a strict Picard stack if it is endowed with a commutativity natural transformation S for the group law SX,Y that induces for each open U ⊂ X the structure of a strict Picard category on G(U). Note that in this section, by “Picard stack”, we always mean a strict (commutative) Picard stack. At this point the reader may wish to review the deﬁnition of a G-2-gerbe from Chapter 2. Proposition 7.2.2. Let G be a gr-stack on X.A 2-gerbe P on X is a G-2-gerbe if and only if its 2-lien is locally equivalent to lien2(G). Proof. If P is a G-2-gerbe, P is locally equivalent to T ors(G). Thus its 2-lien is locally

equivalent to lien2(T ors(G)) = lien2(G). Conversely, let P be a 2-gerbe whose 2-lien is

equivalent, when restricted to some open cover U of X to lien2(G) for some given gr-stack

0 G. We can then choose a second open cover U = (Ui) of X for which there exists a family

of objects xi ∈ GUi . The 2-gerbe P is then locally of the form T ors(Gi) for the tautological

labeling of P deﬁned by Gi = Eq(xi). It follows that lien2(P )|Ui is equivalent to lien2(Gi),

00 0 so that, on the elements Vα of a common reﬁnement U of U and U , we have equivalences

of 2-liens lien2(G)|Vα → lien2(Gi)|Vα . Such an equivalence is deﬁned by sections [η]α on

the open sets Vα of the stack Out(G, Gi). It follows from the deﬁnition of this stack that

these sections lift to sections ηβ : G|Wβ → Gi|Wβ of Eq(G, Gi) on the open sets Wβ of an

000 00 appropriate reﬁnement U of U . Since Gi = Eq(xi), the collection of maps (ηβ) give the G-2-gerbe structure on P . Proposition 7.2.3. Let G be a Picard stack on X. A gerbe P on X is an abelian G-2-

gerbe if and only if lien2(P ) is equivalent to lien2(G).

84 Proof. Suppose P is an abelian G-2-gerbe. Then from the diagram in the deﬁnition of

an abelian G-2-gerbe the terms the terms λij,m ˜ ijk, and {m˜ ijk, gklm} of the 3-cocycle associated to an abelian G-2-gerbe P are all trivial so that its 2-lien is globally equivalent to lien2(G). Conversely suppose P is a 2-gerbe with 2-lien equivalent to lien2(G) for some

Picard stack G. For any object x ∈ GU we can construct as in the proof of the above proposition, an equivalence of 2-liens

lien2(Eq(x)) → lien2(P |U ) → lien2(G|U ) (7–13)

which is described by an outer equivalence Eq(x) → G|U . Since G is now Picard, in the diagram

G|U (7–14) v HH ηx vv HH ηy vv 5=HH vv ηf HH vz v H$ Eq (x) / Eq (y) PU λ PU

of the deﬁnition of an abelian G-2-gerbe, the 2-arrow ηf deﬁnes an equivalence between

such an outer equivalence and an ordinary equivalence of gr-stacks Eq(x) → G|U . The

quasi-inverse ηx : G|U → Eq(x) deﬁnes an abelian G-2-gerbe structure on G. The required commutativity of the above diagram is equivalent to the commutativity of the

corresponding diagram of 2-liens, since it involves Picard categories. This in turn follows by applying the 2-lien 2-functor to the diagram

G (7–15) s KKK sss KK φx ss KKφy ss KKK sy ss K% T ors(Eq(x)) / T ors(Eq(y)) λ

where

φx : G → T ors(Eq(x)) (7–16)

g 7→ Eq(x, g) (7–17)

gives an equivalence between G and a 2-gerbe of torsors.

85 7.3 Strict Picard Stacks and H3

Deﬁnition 7.3.1. We designate by H3(X,L) the set of equivalence classes of 2-gerbes with 2-lien L, and by Hˇ 3(X,L) ⊂ H3(X,L) the set of equivalence classes of connected 2-gerbes with 2-lien L.

Deligne [13] has proved that any strict Picard stack G is 2-equivalent to a 2-term

d complex of abelian sheaves K· = [ K0 / K1 ]. We will prove that Hˇ 3(X, G) is isomorphic to the hypercohomology group Hˇ 3(X,K·).

We begin by describing Deligne’s construction for speciﬁc cases in one direction; that

is, we show how one can associate a strict Picard category (respectively stack) to a given 2-term complex of abelian groups (respectively sheaves). We follow the treatment in [11] and [13].

d Let K. denote a 2-term complex K0 / K1 of abelian groups. Then H is the strict Picard category whose objects are the elements of K1, for any two objects x, y, a morphism between them is an element z ∈ K0 such that dz = x − y in K1, composition of morphisms is addition in K0, and the functor H × H → H is given by addition of objects and morphisms. The associativity and commutativity constraints are the identities. If

. 0 d / 1 K denotes a 2-term complex K K of abelian sheaves, then HK is the prestack

given by the following information: for an open set U ⊂ X, HK (U) is the group category whose objects are the elements of K1(U), for any two objects x, y, a morphism between them is an element z ∈ K0(U) such that dz = x − y in K1(U), composition of morphisms

0 is addition in K (U), and the functor HK (U) × HK (U) → HK (U) is given by addition of objects and morphisms in K0 and K1. Again, the associativity and commutativity constraints are the identities. Then by [13], the stackiﬁcation ch(K) of the prestack HK is called the stack associated to the complex K., where ch(K) is Picard. More precisely,

if C(X) denotes the 2-category of 2-term complexes of abelian sheaves on X, where

morphisms in C(X) are morphisms of complexes, and 2-isomorphism are homotopies

between morphisms, then we have the following result.

