tetrapharmakon.github.io/stuff/repbeamer.pdf The algebra and geometry of categorical groupsThe algebra and geometry of categorical groups FoscoLoregian June27,2013 (sissa) The algebra

The algebra and geometryof categorical groups

Fosco Loregian

June 27, 2013

(sissa) The algebra and geometry of categorical groups June 27, 2013 1 / 34

Algebra


Sketches of a Group

Categorical groups come in various (equivalent) flavours:Groups internal to Gpd (a category with a bifunctor ⋅∶G ×G→ G which turnsit into a group in a way –the Eckmann-Hilton relation– which is compatiblewith composition: (f g) ⋅ (h k) = (f ⋅ h) (g ⋅ k));Crossed modules (=categories internal to Grp, i.e. pairs of groups Gob,Garwith source-target maps Gar Gob, a composition Gar ×Gob

Gar → Gar . . . );

2-groupoids with a single object: f%%

99 g ;

. . .A crossed module consists of a pair of groups G ,E , an action of G on E and amorphism ∂∶E → G such that (Peiffer conditions)

∂(g .e) = g∂eg−1;∂e.f = efe−1.

There is an obvious notion of morphism of crossed modules.


All these different approaches are equivalent: for example

Theorem (Verdier)

There is an equivalence of categories between groups internal to Gpd and crossedmodules.

See [Fo] for a detailed and modern proof.

Sketch of one side of the proof: given a group in Gpd we can define the crossedmodule having G = Ob(G), E =∐X∈Ob(G)G(1,X ), ∂ = t (the targetmap)∶ (e ∶1→ X )↦ X .

Definition (Notation)

We will denote GE ,G the categorical group associated to the crossed module(E ,G , a, ∂), and EG, GG the (two groups of the) crossed module associated to acategorical group G.


Examples

The delooping of an abelian1 group G , i.e. the category BG having a singleobject ∗ and such that BG(∗,∗) = G . Notice that for an abelian group K ,E(BK) = K , G(BK) = 1.The categorical group associated to a category C, having as objects theinvertible endofunctors and as arrows the invertible natural isomorphismsbetween such functors. If C = BK , then the corresponding c.m. isK → Aut(K)∶g ↦ g(−)g−1. There is a functor Cat→ CatGrp which sends acategory to its associated c.g.

Definition (Notation)

A c.g. can be delooped tooa: given G its delooping is the 2-category BG with asingle object ∗ and such that BG(∗,∗) is the category G. Composition is definedas the group operation in (G, ⋅) at the level of 1-cells, and as the composition inthe category G at the level of 2-cells.

aIt is a common procedure in monoidal category theory working in full generality.

1If the interchange law has to be true, we must consider only abelian groups.(sissa) The algebra and geometry of categorical groups June 27, 2013 5 / 34

A categorical group is said to befree if there is at most one arrow between any two distinct objects(so that ∂∶E(G)→ G(G) is mono and E embeds in G as anormal subgroup);transitive if there is an arrow between any two objects (so that∂∶E(G)→ G(G) is epi).intransitive if there are no arrows between any two distinct objects(so that ∂∶E(G)→ G(G) is the trivial map). Example: thewreath product C× ≀ Sym(n).

Recall a couple of notions from 2- and 3-category theory: a 2, 3-category consistsof objects (0-cells), arrows (1-cells) and

2-cells (Example: natural transformations in Cat);3-cells (Example: modifications between natural transformations in thecategory of all 2-categories);composition of 2- and 3-cells.


A natural 2-transformation, between two 2-functors F ,G ∶C→ D consists of a maph∶Ob(C)→Mor(D) such that there is a commutative diagram

C(X ,Y ) //

D(GX ,GY )

C(FX ,FY ) // D(FX ,GY )h(X ,Y )

≅7?

where h(X ,Y ) is a natural iso h(Y ) F (f )⇒ G(f ) h(X ) such thatfor any f ∶X → Y , g ∶Z → X , (h(f ) ∶= h(X ,Y ), h(g) ∶= h(Z ,X ) for short),the following commutes:

G(f ) h(X ) F (g) G(f ) h(X ) F (g) h(X )

h(1X )=1h(X)

h(Y ) F (f ) F (g)h(f )∗F(g)

KS

G(f ) G(g) h(Z)G(f )∗h(g)KS

h(Y ) F (fg)h(fg)

+3 G(fg) h(Z) h(X )


A modification Θ∶h k between two natural 2-transformation(h, h), (k, k)∶F → G , where both F ,G are functors C→ D, consists of a functionΘ assigning to any X ∈ Ob(C) a 2-cell ΘX ∶h(X )⇒ k(X ), in such a way that

