arX

iv:m

ath/

0602

510v

2 [

mat

h.K

T]

23

Jun

2006

REPRESENTATION AND CHARACTER THEORY IN

2-CATEGORIES

NORA GANTER AND MIKHAIL KAPRANOV

1. Introduction

The goal of this paper is to develop a (2-)categorical generalization ofthe theory of group representations and characters. It is classical thata representation of a group G is often determined by its character

χ(g) = tr((g)),

which is a class function on G.Remarkably, generalizations of character theory turn up naturally in

the context of homotopy theory. More precisely, G-equivariant versionsof certain cohomology theories of chromatic level n (see [Rav92] forbackground) lead to n-class functions, i.e., functions

χ(g1, . . . , gn)

defined on n-tuples of commuting elements of G and invariant undersimultaneous conjugation. Thus, in [HKR00] the study of nth MoravaE-theory at a prime p leads to χ(g1, ..., gn) defined on gi of p-powerorder. On the other hand, the construction of equivariant elliptic coho-mology in [Dev96] (which corresponds to n = 2) uses functions χ(g1, g2)as above but without restrictions on the order. The motivation herecomes from equivariant string theory. Earlier 2-class functions came upin Grojnowki’s work on equivariant elliptic cohomology (unpublished).What is lacking in the above approaches is an analog of the notion

of representation which would produce the generalized characters bymeans of some kind of trace construction. In this paper we supplysuch a notion for n = 2. It turns out that the right object to consideris an action of G on a category instead of a vector space or, moregenerally, on an object of a 2-category.

Date: January 22, 2019.Ganter was supported by NSF grant DMS-0504539. Kapranov was supported

by NSF grant DMS-0500565. The paper was completed while Ganter was visitingMSRI.

1

Our approach fits naturally with and was partially motivated by therecent works [TS04], [BDR04] aiming at geometric definitions of ellipticcohomology. All of these works involve 2-categorical constructions.For a subgroup H of a finite group G Hopkins, Kuhn and Ravenel

introduced in [HKR00] a natural induction map from n-class functionson H to those on G and identified it with the transfer map on theMorava En-theories of the classifying spaces of H and G. We definean induction procedure for categorical representations and show thatit produces the map of [HKR00] on the level of characters.

We have recently learned of a work in progress by Bruce Bartlettand Simon Willerton [BW] where, interestingly, the concept of thecategorical trace also appears although the motivation is different.We would like to thank Simon Willerton for his remarks on an earlier

version of this paper, and we would like to thank Haynes Miller, CharlesRezk, Matthew Ando and Alex Ghitza for many helpful conversations.

2. Background on 2-categories

2.1. Recall [ML98] that a 2-category C consists of:

(1) A class of objects Ob C.(2) For any x, y ∈ Ob C a category HomC(x, y). Its objects are

called 1-morphisms from x to y (notation F : x → y). We willalso use the notation 1HomC(x, y) for ObHomC(x, y). For anyF,G ∈ 1HomC(x, y) morphisms from F to G in HomC(x, y) arecalled 2-morphisms from F to G (notation φ : F ⇒ G). Wedenote the set of such morphisms by

2HomC(F,G) = HomHomC(x,y)(F,G).

The composition in the category HomC(x, y) will be denotedby ◦1 and called the vertical composition. Thus if φ : F ⇒ Gand ψ : G⇒ H , then ψ ◦1 φ : F ⇒ H .

(3) The composition bifunctor

HomC(y, z)×HomC(x, y) → HomC(x, z)

(F,G) 7→ F ◦0 G.

Thus if F,G : x → y and φ : F ⇒ G and H,K : y → z andψ : H ⇒ K, then ψ ◦0 ψ : H ◦ F ⇒ K ◦G.

(4) The natural associativity isomorphism

αF,G,H : (F ◦0 G) ◦0 H ⇒ F ◦0 (G ◦0 H),

given for any three composable 1-morphisms F,G,H and satis-fying the pentagonal axiom, see [ML98]

2

(5) For any x ∈ Ob C a 1-morphism 1x ∈ 1HomC(x, x) called theunit moprhism and 2-isomorphisms

ǫφ : 1x ◦ φ⇒ φ, φ : y → x,

ζψ : ψ ◦ 1x ⇒ ψ, ψ : x → z,

satisfying the axioms of [ML98].

We say that C is strict if all the αF,H,H , ǫφ and ζψ are identities (inparticular the source and target of each of them are equal). It is atheorem of Mac Lane and Pare that every 2-category can be replacedby a (2-equivalent) strict one. See [ML98] for details.

2.2. Examples. (a) The 2-category Cat has, as objects, all small cate-gories, as morphisms their functors and as 2-morphisms natural trans-formations of functors. This 2-caetgory is strict. We will use the nota-tion Fun(A,B) for the set of functors between categories A and B (i.e.,1-morphisms in Cat) and NT (F,G) for the set of natural transforma-tions between functors F and G. Most of the examples of 2-categoriescan be embedded into Cat: a 2-category C can be realized as consistingof categories with some extra structure.

(b) Let k be a field. The 2-category 2V ectk, see [KV94] has, asobjects, symbols [n], n = 0, 1, 2, ... For any two such objects [m], [n]the category Hom([m], [n]) has, as objects, 2-matrices of size m byn, i.e., matrices of vector spaces A = ‖Aij‖, i = 1, ..., m, j = 1, ..., n.Morphisms between 2-matrices A and B of the same size are collectionsof linear maps φ = {φij : Aij → Bij}. Composition of 2-matrices isgiven by the formula

(A ◦B)ij =⊕

l

Ail ⊗Blj .

This 2-category is not strict. An explicit strict replacement was con-structed in [Elga].

(c) The 2-category Bim has, as objects, associative rings. If R, S aretwo such rings, then HomBim(R, S) is the category of (R, S)-bimodules.The composition bifunctor

Hom(S, T )×Hom(R, S) → Hom(R, T )

is given by the tensor product:

(M,N) 7→ N ⊗S M.