86 Lemma 7.3.2. ([13], 1.4.13.1) For every Picard stack P there exists a complex K. ∈ C(X) such that P ' ch(K).

When K0 is injective, then we have the following lemma.

Lemma 7.3.3. ([13], 1.4.16.1) Let K. ∈ C(X) and assume that K0 is injective. Then the

prestack HK is already a stack. Let PIC(X) denote the 2-category of Picard stacks on X, and let C0(X) ⊂ C(X) denote the full sub-2-category of 2-term complexes K. with K0 injective. Lemma 7.3.4. ([13], 1.4.17) The construction ch described above induces a 2-equivalence of the 2-categories PIC(X) and C0(X).

We will follow the notation of [11] and (continue to) refer to Deligne’s 2-functor PIC(X) → C(X) by ch. Deligne also gives a characterization of the Picard stack

. . Hom(ch(K), ch(L)), where K , L ∈ C(X). Let ShAb(X) denote the category of abelian sheaves on X.

Lemma 7.3.5. ([13], 1.4.18.1) Assume that L0 is injective. Then we have an equivalence

. . ∼ / ch( τ≤1HomShAb(X)(K ,L ) HomPIC(X)(ch(K), ch(L)) (7–18)

The τ ≤ 1 above refers to a certain truncation operation; see [13] for details. For the rest of this section we will use this point of view while studying G-2-gerbes, where G

is strict Picard. Let P be a connected G-2-gerbe where G is strict Picard. Let K. be the 2-term complex associated to G. We form the double Cˇech complex:

Ci(K0) d / Ci(K1) / 0 (7–19)

δ δ Cij(K0) d / Cij(K1) / 0

δ δ Cijk(K0) d / Cijk(K1) / 0

δ δ Cijkl(K0) d / Cijkl(K1) / 0

87 where δ denotes the Cˇech diﬀerential, and d denotes the complex diﬀerential. This double complex is denoted C.(K.). One can obtain an ordinary complex C. from this double

n L p q p complex as follows. Put C = p+q=n C (K ), with the diﬀerential D = δ + (−1) d in bidegree (p, q). The sign (−1)p is placed here to ensure that D ◦ D = 0. This ordinary complex is called the total complex associated to the double complex C.(K.). Then the Cˇech hypercohomology Hˇ p(K.) of the complex K. is deﬁned to be the cohomology of the

. p n−p total complex C [7]. A degree n cocycle consists of a ﬁnite family cp ∈ C (K ) such that

p p+1 n−p δ(cp) = (−1) d(cp+1) ∈ C (K ). (7–20)

By the deﬁnition of a Picard stack, the 3-cocycle terms λij,m ˜ ijk, and {m˜ ijk, gklm} are trivial. The remaining data (νijkl, gijk) satisfy the relation

νjklmνijlmνijkl = νijkmνiklm. (7–21)

Also, the diagram

gijl xi / xi (7–22)

gikl λij (gjkl) νijkl xi ÑÙ / xi gijk becomes

gijl xi / xi (7–23)

gikl gjkl νijkl xi Ô / xi gijk By Deligne’s construction, when we view this data as taking values in K. instead of

0 1 G, we get that νijkl are sections of K and the gijk are sections of K . Further, by the construction of K., the diagram above says that:

d(νijkl) = gijk + gikl − gjkl − gijl. (7–24)

88 So d(ν) = δ(g). Also, since the ν are sections of K0, the relation

νjklmνijlmνijkl = νijkmνiklm. (7–25)

says that

νjklm + νijlm + νijkl = νijkm + νiklm, (7–26)

νjklm + νijlm + νijkl − νijkm − νiklm = 0, (7–27)

. that is, δ(ν) = 0. Thus the pair (νijkl, gijk) give us a hypercocycle in K . We now discuss the coboundary relations. When G is Picard, these relations simplify

0 to the statement that the cocycles (νijkl, gijk) and (νijkl, γijk) are cohomologous if there exist:

1. A family of objects hij in GUij .

2. A family of morphisms aijk that satisfy:

aijk : γijk ◦ hik ⇒ hij ◦ hjk ◦ gijk (7–28)

in GUijk , for which the following identity is satisﬁed:

0 aijkaiklνijkl = νijklajklaijl (7–29)