G(f ) h(X )G(f )∗ΘA +3 G(f ) k(X )

h(Y ) F (f )

h(f )

KS

ΘB∗F(f )+3 k(Y ) F (f )

k(f )

KS

(hor) 2-cells can be composed: AF&&

G

88 h BU%%

V

99 k C goes to k ⊟ h∶ AUF&&

V G

88 C ;

(ver) 2-cells can be composed: A

F

hEE

H

kG // B goes to k q h∶ A

F&&

H

88 B ;

3-cells Θ,Ψ∶h k compose similarly horizontally and vertically.(sissa) The algebra and geometry of categorical groups June 27, 2013 8 / 34

The Kapranov-Voevodsky category 2-Vect

The category 2-Vect of KV 2-vector spaces hasas objects the set N of (nonzero) natural numbers;as 1-cells A∶N →M, all N ×M matrices with natural coefficients,A = (aij) ∈ NN×M ;

as 2-cells NA&&

B88 Θ B all N ×N matrices of matrices whose

(i , j)-entry is an aij × bij matrix.Example: 2,3 are objects of 2-Vect; a 1-morphism 2→ 3 consists ofan integer valued 2 × 3 matrix, e.g. A = ( 1 2 3

2 1 1 ) or B = ( 1 1 22 2 3 ); a

2-morphism between A and B is, for example, a matrix like

Θ =⎛⎝

a ( b1b2

) ( c11 c12c21 c22c31 c32

)( d11 d12

d21 d22) ( e1 e2 ) ( f1 f2 f3 )

⎞⎠


The category of 2-vector spaces has

An horizontal composition between 2-cells,

NA&&

B88 Θ M

C%%

D

99 Ξ P NCA

))

DB

55 Ξ⊟Θ P where CA,DB are obtained via

product of integer matrices, and Ξ ⊟Θ is the complex matrix whose(i , j)-entry has size ∑ aikckj ×∑bihdhj , hence it is obtained as ⊕M

k=1 Ξij ⊗Θkj .

A vertical composition N

A

ΘDD

C

ΞB // M N

A))

C

55 ΞqΘ M , where

(Ξ qΘ)ij = ΞijΘij (mult. of matrices). It makes sense, because Ξij is aaij × bij -matrix, and Θij is a bij × cij -matrix.

Verify the interchange law:

(Ω ⊟Ψ) q (Ξ ⊟Θ) = (Ξ qΩ) ⊟ (Ψ qΘ)


N

Θ

A // M

Ξ

C // P N

Ξ⊟Θ

CA // P

N

Ψ

B// M

Ω

D// P z→ N

Ω⊟Ψ

DB// P

NE// M

F// P N

FE// P

N

ΨqΘ

A // M

ΩqΞ

C // P N CA //

(Ω⊟Ψ)q(Ξ⊟Θ),(ΞqΩ)⊟(ΨqΘ)

P

z→

NE// M

F// P N

FE// P


The category of 2-vector spaces has

A monoidal product ⊠, such thatN ⊠M is the product of natural numbers;Given A∶N →M, B ∶R → S , A ⊠B ∶NR →MS , where (A ⊠B)

ijkl = aikbjl .

Given Θ,Ψ∶A→ B, (Θ ⊠Ψ)ijkl = Θik ⊗Ψjl (tensor product of matrices).

A monoidal sum such that N ⊞M is the sum of natural numbers, A⊞B is thedirect sum of matrices, and Θ ⊞Ψ is the componentwise direct sum ofmatrices.A direct sum in any 2-Vect(N,M), where (A⊕B)ij = (aij + bij),componentwise sum of integer matrices, and (Θ⊕Ψ)ij = Θij ⊕Ψij , direct sumof complex matrices.A sum, which defines the linear structure on each 2-Vect(N,M), since(Θ +Ψ)ij is obtained summing the matrices Θij ,Ψij (they have the same sizeaij × bij).

Verify: maybe (2-Vect,⊠,⊞) is a rig-category?


Categorical Representations

Definition

We define GL(N) ⊂ 2-Vect(N,N) to be the categorical group associated to thecategory 2-Vect(N,N). It is called the general 2-linear categorical group ofdimension N.

Remark

The crossed module associated to GL(N) is the wreath product C× ≀ Sym(N),where

The map ∂∶ (C×)N → Sym(N) is the trivial one sending everything to id ;the action of Sym(N) on (C×)N permutes the coordinates of a vector.

A (linear) representation of a group G can be regarded as a functor from thedelooping of G , i.e. ρ∶BG → Vect; hence we are led to define a linear2-representation of a categorical group G as a 2-functor R ∶BG→ 2-Vect.

