Free Picard Categories Michael Horst The Ohio State University - PowerPoint PPT Presentation

Free Picard Categories Michael Horst The Ohio State University horst.59@osu.edu https://u.osu.edu/horst.59/ October 28, 2018 Michael Horst OSU

Picard Categories Michael Horst OSU

Picard Categories Groupoid Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Group-like Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Group-like: For all X , there is a Y such that X ⊗ Y ∼ = I ∼ = Y ⊗ X Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Group-like: For all X , there is a Y such that X ⊗ Y ∼ = I ∼ = Y ⊗ X Essential data: Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Group-like: For all X , there is a Y such that X ⊗ Y ∼ = I ∼ = Y ⊗ X Essential data: π 0 ( C ) = Obj ( C ) / ∼ = Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Group-like: For all X , there is a Y such that X ⊗ Y ∼ = I ∼ = Y ⊗ X Essential data: π 0 ( C ) = Obj ( C ) / ∼ = π 1 ( C ) = C ( I , I ) Michael Horst OSU

Picard Categories Groupoid Symmetric monoidal Group-like: For all X , there is a Y such that X ⊗ Y ∼ = I ∼ = Y ⊗ X Essential data: π 0 ( C ) = Obj ( C ) / ∼ = π 1 ( C ) = C ( I , I ) K : π 0 ( C ) → π 1 ( C ), X �→ β X , X ∈ C ( X ⊗ X , X ⊗ X ) ∼ = π 1 ( C ) Michael Horst OSU

Examples Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 2 Top Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 2 Top Z , “Super Integers” Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 2 Top Z , “Super Integers” Obj ( Z ) = Z Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 2 Top Z , “Super Integers” � Z / 2 , if n = m Obj ( Z ) = Z , Z ( n , m ) ∼ = 0 , else Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 2 Top Z , “Super Integers” � Z / 2 , if n = m Obj ( Z ) = Z , Z ( n , m ) ∼ = 0 , else Call Z ( n , n ) = {± 1 n } Michael Horst OSU

Examples ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 2 Top Z , “Super Integers” � Z / 2 , if n = m Obj ( Z ) = Z , Z ( n , m ) ∼ = 0 , else Call Z ( n , n ) = {± 1 n } ( β : n + m → m + n ) = ( − 1 n + m ) nm Michael Horst OSU

Main Result Michael Horst OSU

Main Result Theorem (H) The forgetful functor U : Pic → Grpd has a left adjoint given by Z [ ] : Grpd → Pic . Michael Horst OSU

Main Result Theorem (H) The forgetful functor U : Pic → Grpd has a left adjoint given by Z [ ] : Grpd → Pic . Specifically: For G ∈ Grpd and A ∈ Pic , Pic ( Z [ G ] , A ) ≃ Grpd ( G , A ) as Picard categories, natural in G and A . Michael Horst OSU

Free Picard Category Michael Horst OSU

Free Picard Category For G ∈ Grpd , define Z [ G ] ∈ Pic Michael Horst OSU

Free Picard Category For G ∈ Grpd , define Z [ G ] ∈ Pic k � Obj ( Z [ G ]): n i . G i , for n i ∈ Z and G i ∈ G i =1 Michael Horst OSU

Free Picard Category For G ∈ Grpd , define Z [ G ] ∈ Pic k � Obj ( Z [ G ]): n i . G i , for n i ∈ Z and G i ∈ G i =1 0 � 0 := i =1 Michael Horst OSU

Free Picard Category For G ∈ Grpd , define Z [ G ] ∈ Pic k � Obj ( Z [ G ]): n i . G i , for n i ∈ Z and G i ∈ G i =1 0 � 0 := i =1 Monoidal product: concatenation Michael Horst OSU

Free Picard Category Mor ( Z [ G ]): generated under + and ◦ by Michael Horst OSU

Free Picard Category Mor ( Z [ G ]): generated under + and ◦ by ± 1 n . g : nG → nG ′ , for g : G → G ′ ∈ G Michael Horst OSU

Free Picard Category Mor ( Z [ G ]): generated under + and ◦ by ± 1 n . g : nG → nG ′ , for g : G → G ′ ∈ G β : nG + n ′ G ′ → n ′ G ′ + nG Michael Horst OSU

Free Picard Category Mor ( Z [ G ]): generated under + and ◦ by ± 1 n . g : nG → nG ′ , for g : G → G ′ ∈ G β : nG + n ′ G ′ → n ′ G ′ + nG δ : ( n + Z n ′ ) G → nG + n ′ G Michael Horst OSU

Free Picard Category Mor ( Z [ G ]): generated under + and ◦ by ± 1 n . g : nG → nG ′ , for g : G → G ′ ∈ G β : nG + n ′ G ′ → n ′ G ′ + nG δ : ( n + Z n ′ ) G → nG + n ′ G ζ : 0 Z G → 0 Michael Horst OSU

Free Picard Category These morphisms subject to Michael Horst OSU

Free Picard Category These morphisms subject to ( f . g ) ◦ ( f ′ . g ′ ) = ( f ◦ f ′ ) . ( g ◦ g ′ ), for f , f ′ ∈ Mor ( Z ), g , g ′ ∈ Mor ( G ) Michael Horst OSU

Free Picard Category These morphisms subject to ( f . g ) ◦ ( f ′ . g ′ ) = ( f ◦ f ′ ) . ( g ◦ g ′ ), for f , f ′ ∈ Mor ( Z ), g , g ′ ∈ Mor ( G ) + is functorial with respect to ◦ Michael Horst OSU