. As before, switching to additive notation after viewing the hij and aijk as sections in K , the ﬁrst identity above says that

d(aijk) = γijk + hij − hjk − gijk. (7–30)

d(aijk) = γijk − gijk + δ(hij), (7–31)

So d(aijk) − δ(hij) = γijk − gijk. Similarly, the second identity says that

0 aijk + aikl + νijkl = νijkl + ajkl + aijl (7–32)

89 so

0 νijkl − νijkl = ajkl + aijl − aijk − aikl = δ(aijk). (7–33)

0 Thus the coboundary conditions for the two cocycle pairs (νijkl, gijk) and (νijkl, γijk) in G translate exactly into hyper-coboundary conditions in K.. Thus we get a well-deﬁned map:

Hˇ 3(X, G) → Hˇ 3(X,K.) (7–34)

that sends the cohomology class of each pair (νijkl, gijk) to its corresponding class

(νijkl, gijk).

To see that this map gives an equivalence, observe that any hypercocycle (νijkl, gijk) in Hˇ 3(X,K.) satisﬁes

νjklm + νijlm + νijkl − νijkm − νiklm = 0, (7–35)

and

d(νijkl) = gijk + gikl − gjkl − gijl. (7–36)

Since this cocycle takes values in K., by Deligne ([13]), we can view it as a cocycle pair

(νijkl, gijk) that takes values in its associated Picard stack G and satisﬁes the relations

gijl xi / xi (7–37)

gikl gjkl νijkl xi Ô / xi gijk

and

νjklmνijlmνijkl = νijkmνiklm. (7–38)

But by Breen [4], Theorem 5.6, such a cocycle with values in G has an associated G-2-gerbe P, where P is constructed by precisely reversing the construction which

associates a nonabelian 3-cocycle (νijkl, m˜ ijk, gijk, λij) to a G-2-gerbe P, of which the

90 ˇ 3 ˇ 3 . G-valued cocycle pair (νijkl, gijk) is a special case. Thus the map H (X, G) → H (X,K ) is an isomorphism.

91 CHAPTER 8 CONCLUSION

We summarize the results of this thesis. The goal of this project was to deﬁne the

2-lien of a 2-gerbe. To this end, we needed to define the notion of a quotient by an action of a group stack on a stack. In chapter 3 we solved this problem locally: that is, we showed that coequalizers were representable in the 2-category CAT of (small) categories by explicitly constructing a coequalizer for (small) pair of arrows. In chapter 4 we defined what it means to have an inner equivalence of a group category. We then gave a definition for the action of a group category on a category, and defined the quotient by such an action using the coequalizer constructed in chapter 3. In chapter 5 we carried out the stackification of these concepts: that is, we showed the representability of coequalizers

in the 2-category ST ACKS, and gave a definition for the action of a group stack on a stack, and the quotient by such an action. In chapter 6 we finally defined the 2-lien of a space X. We showed how every 2-gerbe gives us such a 2-lien and did the work to establish that this 2-lien is in fact given up to canonical 2-equivalence. We then gave a more concrete description of a 2-lien using cocycles. Finally in chapter 7 we proved some results about G-2-gerbes whose 2-lien arises from a strict Picard stack. We recalled Deligne’s correspondence between strict Picard stacks G and 2-term complexes of abelian sheaves K.. We then proved that the set of equivalence classes Hˇ 3(X, G) of connected

gerbes with 2-lien G is isomorphic to the hypercohomology group Hˇ 3(X,K·). There are some natural directions that one could take as a continuance of this project. The above cohomological result of chapter 7 has only been proven for connected 2-gerbes, and it would be nice to prove it after removing the connectedness hypothesis. The proof would involve working with hypercovers (open covers of each open set) but the result would still seem to be true for G Picard. Then as a next step, one could relax the strict Picard assumption and only assume that the 2-lien G is braided, and try to ﬁnd a nice

cohomological description for 2-gerbes with 2-lien G.

92 REFERENCES [1] A.L. Carey and M.K. Murray, Fadeev’s anomaly and bundle gerbes, (Lett. Math. Phys. 37, no 1, 1996, pp. 29-36).

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[18] G. Laumon and L. Moret-Bailly, Champs Alg´ebriques, Springer-Verlag 2000.

[19] S. MacLane, Categories for the Working Mathematician, Graduate Texts in Math., Springer-Verlag 1971.

[20] I. Moerdijk, Introduction to the Language of Stacks and Gerbes, (in Course Notes, University of Utrecht, arXiv:math.AT/0212266 2002).

[21] M.K. Murray, Bundle Gerbes, (J. London Math. Soc. 54, no 2, 1996, pp. 403-416). [22] M. Romagny, Group Actions on Stacks and Applications, (Michigan Math. J. 53, no 1, 2005, pp. 209-236). BIOGRAPHICAL SKETCH I was born in Mumbai, India. I graduated from Truman State University in May 2001 and started graduate school at UF in the fall. I started working with Dr. Crew in 2004.