The integer R(∗) = N is called the dimension of the representation.


The category Rep2-Vect(G) = Fun(BG,2-Vect) is a 2-category where 1-cells arecalled intertwiners between representations, and 2-cells are called 2-intertwiners:these are, respectively, natural transformations between representationR,T ∶BG→ 2-Vect, and modification between natural transformationsh, k ∶R ⇒ T .

Rep2-Vect(G) becomes a monoidal categorywith respect to a monoidal product of representations, obtainedcomponentwise as (R, R), (T , T )↦ R ⊠T (X ) = R(X ) ⊠T (X ), with asuitable coherer R ⊠T .With respect to a monoidal sum of representations obtained componentwiseas (R, R), (T , T )↦ R ⊞T (X ) = R(X ) ⊞T (X ) (the coherer of this monoidalstructure is trivial).

Both monoidal structure extend by (2-)functoriality to 1- and 2-intertwiners,following the rules of the former definition for ⊠ and ⊞.


Strict representations

We can now turn on the study of strict 2-representations of categorical groups:these are strict functors ρ∶BG→ 2-Vect, which can be regarded as morphisms ofc.g.

ρ∶G→ GL(ρ(∗)) = GL(N)

hence equivalently as morphisms of crossed modules EG //

ρt

GGρb

(C×)N1// Sym(N)

.

1-intertwiners between strict representations ρN , σM can be described as arrowsh∗ ∈ 2-Vect(N,M) with h∶X ↦ h(X ) ∈ 2-Vect(h∗ ρ(X ), σ(X ) h∗), subject tcertain coherence conditions:

h∗ρ(XY )h(XY ) +3 σ(XY )h∗ h∗ρ(1) = h∗

1h∗

h∗ρ(X )ρ(Y )h(X)ρ(Y )

+3 σ(X )h∗ρ(Y )σ(X)h(Y )

+3 σ(X )σ(Y )h∗ σ(1)h∗ = h∗


Strict representations

We can now turn on the study of strict 2-representations of categorical groups:these are strict functors ρ∶BG→ 2-Vect, which can be regarded as morphisms ofc.g.

ρ∶G→ GL(ρ(∗)) = GL(N)

hence equivalently as morphisms of crossed modules EG //

ρt

GGρb

(C×)N1// Sym(N)

.

1-intertwiners between strict representations ρN , σM can be described as arrowsh∗ ∈ 2-Vect(N,M) with h∶X ↦ h(X ) ∈ 2-Vect(h∗ ρ(X ), σ(X ) h∗), subject tcertain coherence conditions:

X

f

ρ(X )h∗h(X) +3

ρ(f )∗h∗

h∗σ(X )

h∗∗σ(f )

Y ρ(Y )h∗h(Y )

+3 h∗σ(Y )


Example: G(3, 2)

Let (C3,C2, ∂, a) the crossed module where C2 = ±1, C3 = 1, x , x−1,∂∶C3 → C2 is the trivial map and the action a∶C2 C3 coincides with theinversion x ↦ x−1. Let G(3,2) be the associated categorical group.

We can classify all the 1- and 2-dimensional 2-representations of G(3,2):

V(1) Trivial 1-dimensional representation: C3ρt

∂ // C2ρb

C×

1// 1

There are no other 1-dimensional representations, since ρt is determined by acharacter of C3, which is forced to be the trivial one by the commutativitycondition.

V(2) Trivial 2-dimensional representation: C3

ρt

∂ // C2

ρb

(C×)21// C2

: V(2) ≅ V(1) ⊞V(1).


All the other 2-dimensional representations of G(3,2) are classified by the ring ofcharacters of C3: ⎧⎪⎪⎨⎪⎪⎩

ρb(±1) = ±1 ∈ C2

ρt(x) = (ξ(x), ψ(x))

where ξ,ψ are complex characters of C3: but now ξ = ψ−1, by the commutativityrelation for (ρt , ρb) to be a morphism of crossed modules. So ρ is completelydetermined by ξ.

One can also classify all the 1- and 2-intertwiners between these representations:it is done in [Mac] pp. 21-25.Hao about the structure of monoidal bicategory of Rep(G(2,3))? As an example:the trivial 1-dimensional character/representation acts as a neutral element for ⊠.


Characters

A character of a categorical grup G is a 1-dimensional2-representation ρ∶G→ GL(1).

These representations are completely classified by a complex characterξρ of the group EG of the associated crossed module, EG C×. Anintertwiner (h, h) between two such characters is either zero or it iscompletely determined by a representation h of the group GG, actingas

h(X ) = ξ)ρ(e)−1ξσ(e)where e is such that ∂e = X ∈ GG.A 2-intertwiner Θ∶ (h, h)→ (k , k) consists simply of a classicalintertwiner between the two representation h and k .