This 2-category is also not strict.3

Relation to Cat: To a ring R we associate the category Mod − R ofright R-modules. Then each (R, S)-bimodule M defines a functor

Mod −R →Mod − S, P 7→ P ⊗RM.

The 2-category 2V ectk is realized inside Bim by associating to [m]the ring k⊕m. An m by n 2-matrix is the same as a (k⊕m, k⊕n)-bimodule.We will denote by Bimk the sub-2-category in Bim formed by k-

algebras as objects and the same 1- and 2-morphisms as in Bim.

(d) Let X be a CW-complex. The Poincare 2-category Π(X) has, asobjects, points of X , as 1-morphisms Moore paths [0, t] → X and as2-morphisms homotopy classes of homotopies between Moore paths.We will occasionally use the concept of a (lax) 2-functor Φ: C → D

betweeen 2-categories C and D. Such a 2-functor consists of mapsObC → ObD, 1 − MorC → 1 − MorD and 2 − MorC → 2 − MorDpreserving the composition of 2-morphisms and preserving the compo-sition of 1-morphisms up to natural 2-isomorphisms. See [ML98, ??]for Details.

2.3. 2-categories with extra structure. We recall the definition ofan enriched category from [ML98] or [Kel82]. Let (V,⊗, S) be a closedsymmetric monoidal category, so ⊗ is the monoidal operation and S isa unit object.

Definition 2.1. A V-category C is defined in the same way as a cate-gory, with the morphism sets replaced by objects hom(X, Y ) of V andcomposition replaced by V-morphisms

hom(X, Y )⊗ hom(Y, Z) → hom(X,Z),

with units

1X : S → hom(X,X),

such that the usual assiociativity and unit diagrams commute.

Example 2.2. Categories enriched over the category of abelian groupsare commonly known as pre-additive categories. By an additive cate-gory one means a pre-additive category possessing finite direct sums.If k is a field, categories enriched over the category of k-vector spacesare known as k-linear categories.

Example 2.3. As is well known, a strict 2-category is the same as acategory enriched over the category of small categories with ⊗ beingthe direct product of categories.

4

Definition 2.4. Let V be a category. A strict 2-category C enrichedover V, or shorter a V-2-category, is a category enriched over the cat-egory of small V-categories.

Definition 2.5. We define a strict pre-additive 2-category to be a2-category enriched over the category of abelian groups. Let k be afield. Then a (strict) k-linear 2-category is defined to be a 2-categoryenriched over the category of k-vector -spaces. Weak additive and k-linear categories are defined in a similar way.

We will freely use the concept of a triangulated category [Nee01],[GM03]. If A is triangulated, then we denote by X [i] the i-fold iteratedshift (suspension) of an object X in A. We will denote

Hom•A(X, Y ) =

⊕

i

HomA(X, Y [i]).

We will call a 2-category C triangular if each HomC(x, y) is madeinto a triangulated category and the composition functor is exact ineach variable.

2.4. Examples. (a) The 2-category Bim is additive. The 2-categories2V ectk,Bimk are k-linear.

(b) Define the 2-category DBim to have the same objects as Bim,i.e., associative rings. The category HomDBim(R, S) is defined to bethe derived category of complexes of (R, S)-bimodules bounded above.The composition is given by the derived tensor product:

(M,N) 7→ N ⊗LS M.

This gives a triangular 2-category.

(c) The 2-category Vark has, as objects, smooth projective algebraicvarieties over k. If X, Y are two such varieties, then

HomVark(X, Y ) = DbCoh(X × Y )

is the bounded derived category of coherent sheaves on X × Y . IfK ∈ DbCoh(Y ×Z) and L ∈ DbCoh(X × Y ), then their composition isdefined by the derived convolution

K ∗ L = Rp13∗(p∗12L ⊗L p∗23K),

where p12, p13, p23 are the projections of X × Y × Z to the products oftwo factors. This again gives a triangular 2-category.

5

Relation to Cat: To every varietyX we associate the categoryDbCoh(X).Then every K ∈ DbCoh(X × Y ) defines a functor

FK : DbCoh(X) → DbCoh(Y ), F 7→ Rp2∗(p∗1F ⊗L K),

and FK∗L is naturally isomorphic to FK ◦FL. It is not known, however,whether the natural map

HomDbCoh(X,Y )(K,L) → NT (FK, FL)

is a bijection for arbitrary K,L. So in practice the source of this mapis used as a substitute for its target.

(d) The 2-category CWk has, as objects, finite CW-complexes. Forany two such complexes X, Y the category HomCW(X, Y ) is definedto be DbConstr(X × Y ), the bounded derived category of (R-) con-structible sheaves of k-vector spaces on X × Y , see [KS94] for back-ground on constructible sheaves. The composition is defined simi-larly to the above, with sheaf-theoretic direct images p−1

ij instead ofO-module-theoretic direct images p∗ij . This is a triangular 2-category.

(e) Let Ab denote the category of all abelian categories. For any suchcategories A,B the category Fun(A,B) is again abelian: a sequenceof functors is exact if it takes any object into an exact sequence. Sowe have a triangular 2-category DAb with same objects as Ab butHom(A,B) = DbFun(A,B).

3. The categorical trace

3.1. The main definition. As motivation consider the 2-category2V ectk. In this situation there is a naıve way to define the “trace”of a 1-automorphism, namely as direct sum of the diagonal entries ofthe matrix. This naıve notion of trace is equivalent to the followingdefinition that makes sense in any 2-category C:

Definition 3.1. Let C be a 2-category, x an object of C and F : x→ xa 1-endomorphism of x. The categorical trace of F is defined as

Tr(F ) = 2-Hom(1x, F ).

If C is triangular, we write

Tri(F ) = Tr(F [i]), i ∈ Z, Tr•(F ) =⊕

i

Tri(F ).

6

Remark 3.2 (Functoriality). Note that for each x, the categoricaltrace defines a functor

Tr: 1-End(x) → Set

φ ∈ 2-Hom(F,G) 7→ φ∗,

where

φ∗ : Tr(F ) → Tr(G)

is given by composition with φ. A priori, Tr is set valued, but if weassume C to be enriched over a category V (cf. Definition 2.4), Tr takesvalues in V. We will often assume that C is k-linear for a fixed field k.