Free Picard Category These morphisms subject to ( f . g ) ◦ ( f ′ . g ′ ) = ( f ◦ f ′ ) . ( g ◦ g ′ ), for f , f ′ ∈ Mor ( Z ), g , g ′ ∈ Mor ( G ) + is functorial with respect to ◦ Braided hexagon That β, δ, and ζ are monoidal natural β ◦ β = Id Michael Horst OSU

Free Picard Category These morphisms subject to ( f . g ) ◦ ( f ′ . g ′ ) = ( f ◦ f ′ ) . ( g ◦ g ′ ), for f , f ′ ∈ Mor ( Z ), g , g ′ ∈ Mor ( G ) + is functorial with respect to ◦ Braided hexagon That β, δ, and ζ are monoidal natural β ◦ β = Id ( n + Z n ′ + Z + n ′′ ) G δ ( n + Z n ′ ) G + n ′′ G δ δ +Id nG + ( n ′ + Z n ′′ ) G nG + n ′ G + n ′′ G Id+ δ Michael Horst OSU

Free Picard Category β Z G ( n ′ + Z n ) G ( n + Z n ′ ) G δ δ nG + n ′ G n ′ G + nG β Michael Horst OSU

Free Picard Category β Z G ( n ′ + Z n ) G ( n + Z n ′ ) G δ δ nG + n ′ G n ′ G + nG β δ (0 Z + Z n ) G 0 Z G + nG = ζ +Id nG 0 + nG = Michael Horst OSU

Free Picard Category β Z G ( n ′ + Z n ) G ( n + Z n ′ ) G δ δ nG + n ′ G n ′ G + nG β δ (0 Z + Z n ) G 0 Z G + nG = ζ +Id nG 0 + nG = Note: nG + ( − n ) G ∼ = ( n − n ) G = 0 Z G ∼ = 0 Michael Horst OSU

Proof Highlights Michael Horst OSU

Proof Highlights Grpd ( G , A ) ∋ F �→ F ∈ Pic ( Z [ G ] , A ) Michael Horst OSU

Proof Highlights Grpd ( G , A ) ∋ F �→ F ∈ Pic ( Z [ G ] , A ) F ( n . G ) = � | n | sgn( n ) F ( G ) Michael Horst OSU

Proof Highlights Grpd ( G , A ) ∋ F �→ F ∈ Pic ( Z [ G ] , A ) F ( n . G ) = � | n | sgn( n ) F ( G ) �� n i . G i + � n ′ = F ( � n i . G i ) + F �� n ′ � � j . G ′ j . G ′ F j j Michael Horst OSU

Proof Highlights Grpd ( G , A ) ∋ F �→ F ∈ Pic ( Z [ G ] , A ) F ( n . G ) = � | n | sgn( n ) F ( G ) �� n i . G i + � n ′ = F ( � n i . G i ) + F �� n ′ � � j . G ′ j . G ′ F j j F (1 n . g : n . G → n . G ′ ) = � | n | sgn( n ) F ( g ) Michael Horst OSU

Proof Highlights Grpd ( G , A ) ∋ F �→ F ∈ Pic ( Z [ G ] , A ) F ( n . G ) = � | n | sgn( n ) F ( G ) �� n i . G i + � n ′ = F ( � n i . G i ) + F �� n ′ � � j . G ′ j . G ′ F j j F (1 n . g : n . G → n . G ′ ) = � | n | sgn( n ) F ( g ) F ( − 1 1+ n . Id G ) = K F ( G ) + � | n | sgn( n )Id F ( G ) Michael Horst OSU

Proof Highlights Pic ( Z [ G ] , A ) ∋ F �→ u F ∈ Grpd ( G , A ) Michael Horst OSU

Proof Highlights Pic ( Z [ G ] , A ) ∋ F �→ u F ∈ Grpd ( G , A ) u F ( G ) = F (1 . G ) Michael Horst OSU

Proof Highlights Pic ( Z [ G ] , A ) ∋ F �→ u F ∈ Grpd ( G , A ) u F ( G ) = F (1 . G ) u F ( g : G → G ′ ) = F (1 1 . g ) Michael Horst OSU

Proof Highlights Pic ( Z [ G ] , A ) ∋ F �→ u F ∈ Grpd ( G , A ) u F ( G ) = F (1 . G ) u F ( g : G → G ′ ) = F (1 1 . g ) u ( F ) = F Michael Horst OSU

Proof Highlights Pic ( Z [ G ] , A ) ∋ F �→ u F ∈ Grpd ( G , A ) u F ( G ) = F (1 . G ) u F ( g : G → G ′ ) = F (1 1 . g ) u ( F ) = F �� � F ( n . G ) ∼ | n | sgn( n ) . G F ( δ ) F = Michael Horst OSU

Proof Highlights Pic ( Z [ G ] , A ) ∋ F �→ u F ∈ Grpd ( G , A ) u F ( G ) = F (1 . G ) u F ( g : G → G ′ ) = F (1 1 . g ) u ( F ) = F �� � F ( n . G ) ∼ ∼ | n | sgn( n ) . G = � | n | F (sgn( n ) . G ) F ( δ ) F = ∼ = � | n | sgn( n ) F (1 . G ) = u F ( n . G ) Michael Horst OSU

Group rings and the free module Michael Horst OSU

Group rings and the free module Conjecture For G ∈ Pic , Z [ G ] categorifies the group ring, in that Z [ ] CMon( Pic , ∗ ) Pic π 0 ⊣ ⊣ π 0 CMon( Ab , ⊗ ) Ab Z [ ] Michael Horst OSU