Geometry


Classical Čech theory

Suppose M is a manifold, and G a topological group;

For any covering Uii∈I = U of M we can organize isomorphism classes of principalG -bundles in a set H1

(U,G),

which is in bijection with

1-cocycles

gij ∶Ui∩Uj→G /being

cohomologus

where “being cohomologous” for two cocycles gij ,g ′ij (i.e. functions satisfying thecocycle condition for any i , j , k ∈ I ) means that there is a family of functions fisuch that gij fj = fig ′ij .

If M can be “nicely” covered, then H1(U,G) = H1

(M,G) does not depend on thechoice of a good cover U; otherwise one obtains H1

(M,G) as a limit limÐ→U

H1(U,G)

over all covers U ∈ cov(M) (notice that good covers are cofinal in this family).

(Brown) The functor H1(−,G) is representable in the homotopy category of spaces,

and its representing objects is the classifying space of G : H1(M,G) ≅ [M,BG].


Can one obtain a similar theory for (whatever they are) principaleG-2-bundles for a (topological) categorical group G?

First of all the basic theory extends verbatim adding everywhere the word“topological”:

A topological categorical group is an internal group in TopGpd, the categoryof toplogical groupoids;A topological crossed module consists of an internal category in TopGrp, thecategory of topological groups;Verdier’s theorem applies as well: (topological) crossed modules areequivalent to (topological) categorical groups.However a new equivalent characterization of a c.g. can be used to betterunderstand the topological behaviour of such things. . .

(Topological) categorical groups are (topological) 2-groupoids with a single object.


Let G be a c.g., (EG,GG) be its associated crossed module. As we have seenbefore, G can be delooped to a 2-category with a single object ∗, where

1-cells are objects of G and an arrow g → g ′ is h ∈ E such that ∂(h)g = g ′.Composition of 1-cells amounts to the monoidal product, and inversionequals dualization,horizontal and vertical composition are obtained thanks to Peiffer relationsbetween ∂∶E → G and a∶G E , and the interchange law holds because ∂and a are suitably compatible.

g$$

∂(h)g

:: h f$$

∂(k)f

:: k $$

:: EE

// $$

::

This category is easily seen to be a groupoid!


Segal fundamental Lemma

Čech cocycles are functors!

Let Ui = U be a covering of the base space M; then we can define agroupoid U having objects ∐i∈I Ui , and a unique arrow (x , i)→ (x , j)iff x ∈ Ui ∩Uj ; define a functor g ∶ U→ G ∶ (x , i)↦ ∗ (the unique objectof G regarded as a category), and sending an arrow x ∈ Uij to anelement gij of G ;

cocyclesgij

←→ functorsg ∶U→G

Functoriality conditions for g are precisely cocycle conditions forgij;two cocycles are cohomologous iff their corresponding functors arenaturally isomorphic.


We can categorify Segal’s fundamental Lemma:A Čech cocycle for a categorical group consists in this setting of apseudofunctor g ∶ U→ G, where U is the groupoid of the Segal’s constructionregarded as a 2-category having only identity 2-cells, and G a 2-groupoidwith a single object .

Pseudofunctoriality now translates into some coherence conditions having theseshapes: a suitable tetrahedron for the cocycle cond.,

gkl

gjk

gik

gklgik =

gil gil

hijl

hijk" hjkl

4<hikl

gjl ⋅hijk +3

hjkl ⋅gik

hijl

hikl

+3


and the prism

x

nm

fe

g

y

p

z

α

β

δ

em

γ

V^

which expresses the compatibility between cohomologus cocycles/isomorphicfunctors U→ G,


and is equivalent to

nm

p .

ζ

with the additional coherence condition

znmγ⋅m +3

z ⋅ζ

fxmf ⋅β +3 fey

α⋅y

zpδ

+3 gy


These conditions are exactly the coherence conditions which in [Ba]define principal G-2-bundles over a manifold M:

Let (E ,G , ∂, a) be the crossed module associated to G, andUi = U a cover of M.Then a cocycle subordinated to U consists of maps gij ∶Uij → Gsuch that

A weak cocycle condition is satisfied:

∂(hijk)gijgjk = gik

for some hijk ∶Ui ∩Uj ∩Uk → H.Similarly, we say two weak cocycles (gij ,hijl) and (g ′ij ,h′ijl) arecohomologous if there is a family of maps fi ∶Ui → G ,kij ∶Ui ∩Uj → H such that

∂(kij)gij fj = fig ′ij .