3.2. Examples of the categorical trace. Our first example is themotivational example mentined above.

Example 3.3 (2-vector spaces). Let C = 2-Vectk and x = [n]. ThenF is an n × n matrix A = (Aij), where the Aij are vector spaces. Inthis case,

Tr(A) =

n⊕

i=1

Aii.

Example 3.4 (Categories). Let C = Cat and x = A a category, soF : A → A is an endofunctor. Then Tr(F ) = NT (IdA, F ) is the set ofnatural transformations from the identity functor to F .

Example 3.5 (Bimodules). Let C = DBim, so that x = R is a ring,and F =M is an R-bimodule. Then

Tr•(F ) = Ext•R⊗Rop(R,M)

is the Hochschild cohomology of M.

Example 3.6 (Varieties). C = Vark, x = X a variety, F = K acomplex of coherent sheaves on X ×X . Then

Tr•(F ) = H•(X, i!(K)).

here i : X → X ×X is the diagonal embedding, i! is the roiht adjointof i∗, and H is the hypercohomology. In particular, if K is a vectorbundle on X ×X situated in degree 0, then

Tr•(F ) = H•(X,K|∆)

is the cohomology of the restriction of K to the diagonal.7

3.3. The center of an object. The set Tr(1x) will be called thecenter of x and denoted Z(x). It is closed under both compositions ◦0and ◦1. The following fact is well known [ML98]

Proposition 3.7. The operations ◦0 and ◦1 on Z(x) coincide and makeit into a commutative monoid.

Thus, if C is pre-additive, then Z(x) is a commutative ring and foreach F : x→ x the group Tr(F ) is a Z(x)-module.

3.4. Examples. (a) If C = Ab and x = A is an abelian category, thenZ(A), i.e., the ring of natural transformations from the identity functorto itself is known as the Bernstein center of A, see [Ber84]

(b) If C = Π(X) is the Poincare 2-category of a CW-complex X , thenZ(x) = π2(X, x) is the second homotopy group. Proposition 3.7 is thecategorical analog of the commutativity of π2.

3.5. Conjugation invariance of the categorical trace. In this sec-tion we assume for simplicity that the 2-category C is strict. Recall[ML98] that a 1-morphism G : y → x is called an equivalence if thereexist a 1-morphism H : x→ y called quasi-inverse and 2-isomorphismsu : 1x ⇒ GH , v : 1y ⇒ HG. For any object x, the 1-morphism G = 1xis an equivalence with H = 1x and u, v the isomorphisms from 2.1 (4).If G and G′ are composable 1-morphisms, which are equivalences withquasi-inverses (H, u, v) and (H ′, u′, v′) respectively, then G′ ◦0 G is anequivalence with quasi-inverse

(1) (H ◦0 H′, (G′ ◦0 u ◦0 H

′) ◦1 u′, (H ◦0 v

′ ◦0 G) ◦1 v) .

Proposition 3.8 (Conjugation invariance). (a) Let F : x → x be a1-endomorphism and G : x→ y an equivalence. Then the rule

(φ : 1x ⇒ F ) 7→ (G ◦0 φ ◦0 H) ◦1 u

defines a bijection of sets

ψ(G,H, u, v): Tr(F ) → Tr(GFH).

By abuse of notation, we will write ψ(G) when H, u and v are clearfrom the context.(b) Assume that G and G′ are 1-endomorphisms of x and that both ofthem are equivalences. Then we have

ψ(G′ ◦0 G) = ψ(G′) ◦ ψ(G).

(c) We have ψ(1x) = id.8

Proof. (a) To explain the formula, note that we can view u as a 2-morphism 1y ⇒ GH = G ◦ 1x ◦H , while

G ◦0 φ ◦0 H : G ◦ 1x ◦H ⇒ G ◦ F ◦H.

Since u is a 2-isomoprphism, composing with u is a bijection.,

(b) This follows from the definition of ψ(G) together with (1).

(c) Obvious. �

Proposition 3.9. Let C be an additive 2-category and F,G : x→ x betwo 1-moprhisms. Then

Tr(F ⊕G) = Tr(F )⊕ Tr(G).

This is obvious as H = HomC(x, x) is an additive category so

HomH(1x, F ⊕G) = HomH(1x, F )⊕HomH(1x, G).

4. 2-representations and their characters

4.1. 2-representations. Let G be a group. We view G as a 2-categorywith one object, pt, the set of 1-morphisms Hom(pt, pt) = G and allthe 2-morphisms being the identities of the above 1-morphisms.objects and one morphisms as above and only identity 2-morphisms.

Definition 4.1. Let C be a 2-category. A representation of G in C isa lax 2-functor from G to C. This spells out to the following data:

(a) an object c of C,(b) for each element g ∈ G, a 1-automorphism ρ(g) of V ,(c) for any pair of elements (g, h) of G a 2-isomorphism

φg,h : (ρ(g) ◦ ρ(h))∼=

=⇒ ρ(gh)

(d) a 2-isomorphism

φ1 : ρ(1)∼=

=⇒ idc

such that the following conditions hold

(e) for any g, h, k ∈ G we have

φ(gh,k)(φg,h ◦ ρ(k)) = φ(g,hk)(ρ(g) ◦ φh,k)

(associativity); we also write φg,h,k,(f) we have

φ1,g = φ1 ◦ ρ(g) and φg,1 = ρ(g) ◦ φ1.

Note that this definition is the special case of a discrete 2-group ofElgueta’s definition [Elgb, Def.4.1]. 2-representations of G in C form a2-category 2RepC(G), see [BW] for details.

