Categorical tori and their representations A report on work in progress Nora Ganter Workshop on Infinite-dimensional Structures in Higher Geometry and Representation Theory Hamburg, February 2015

Crossed modules and categorical groups following Noohi (Strict) categorical groups Crossed modules are (strict) monoidal groupoids consist of a group G , a right G - module A and a homomorphism G 1 ψ : A − → G with s t ψ ( a g ) g − 1 ψ ( a ) g = G 0 a − 1 ba . ψ ( a ) · b = with invertible objects (w.r.t. • ). A crossed module ( G , A , ψ ) en- The crossed module of the cate- gorical group G above is codes a strict categorical group G = G 0 G ⋉ A = ker ( s ) pr 1 A pr 1 · ψ g − 1 • a • g a g = G ψ = t . group multiplication gives • and ( g ψ ( b ) , a ) ◦ ( g , b ) = ( g , ab ).

Example: the crossed module of a categorical torus ∨ and a bilinear form J on Λ ∨ . From Two ingredients: A lattice Λ this, we form the crossed module ψ ∨ × U (1) ∨ ⊗ Z R Λ t := Λ m , ( m , z ) ∨ × U (1) is given by where the action of x ∈ t on Λ ( m , z ) x = ( m , z · exp( J ( m , x ))) .

Categorical tori The categorical torus T is the strict monoidal category with objects: t , z ∨ , z ∈ U (1) , arrows: x − − → x + m , x ∈ t , m ∈ Λ composition: the obvious one, multiplication: addition on objects and on arrows zw exp( J ( m , y )) z w ( x x + m ) • ( y y + n ) = ( x + y x + y + m + n ) .

Classification Schommer-Pries, Wagemann-Wockel, Carey-Johnson-Murray-Stevenson-Wang Up to equivalence, the categorical torus T only depends on the even symmetric bilinear form I ( m , n ) = J ( m , n ) + J ( n , m ) . More precisely, ∨ , Z ) S 2 = H 4 ( BT ; Z ) ∼ = H 3 − I ∈ Bil ev (Λ gp ( T ; U (1)) classifies the equivalence class of the extension pt / / U (1) − → T − → T . Examples: 1. T max ⊂ G maximal torus of a simple and simply connected ∨ coroot lattice, I bas basic bilinear form, compact Lie group, Λ 2. (Λ Leech , I ) or another Niemeyer lattice.

Aussie-rules Lie group cohomology H 3 H 3 ( BT • ; U (1)) ˇ gp ( T ; U (1)) = and − I corresponds to the ˇ Cech-simplicial 3-cocycle T × T × T 1 d = 3 T × T exp( − J ( m , y )) d = 2 T 1 d = 1 ∨ × Λ ∨ ) d ∨ ) d t d ( t × Λ ( t × Λ where the non-trivial entry is short for (( x , m ) , ( y , n )) �− → exp( − J ( m , y )) .

Autoequivalences of the category of coherent sheaves � T = Hom ( T , U (1)) spec C [ � T C = T ] C [ � T ] − mod fin C ohT C ≃ Bimod fin 1 Aut ( C ohT C ) ≃ [Deligne]. C [ � T ] Inside 1 Aut ( C oh ( T C )), we have the full subcategory of direct image functors f ∗ of variety automorphisms f . This categorical group belongs to the crossed module 1 C [ � T ] × Aut var ( T C ), T ] × by precomposition, ϕ �→ ϕ ◦ f . where f acts on C [ �

The basic representation of a categorical torus The basic representation of T is the strict monoidal functor ̺ bas : T 1 Aut ( C oh ( T C )). induced by the map of crossed modules ( m , z ) z · e 2 π iJ ( m , − ) ∨ × U (1) C [ � Λ T ] × r bas : ψ 1 Aut var ( T C ) t mult exp( x ) . x

The involution ι The involution ι of T , sending t to t − 1 lifts to an involution of T , given by the map of crossed modules ( m , z ) ( − m , z ) ∨ × U (1) ∨ × U (1) Λ Λ ι : ψ 1 t t x − x . This gives rise to an action of the group {± 1 } by (strict monoidal) functors on the category T .

Extraspecial categorical 2-groups The fixed points of ι on T form the elementary abelian 2-group T {± 1 } = T [2] ∼ ∨ / 2Λ ∨ . = Λ The categorical fixed points (or equivariant objects) of ι on T form an extension → T {± 1 } − → � pt / / U (1) − T [2] of the extraspecial 2-group � T [2] with Arf invariant φ ( m ) = 1 ∨ . 2 I ( m , m ) mod 2Λ Example: In the example of the Leech lattice, � T [2] is the subgroup of the Monster that is usually denoted 2 1+24 .

1Automorphisms of the basic representation Let T C ⋊ {± 1 } be the categorical group of the crossed module ∨ × C × t C ⋊ {± 1 } Λ ( m , z ) ( m , 1), where − 1 acts on everything by ι . Extend the basic representation to ̺ bas : T C ⋊ {± 1 } 1 Aut ( C oh ( T C )), by setting r bas ( − 1) := ι . So, ̺ bas ( − 1) := ι ∗ . Theorem: The 1automorphisms of this ̺ bas form the extraspecial categorical 2-group T {± 1 } . C

Normalizers Let ̺ : H − → G = GL ( V ) be a representation of a group H on some vector space. Then Aut ( ̺ ) = C ( ̺ ) = { g ∈ G | c g ◦ ̺ = ̺ } is the centralizer of (the image of) ̺ in G . Here c g is conjugation by g . Definition [Dror Farjoun, Segev]: The injective normalizer of ̺ is the subgroup of Aut ( H ) × G defined as N ( ̺ ) = { ( f , g ) | c g ◦ ̺ = ̺ ◦ f } . If ̺ is injective, this is the normalizer of its image.

Towards the refined Monster? (In progress) Theorem: The 1automorphisms of T form an extension ∨ , I ) . pt / / Λ − → 1 Aut ( T ) − → O (Λ ∨ , I ) is the group of linear isometries of (Λ ∨ , I ). Here O (Λ Example: the Conway group O (Λ ∨ Leech , I ) = Co 0 . In spirit, the subgroup of the Monster, known as � 2 1+24 . Co 1 = T [2] ⋊ ( Co 0 / {± id } ) wants to parametrize the isomorphism classes of some categorical variant of normalizer of ̺ bas : T C ⋊ {± 1 } − → 1 Aut ( C oh ( T C )).

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