“Being cohomologous” is now an equivalence relation on the setZ(U,G) of cocycles subordinated to U, for the categorical group G;we define H1(U,G) to be the quotient of Z(U,G) by this relation,and

H1(M,G) ∶= limÐ→U∈cov(M)

H1(U,G).

Main Theorem: Let G a well pointed2 topological c.g. and M amanifold. Then

H1(M,G) ≅ [M,B∣G∣]where B∣G∣ is the classifying space of the topological group ∣G∣,defined as the geometric realization of the nerve of the category G.

2This is a technical condition for a topological c.g. in which the groups EG,GG have thehomotopy type of a cw-complex. A topological group, or a pointed space, is well-pointed if1G→ G is a closed cofibration; a c.g. is well pointed, by definition, if EG,GG are well pointed.


The nerve-realization paradigm

Theorem (General nonsense adjunction)

Given a functor F ∶C→ D to a cocomplete category, it is always possible to exhibit anextension of F to the category C of presheaves of sets on C, F = LanyF ∶ C→ D (this issort of a universal property of C). Moreover this functor has a right adjoint S .

Apply the previous theorem to the following two cases:

∆ //i //

y

Cat

N~~

∆

>> ∆ρ//

y

Top

~~∆

R>>

⊳ i ∶∆→ Cat regards the totally ordered set 0 < 1 < ⋅ ⋅ ⋅ < n as a category;⊳ y is the Yoneda embedding ∆ ∆∶ [n]↦∆(−, [n]);⊳ ρ∶∆→ Top “realizes” each 0 < 1 < ⋅ ⋅ ⋅ < n as the standard n-simplex embedded inRn+1.

Cat // ∆ // Top

G // N(G) // R(N(G)) = ∣G∣


The proof of the Main Theorem exploits the following three lemma:1. Let G a topological well pointed c.g.; then

∣G∣ is (a topological group and) again well pointed;There exist a c.g. G and a short exact sequence of topological groups1→ H → ∣G∣→ ∣G∣→ 1 where H is the domain of ∂ in the crossed moduleassociated to G;G ≅ G ⋉ EH, where EH is the universal principal H-bundle over BH(corresponds to 1BH in [BH,BH] ≅ H1

(BH,H)).

2. Any short exact sequence σ∶1→ H⊴Ð→ G → K → 1 gives a crossed module

Gσ = (G ,H) where the action of G is by conjugation and ∂ is the inclusionH G ; then there is an isomorphism

H1(M,G(G ,H)) ∶= H1(M,Gσ) ≅ H1(M,K)

3. Let 1→ G→ H→ K→ 1 be a s.e.s. of categorical groups (it means that theunderlying s.e.s. of topological groups obtained via associated crossedmodules are exact): then there is a short exact sequence of pointed sets

H1(M,G)→ H1(M,H)→ H1(M,K)


We know that there exists an isomorphims in classical Čech theory

H1(M, ∣G∣) ≅ [M,B∣G∣]hence to prove the Theorem it suffices to show that H1(M,G) ≅ H1(M, ∣G∣).Now Lemma 1 entails that there exists a short exact sequence σ

1→ H → G ⋉ EH → ∣G∣→ 1

and Lemma 2 entails that H1(M,Gσ) ≅ H1(M, ∣G∣). So to build the desiredisomorphism it suffices to show that

H1(M,Gσ) ≅ H1(M,G)Now consider the short exact sequence of crossed modules

1 // 1

// H

// H

// 1

(∮ )1 // EH // G ⋉ EH // G // 1

Lemma 3 entails that H1(M,EH)→ H1(M,Gσ)→ H1(M,G) is exact. But

H1(M,EH) is zero, because EH is contractible, and the sequence ∮ is splitexact.


John C. Baez, Danny Stevenson, The Classifying Space of a Topological2-Group, http://arxiv.org/abs/0801.3843.

Toby Bartels, Higher gauge theory I: 2-Bundles,http://arxiv.org/abs/math/0410328.

John W. Barrett, Marco Mackaay, Categorical representations of categoricalgroups, http://arxiv.org/abs/math/0407463.

. Forrester-Barker, Group objects and internal categories,http://arXiv.org/abs/math/0212065v1.


http://arxiv.org/abs/0801.3843

http://arxiv.org/abs/math/0410328

http://arxiv.org/abs/math/0407463

http://arXiv.org/abs/math/0212065v1

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tetrapharmakon.github.io/stuff/repbeamer.pdf The algebra and geometry of categorical groupsThe algebra and geometry of categorical groups FoscoLoregian June27,2013 (sissa) The algebra