9

4.2. The category of equivariant objects. Consider the particularcase of Definition 4.1 when C = Cat is the 2-category of (small) cate-gories. Then a 2-representation of G in C is the same as an action ofG on a category V so that each

ρ(g) : V → V

is a functor and each φg,h is a natural transformation.We denote the category of G-equivariant objects in V by VG. By

definition, an object of VG is a system

(X, ǫg : X → ρ(g)(X), g ∈ G) ,

where X ∈ Ob(V) and the ǫg are isomorphisms satisfying the followingcompatibility condition: for any g, h ∈ G the diagram

Xǫg

//

ǫgh

��

ρ(g)(X)

ρ(g)(ǫh)��

ρ(gh)(X) ρ(g)(ρ(h)(X))φg,h,X

oo

is commutative. This concept is analogous to the concept of the sub-space of G-invariants in the usual representation theory in the followingsense:

Proposition 4.2. Let W be a category. Define the trivial action ofG on W by taking all ρ(g) and φg,h to be the identities. Then anyG-functor from W to V factors through the forgetful functor

VG → V.

Proposition 4.3. Let k be a field, and let C be the 2-category of addi-tive k-linear categories. Then

VG ∼= Hom2RepC(G) (Vectk,V) ,

where Vectk is the category of finite dimensional k-vector spaces withthe trivial G-action.

Further, in the situation of the above Proposition (i.e., V being anadditive k-linear category), the category VG is in fact a module categoryover the monoidal category

(Rep(G),⊗) .

In other words, if X is a G-equivariant object and V is a representationof G, then V ⊗ X is again an equivariant object. Module categoriesover Rep(G) have been studied by Ostrik [Ost01] and provide an alter-native approach to the categorical generalization of G-representations.

10

It seems that in general, the passage from V to VG leads to some lossof information. However, in some particular cases, the two approachesare equivalent, see below.

4.3. Characters of 2-representations. We are now ready to definethe categorical character of such a 2-representation. To motivate thedicussion of this section, we start with a reminder of classical charactertheory.

4.3.1. Group characters and class functions. We fix a field k of char-acteristic 0 containing all roots of unity. Let G be a group. Recall thata function f : G → k is called a class function if it is invariant underconjugation.

Notation 4.4. We denote by Cl(G) the space of class functions. Letalso Rep(G) be the category of finite-dimensional representations of Gover k. Following the usual notation in the literature, we write R(G)for its Grothendieck ring K(Rep(G)).

If ρ : G→ Aut(V ) is a representation, then its character

χV : G → k

g 7→ Trace(ρ(g))

is a class function. The following is well known [Ser77].

Proposition 4.5. If G is finite, then the correspondence V 7→ χVinduces an isomorphism of rings

R(G)⊗ k → Cl(G).

4.3.2. The categorical character. The classical definitions discussed inthe previous section suggest the following analogues for 2-representations:

Definition 4.6. Let ρ be a 2-representation of G. We define the cat-egorical character of ρ to be the assignment

g 7→ Tr(ρ(g)).

We now discuss the sense in which the categorical character is a classfunction. First we recall the definition of the inertia groupoid of G:

Definition 4.7. Let G be a group. The inertia groupoid Λ(G) of Gis the category that has as objects the elements of G and a morphismfrom f to k for each g with k = gfg−1.

11

Proposition 4.8. Let C be a 2-category and let ρ be a representationof G in C. Then the categorical character of ρ is a functor from theinertia groupoid Λ(G) to the category of sets:

Tr(ρ): Λ(G) → Set.

If C is enriched over a category V, then this functor takes values in V.In other words, for any f, g ∈ G there is an isomorphism

ψ(g) = ψf (g): Tr(ρ(f)) → Tr(ρ(gfg−1)),

and these isomorphisms satisfy

(a) ψ(gh) = ψ(g) ◦ ψ(h) and(b) ψ(1) = idρ(f).

Proof. Pick f, g ∈ G and write F for ρ(f), G for ρ(g), H for ρ(g−1)and define

u: 1c → GH

as the composite of maps from Definition 4.1

u := φ−1g,g−1φ

−11 .

With this notation, Proposition 3.8 (a) implies the existence of anisomorphism

ψ′ : Tr(ρ(f))∼=

−→ Tr(ρ(g)ρ(f)ρ(g−1)).

Composed with Tr(φg,f,g−1) this gives the desired map ψ(g). Properties(a) and (b) of ψ(g) follow from Proposition 3.8 (b) and (c). �

Remark 4.9. In the case when G is a topological or algebraic group,under natural assumptions on ρ and C we have that Tr(()ρ) is a sheafon G equivariant under conjugation.

Definition 4.10. If ρ is a 2-representation in a k-linear linear 2-category with finite-dimensional 2-Hom(φ, ψ), we define the categoricalcharacter of ρ to be the function χρ on pairs of commuting elementsgiven by

χρ(g, h) = trace (ψ(h)|Tr(ρ(g))) .

Note that

χρ(s−1gs, s−1hs) = χρ(g, h).

12

5. Examples

5.1. 1-dimensional 2-representations. Let k be a field and

c : G×G→ k∗

be a 2-cocycle, i.e., it satisfies the identity

c(gh, k)c(g, h) = c(g, hk)c(h, k).

We then have an action ρ = ρc of G on V ectk. By definition, for g ∈ Gthe functor ρ(g) : V ectk → V ectk is the identity, while

φg,h : id = ρ(g) ◦ ρ(h) ⇒ ρ(gh) = id

is the multiplication with c(g, h), and φ1 is the multiplication by c(1, 1).The cocycle condition for c is equivalent to Condition (e) of Definition4.1, while Condition (f) follows because

c(1, 1g) · c(1, g) = c(1, 1) · c(1, g)

implies that

c(1, g) = c(1, 1)

and similarly

c(g, 1) = c(1, 1).

Cohomologous cocycles define equivalent 2-representations, and it iseasy to see that H2(G, k∗) is identified with the set of G-actions onVectk modulo equivalence. Compare [Kap95].

We now find the categorical character and the 2-character of ρc. Firstof all, the functor ρc(g) being the identity,

Tr(ρc(g)) = k.

Next, the equivariant structure on Tr(ρc) was defined in the proofs ofpropositions 3.8 (a) and 4.8 to be the composition

Tr(ρc(f))u

−→ Tr(ρc(g)ρc(f)ρc(g−1)) −→ Tr(ρc(gfg

−1)).

Here u is induced by the 1-composition with

u: 1Vectk ⇒ GH,

where G = ρc(g) and H = ρc(g−1). In our case,

u = c(g, g−1)−1

(multiplication with a scalar). The second map is induced by

φg,f,g−1 = c(g, f)c(gf, g−1),

see Definition 4.1 (e). As a result we have13

Proposition 5.1. For any two commuting elements f, g ∈ G, we have

χρc(f, g) = c(g, f)c(gf, g−1)c(g, g−1)−1.

Notice also the following:

Proposition 5.2 (compare [Elgb]). Let ρc be the one-dimensional 2-representation of G on V = Vectk corresponding to c. Then objects ofVG are the same as projective representations of G with central chargec.

5.2. Representations on 2-vector spaces. 2-representations ρc fromSection 5.1 can be viewed as acting on the 1-dimensional 2-vector space[1]. More precisely, let

1 → k∗ → Gπ→ G→ 1

be the central extension corresponding to the cocycle c. For everyg ∈ G the set π−1(g) is the a k∗-torsor, and therefore

Lg := π−1(g) ∪ {0}

is a 1-dimensional k-vector space. The group structure on G inducesisomorphisms

Lg ⊗k Lh → Lgh.

It follows that associating to g ∈ G the 2-matrix ‖Lg‖ of size 1 × 1gives a 2-representation of G on [1] ∈ Ob(2-Vectk)More generally, a 2-representation ρ of G on [n] consists a datum,

for each g ∈ G, of a quasi-invertible 2-matrix ρ(g) = ‖ρ(g)ij‖ of sizen × n, with ρ(g)ij belong to Vectk plus the data φg,h as in Definition4.1.

Lemma 5.3. A 2-matrix V = ‖Vij‖ of size n× n is quasi-invertible ifand only if there is a permutation σ ∈ Σn such that Vij = 0 for i 6= σ(j)and dim(Vi,σ(i)) = 1.

It follows that an n-dimensional 2-representation of G defines a ho-momorphism G → Σn plus some cocycle data. This is naturally ex-pained in the context of induced 2-representations, cf. Section 7 below.

5.3. Constructible sheaves. Let X be a finite CW-complex actedupon by G. We view each g ∈ G as a homomeorphism g : X → X . LetA be the category DbConstr(X), see Section 2.4 (c). The base field kwill be taken to be the field C of complex numbers, for simplicity. Wehave then an action ρ of G on A given by

ρ(g)(F) = g−1(F).14

As in Examples 2.4 (b), (c), it is more practicable to lift this action on acategory to an action on an object X of the 2-category CWC. Namely,for g ∈ G we denote by Γ(g) ⊂ X × X its graph and associate to gthe constructible sheaf Kg = CΓ(g), the constant sheaf on Γ(g). Notethat ρ(g) = FKg

is the functor associated to Kg. It is clear that thecorrespondence g 7→ Kg gives an action of G on the object X which wedenote ρ.

Proposition 5.4. Assume that X is a smooth oriented manifold of realdimension d. Then the categorical character of ρ is found as follows:

Tr•(ρ(g)) = H•+codimXg

(Xg,C),

where Xg ⊆ X is the fixed point locus of g.

Proof. As in Example 3.6, we denote by i : X → X ×X the diagonalembedding and we have

Tr(ρ(g)) = Tr(FKg)−H

•(X, i!(Kg))

and

i!(Kg) = RΓ∆(Kg) = CXg [codimXg].

Here ∆ = i(X) is the diagonal in X ×X . �

In fact, the character sheaves of Lustig can be obtained in a similarway.

6. Representations of finite groupoids

Definition 6.1. Let G be a finite groupoid. A representation overk of G is a functor from G to Vectk. A morphism between two G-representations is, a natural transformation between them. We denotethe category of G-representations over k by Rep(G). Direct sum and(objectwise) tensor product make Rep(G) into a bimonoidal category,so that its Grothendieck construction R(G) is a ring, the representationring of G.

Definition 6.2. Let α: H → G be a map of finite goupoids. Thenprecomposition with α defines a functor

res |α : Rep(G) → Rep(H).

If H is a subgroupoid of G and α is its inclusion we also denote res |αby res |GH .

15

Definition 6.3. Let G be a finite groupoid. The groupoid algebrak[G] has as underlying k-vector space the vector space with one basis-element eg for each morphism g of G. The algebra structure is givenby

eg · eh =

{egh if g and h are composable

0 else

The categories of G-representations over k and of k[G]-modules arethen equivalent.

Proposition 6.4. If α: H → G is an equivalence of groupoids, then

res |α : Rep(G) → Rep(H)

is an equivalence of categories.

Corollary 6.5. If the groupoids G and H are equivalent, then theirgroupoid algebras are Morita equivalent.

Observation 6.6. Every finite groupoid is equivalent to a disjointunion of groups.

Corollary 6.7. The groupoid algebra of a finite groupoid G is semi-simple. Thus there is a unique decomposition of representations of Ginto irreducibles.

Definition 6.8. Let α: H → G be a map of groupoids, and let V bea representation of H . Viewing V as a k[H ]-module, we define theinduced G-representation of V by

ind |α(V ) := k[G]⊗k[H] V.

We will sometimes write ind |GH for ind |α, if the map is obvious.

Note that ind |α is left adjoint to res |α.Let

α: H → G

be faithful and essentially surjective, and let ρ be a representation ofH . By Observation 6.6, we may assume that G and H are disjointunions of groups. Then a representation of G can be described onegroup at a time, so we may as well assume that G is a single group and

α: H1 ⊔ · · · ⊔Hn → G

is given by a (nonempty) set of injective group maps α1, . . . , αn.In this situation, the induced representation of ρ along α is isomor-

phic to

ind |α(ρ1, . . . ρn) :=⊕

ind |αiρi.

16

Note also that away from the essential image of α, the induced rep-resentation along α is zero.

Definition 6.9. Let G be a groupoid. The inertia groupoid Λ(G) ofG has as objects the automorphisms of G and one morphism from fto k for every morphism g of G with k = gfg−1. A class function on agroupoid G is a function defined on isomorphism classes of Λ(G). Letρ be representation of a finite groupoid G and assume that our field khas all roots of unity. Then we define the character of ρ to be the kvalued class function on G given by

χρ([g]) = Trace(ρ(g)).

Remark 6.10. Sending a representation to its character is a ring map

R(G) → Cl(G; k).

By Observation 6.6 and Proposition 4.5 it becomes an isomorphismafter tensoring with k.

Proposition 6.11. Assume that α is a faithful functor, let x be anobject of G and g ∈ HomG(x, x). Then the character of the inducedrepresentation evaluated at g is given by the formula

χind(x, g) =∑

y∈H0

1

| orbitH(y)|

1

|AutH(y)|

∑

s∈G1|sx=y

sgs−1∈AutH (y)

χ(y, sgs−1).

Here orbitH(y) is the H-isomorphism class of y, and the second sum isover all morphisms s of G with source x and target y that conjugate ginto a morphism in the image of α|AutH (y).

Proof : Let orbitG(x) denote the G-isomorphism class of x, and let

R = {y1, . . . , yn}

be a system of representatives for the H-isomorphism classes mappingto orbitG(x) under α. For each j, pick an sj with sjx = α(yj). Denoteα|AutH (yj) by αj. We have

χind =

n∑

j=1

χind |αj

and

χind |αj(x, g) = χind |αj

(sjx, sjgs−1j ).

17

By the classical formula for the character of an induced representationof a group, we have

χind|αj(sjx, sjgs

−1j ) =

1

|AutH(yj)|

∑

t∈AutG(yj)

tsjgs−1j t−1∈AutH (yj)

χ(yj, tsjgs−1j t−1)

=1

|AutH(yj)|

∑

sx=yj

sgs−1∈AutH (yj)

χ(yj, sgs−1).

The first sum in the Proposition is over all objects y of H . If α(y)is not isomorphic to x, then the second sum is empty. For the onesisomorphic to x, we are double counting: rather than just having onesummand for the representative yj, we have the same summand forevery element in its H-orbit. �

Corollary 6.12. Consider an inclusion of groups H ⊆ G and theinduced map of inertia groupoids α: Λ(H) → Λ(G). Let ρ be a repre-sentation of Λ(H). Then the character of the induced representationind |αρ evaluated at a pair of commuting elements of G is given by theformula

χ(g1, g2) =∑

h1∈H

1

|[h1]H |

1

|CH(h1)|

∑

sg1s−1=h1

sg2s−1∈CH (h1)

χ(sg1s−1, sg2s

−1)

=1

|H|

∑

s(g1,g2)s−1

∈H×H

χ(sg1s−1, sg2s

−1).

Here [h1]H is the conjugacy class of h1 in H while CH(h1) is the cen-tralizer of h1 in H.

7. Induced 2-representations

Definition 7.1. Let H ⊆ G be an inclusion of finite groups. Let

ρ: H → Fun(V,V)

be an action ofH on a category V. Let ind |GH(V) be the category whoseobjects are maps

f : G→ ObV

together with an isomorphism for every g ∈ G and h ∈ H

ug,h : f(gh) → ρ(h−1)(f(g))18

satisfying the following condition: For every g ∈ G and every h1, h2 ∈

H , the diagram f(gh1h2)ugh1,h2

//

ug,h1h2

��

ρ(h−11 )(f(gh2))

ρ(h−12 )ug,h1

��

ρ((h1h2)−1)(f(g)) ρ(h−1

2 )ρ(h−11 )(f(g))

φh−11

,h−12

oo

commutes.

An explicit description of ind |GHV is given as follows (compare thisto the definition of the induced representation in [Ser77]):Let m be the index of H in G. The underlying category of ind |GHV is

then identified with Vm. Such an identification is obtained by pickinga system of representatives

R = {r1, . . . rm}

of left cosets of H in G and associating to every map f as above thesystem (f(r1), ..., f(rm)).We view ind |GHρ(g) as m×m matrix whose entries are functors from

V to V. Then

(ind |GHρ(g)

)ij=

{ρ(h) if grj = rih, h ∈ H

0 else.

Note that in each row and each column, there is exactly one blockentry, and that therefore such a matrix gives a functor from Vm to Vm.Composition:(ind |GHρ(g1)

)◦1

(ind |GHρ(g2)

)ik=

=

{ρ(h1) ◦1 ρ(h2) if g1rj = rih1 and g2rk = rjh2

0 else.

In the case that this is not zero,(ind |GHρ(g1g2)

)ik= ρ(h1h2),

since in this caseg1g2rk = rih1h2.

On this block the composition isomorphism is given by the 2-isomorphism

ρ(h1) ◦1 ρ(h2) ⇒ ρ(h1h2).

Similarly, the isomorphism

ind |GHρ(1) ⇒ 1Vm

is given by the corresponding map for ρ.19

Proposition 7.2 (compare [Ost01, Ex.3.4.,Th.2] and [Elgb, 6.5]). LetG be a finite group, and let V = Vect⊕nk . Then any 2-representation ρof G in V is isomorphic to a direct sum

m⊕

i=1

ind |GHiρωi,

where the Hi are subgroups of G, ωi ∈ H2(Hi, k∗), and ρωi

is the 1-dimensional 2-representation corresponding to ωi. Moreover, the sys-tem of (Hi, ωi) is determined by the G-action on V uniquely up toconjugation.

Proof : By Lemma 5.3, ρ defines a homomorphism from G to Σn, i.e.,a G-action on the set {1, . . . , n}. Let O1, . . . , Om be the orbits of thisaction. It follows that, after renumbering of 1, . . . , n, the 2-matricesρ(g) become block diagonal with blocks of sizes |O1|, . . . , |Om|. Hence

ρ ∼=

m⊕

i=1

ρi.

Let Hi be the stabilizer of an element of Oi. For h ∈ Hi, the 2-matrixρi(h) is diagonal with the same 1-dimensional vector space Li(h) onthe diagonal. We conclude that

ρi ∼= ind |GHi(ρωi

),

where ρωiis the 1-dimensional 2-representation of Hi corresponding to

the system (Li(h), h ∈ Hi). �

Remark 7.3. It follows that in the particular case of the proposition,the approach of Ostrik is equivalent to ours. In fact, the category

(ind |GH (ρω)

)G, H ⊆ G, ω ∈ H2(H, k∗),

is identified, as a Rep(G) module category, with the category of pro-jective representations of H with the central charge ω.

7.1. The character of the induced 2-representation. The aim ofthis section is to prove the following theorem.

Theorem 7.4. Let V be k-linear. The categorical trace Tr takes in-duced 2-representations into induced representations of groupoids. Thatis,

Tr(ind |GHρ)∼= ind |

Λ(G)Λ(H)(Tr(ρ)),

as representations of Λ(G).20

Corollary 7.5 (compare [HKR00, Thm D]). Assume that the Tr(ρ(h))are finite dimensional. Let χ denote the 2-character of ρ. The 2-character of the induced representation is given by

χind(g, h) =1

|H|

∑

s−1(g,h)s∈H×H

χ(s−1gs, s−1hs).

Proof. of Theorem 7.4 We want to compute

χind := Tr(ind |GHρ)

as representation of

Λ(G) ≃∐

[g]G

CG(g).

Let R be a system of representatives of G/H , and fix g ∈ G. Theunderlying vector space of χind(g) is the sum over all r ∈ R whichproduce a diagonal block entry in indρ(g),

(2) χind(g) =⊕

r−1gr∈H

Tr(ρ(r−1gr)),

(compare [Ser77]). We need to determine the action of CG(g) onχind(g). For this purpose, we replace our system of representativesR in a convenient way: The decomposition

[g]G ∩H = [h1]H ∪ · · · ∪ [hl]H

induces a decomposition

{r ∈ R | r−1gr ∈ H} =

l⋃

i=1

Ri,

withRi := {r ∈ R | r−1gr ∈ [hi]H}.

We fix i, pick ri ∈ Ri, and write hi := ri−1gri.

Lemma 7.6. We can replace the elements of Ri in such a way thatleft multiplication with r−1

i maps Ri bijectively into a system of repre-sentatives of

CG(hi)/CH(hi).

Proof. If r ∈ Ri satisfies

r−1gr = h−1hih,

we replace r by rh−1, which represents the same left coset of G/H asr does. Note that

(3) (rh−1)−1grh−1 = hi.21

We have

(r−1i rh−1)

−1hi(r

−1i rh−1) = hi,

therefore r−1i (rh−1) ∈ CG(hi). Assume now that we have replaced Ri

in this way. To prove that left multiplication with r−1i is injective, let

r 6= r′ ∈ Ri. Then

(r−1i r)−1r−1

i r′ = r−1r′

is not in H , and therefore r−1i r′ and r−1

i r are in different left cosets ofCH(hi) in CG(hi). To prove surjectivity, let g ∈ CG(hi). Write

rig = rh

with r ∈ R and h ∈ H . Then

r−1gr = hg−1ri−1grigh

−1 = hg−1high−1 = hhih

−1.

Therefore, r is in Ri, and it follows from the identity (3) that

r−1gr = hi.

Thus r−1i r = gh is in the same left coset of CH(hi) in CG(hi) as g is. �

Let αi denote the composition

CH(hi) → CG(hi)→CG(g),

where the second map is conjugation by r−1i . Recall that as represen-

tation of CG(g),

(ind |

Λ(G)Λ(H)π

)(g) =

l⊕

i=1

ind |αiπ(hi).(4)

Lemma 7.7. As a representation of CG,

χind(g) ∼=

l⊕

i=1

ind |αiTr(ρ(hi)).

Proof. Let f ∈ CG(g), and let r ∈ Ri. Write

fr = rh,

with r ∈ R and h ∈ H . We claim that r is also in Ri and that h is inCH(hi). This follows from

r−1gr = hr−1f−1gfrh−1 = hr−1grh−1 = hhih−1 = hi

as in the proof of Lemma 7.6. We are now ready to compute the blockentry corresponding to (r, r) of

indρ(f−1) ◦1 indρ(g) ◦1 indρ(f) :

22

fr = rh gives (indρ(f))rr = ρ(h),

gr = rhi gives (indρ(g))rr = ρ(hi),

f−1r = rh−1 gives (indρ(f−1))rr = ρ(h−1)

and all other block entries in these rows and columns are zero. Thus

(5)(indρ(f

−1) ◦1 indρ(g) ◦1 indρ(f))rr= ρ(h−1) ◦1 ρ(hi) ◦1 ρ(h),

and the 2-morphism from (5) to

(indρ(g))rr = ρ(hi)

is the 2-morphism

ρ(h−1) ◦1 ρ(hi) ◦1 ρ(h) ⇒ ρ(hi)

corresponding to h−1hih = hi. This proves that the action of CG(g) onχind(g) decomposes into actions on

⊕

r∈Ri

ρ(hi).

More precisely, if fr = rh, then f maps the summand correspondingto r to the one corresponding to r by

h: ρ(hi) → ρ(hi).

But

fr = rh ⇐⇒ (ri−1fri)(r

−1i r) = (r−1

i r)h,

and the action of

ri−1fri ∈ CG(hi)

on

ind |CG(hi)CH (hi)

ρ(hi)

is given by

h: r−1i rρ(hi) → r−1

i rρ(hi).

�

This completes the proof of Theorem 7.4. �

23

8. Comparison with Hopkins-Kuhn-Ravenel character

theory

8.1. Inertia orbifolds. Our 2-character map

(6) Tr: 2 Rep(G) → Rep(Λ(G))

should be compared to the orbifold Chern character map defined byAdem-Ruan and interpreted by Moerdijk [Moe02, p. 18] as a map

(7) K•(M)⊗ C →∏

i

H2i+•(Λ(M),C),

where M is an orbifold with a compact quotient space and Λ(M) isits inertia orbifold. More precisely, if M = •//G, then K(M) is theGrothendieck group of Rep(G). This suggests that 6 has a generaliza-tion for an arbitrary orbifold M as above, yielding a transformation

2Korb(M) → Korb(Λ(M)).

Here 2Korb(M) is a (yet to be defined) orbifold/equivariant version ofthe 2-vector bundleK-theory of Baas et al. Recall that the non-orbifold2K is interpreted as some approximation to the elliptic cohomology.Therefore the orbifold version is to be regarded as a geometric versionof equivariant elliptic cohomology, thus making more precise our pointin the introduction.Note that inertia orbifolds also turn up in the original paper [HKR00],

where working at chromatic level n requires using n-fold iterated inertiaorbifolds Λn(M).

8.2. The Todd genus of Xg. Let G act on a compact d-dimensionalcomplex manifold X . For g ∈ G, the fixed point locus Xg is then acompact complex submanifold. Consider the graded vector spaces

T(g) := H•(Xg,O),

where O is the sheaf of holomorphic functions on Xg. Clearly, theseare conjugation equivariant, i.e., they form a representation of Λ(G).Its character is a 2-class function

χX(g, h) = Trace(h|T(g)).

This function takes values in the cyclotomic ring Z[ζN ], where ζN isan N th root of 1 and N is the order of G. Let [X ] ∈ MU(BG) denotethe image of the equivariant cobordism class of X . Fix a prime p, andlet E = E2 be the second Morava E-theory at p. Recall that E comeswith a canonical natural transformation of cohomology theories

φ: MU∗(−) → E∗(−).24

Let now G be finite. In this situation Hopkins, Kuhn and Ravenelconstructed a map

a: E∗(BG) → 2Cl(G,D),

for a certain ring D, see [HKR00].Question 1: Is there a natural 2-representation in some category of

sheaves associated to X whose categorical character is the class sheafT?Question 2: What is the relationship between a(φ[X ])(g, h) and

χX(g, h)?

References

[BDR04] N. A. Baas, B. I. Dundas, and J. Rognes. Two-vector bundles and formsof elliptic cohomology. In U. Tillmann, editor, Proceedings of the 2002 Ox-ford Symposium in Honour of the 60th Birthday of Graeme Segal. Cam-bridge University Press, 2004.

[Ber84] J. N. Bernstein. Le “centre” de Bernstein. In Representations of reductivegroups over a local field, Travaux en Cours, pages 1–32. Hermann, Paris,1984. Edited by P. Deligne.

[BW] Bruce Bartlett and Simon Willerton. Work in progress.[Dev96] Jorge A. Devoto. Equivariant elliptic homology and finite groups. Michi-

gan Math. J., 43(1):3–32, 1996.[Elga] Josep Elgueta. A strict totally coordinatized version of Kapranov and

Voevodsky’s 2-category 2Vect. arXiv: math.CT/0406475.[Elgb] Josep Elgueta. Representation theory of 2-groups on Kapranaov and Vo-

evodsky’s 2-vector spaces. arXiv: math.CT/0408120.[GM03] Sergei I. Gelfand and Yuri I. Manin. Methods of homological algebra.

Springer Monographs in Mathematics. Springer-Verlag, Berlin, secondedition, 2003.

[HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Gen-eralized group characters and complex oriented cohomology theories. J.Amer. Math. Soc., 13(3):553–594 (electronic), 2000.

[Kap95] M. M. Kapranov. Analogies between the Langlands correspondence andtopological quantum field theory. In Functional analysis on the eve of the21st century, Vol. 1 (New Brunswick, NJ, 1993), volume 131 of Progr.Math., pages 119–151. Birkhauser Boston, Boston, MA, 1995.

[Kel82] Gregory Maxwell Kelly. Basic concepts of enriched category theory, vol-ume 64 of London Mathematical Society Lecture Note Series. CambridgeUniversity Press, Cambridge, 1982.

[KS94] Masaki Kashiwara and Pierre Schapira. Sheaves on manifolds, volume292 of Grundlehren der Mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences]. Springer-Verlag, Berlin, 1994. Witha chapter in French by Christian Houzel, Corrected reprint of the 1990original.

25

[KV94] M. M. Kapranov and V. A. Voevodsky. 2-categories and Zamolodchikovtetrahedra equations. In Algebraic groups and their generalizations: quan-tum and infinite-dimensional methods (University Park, PA, 1991), vol-ume 56 of Proc. Sympos. Pure Math., pages 177–259. Amer. Math. Soc.,Providence, RI, 1994.

[ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5of Graduate Texts in Mathematics. Springer-Verlag, New York, secondedition, 1998.

[Moe02] Ieke Moerdijk. Orbifolds as groupoids: an introduction. In Orbifolds inmathematics and physics (Madison, WI, 2001), volume 310 of Contemp.Math., pages 205–222. Amer. Math. Soc., Providence, RI, 2002.

[Nee01] Amnon Neeman. Triangulated categories, volume 148 of Annals of Math-ematics Studies. Princeton University Press, Princeton, NJ, 2001.

[Ost01] Viktor Ostrik. Module categories, weak Hopf algebras and modular in-variants. arXiv:math.QA/0111139, 2001.

[Rav92] Douglas C. Ravenel. Nilpotence and periodicity in stable homotopy theory,volume 128 of Annals of Mathematics Studies. Princeton University Press,Princeton, NJ, 1992. Appendix C by Jeff Smith.

[Ser77] Jean-Pierre Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977. Translated from the second French edition byLeonard L. Scott, Graduate Texts in Mathematics, Vol. 42.

[TS04] P. Teichner and S. Stolz. What is an elliptic object? In U. Tillmann,editor, Proceedings of the 2002 Oxford Symposium in Honour of the 60thBirthday of Graeme Segal. Cambridge University Press, 2004.

26