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:

π0(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:

π0(C ) = Obj(C ) / ∼ = π1(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:

π0(C ) = Obj(C ) / ∼ = π1(C ) = C (I, I) K : π0(C ) → π1(C ), X → βX,X ∈ C (X ⊗ X, X ⊗ X) ∼ = π1(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: π0(Pic(R)) = pic(R)

Michael Horst OSU

Examples

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω2Top

Michael Horst OSU

Examples

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω2Top Z, “Super Integers”

Michael Horst OSU

Examples

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω2Top Z, “Super Integers”

Obj(Z) = Z

Michael Horst OSU

Examples

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω2Top Z, “Super Integers”

Obj(Z) = Z, Z(n, m) ∼ =

• Z/2,

if n = m 0, else

Michael Horst OSU

Examples

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω2Top Z, “Super Integers”

Obj(Z) = Z, Z(n, m) ∼ =

• Z/2,

if n = m 0, else Call Z(n, n) = {±1n}

Michael Horst OSU

Examples

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω2Top Z, “Super Integers”

Obj(Z) = Z, Z(n, m) ∼ =

• Z/2,

if n = m 0, else Call Z(n, n) = {±1n} (β : n + m → m + n) = (−1n+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 Obj (Z[G ]):

k

• i=1

ni.Gi, for ni ∈ Z and Gi ∈ G

Michael Horst OSU

Free Picard Category

For G ∈ Grpd, define Z[G ] ∈ Pic Obj (Z[G ]):

k

• i=1

ni.Gi, for ni ∈ Z and Gi ∈ G

0 :=

• i=1

Michael Horst OSU

Free Picard Category

For G ∈ Grpd, define Z[G ] ∈ Pic Obj (Z[G ]):

k

• i=1

ni.Gi, for ni ∈ Z and Gi ∈ G

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

±1n.g : nG → nG ′, for g : G → G ′ ∈ G

Michael Horst OSU

Free Picard Category

Mor (Z[G ]): generated under + and ◦ by

±1n.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

±1n.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

±1n.g : nG → nG ′, for g : G → G ′ ∈ G β : nG + n′G ′ → n′G ′ + nG δ : (n +Z n′)G → nG + n′G ζ : 0ZG → 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 (n +Z n′)G + n′′G nG + (n′ +Z n′′)G nG + n′G + n′′G

δ δ δ+Id Id+δ

Michael Horst OSU

Free Picard Category

(n +Z n′)G (n′ +Z n)G nG + n′G n′G + nG

βZG δ δ β

Michael Horst OSU

Free Picard Category

(n +Z n′)G (n′ +Z n)G nG + n′G n′G + nG

βZG δ δ β

(0Z +Z n)G 0ZG + nG nG 0 + nG

δ = ζ+Id =

Michael Horst OSU

Free Picard Category

(n +Z n′)G (n′ +Z n)G nG + n′G n′G + nG

βZG δ δ β

(0Z +Z n)G 0ZG + nG nG 0 + nG

δ = ζ+Id =

Note: nG + (−n)G ∼ = (n − n)G = 0ZG ∼ = 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)

F ni.Gi + n′

j.G ′ j

• = F ( ni.Gi) + F

n′

j.G ′ j

• Michael Horst

OSU

Proof Highlights

Grpd(G , A ) ∋ F → F ∈ Pic(Z[G ], A )

F(n.G) =

|n| sgn(n)F(G)

F ni.Gi + n′

j.G ′ j

• = F ( ni.Gi) + F

n′

j.G ′ j

• F(1n.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)

F ni.Gi + n′

j.G ′ j

• = F ( ni.Gi) + F

n′

j.G ′ j

• F(1n.g : n.G → n.G ′) =

|n| sgn(n)F(g)

F(−11+n.IdG) = KF(G) +

|n| sgn(n)IdF(G)

Michael Horst OSU

Proof Highlights

Pic(Z[G ], A ) ∋ F → uF ∈ Grpd(G , A )

Michael Horst OSU

Proof Highlights

Pic(Z[G ], A ) ∋ F → uF ∈ Grpd(G , A )

uF(G) = F(1.G) Michael Horst OSU

Proof Highlights

Pic(Z[G ], A ) ∋ F → uF ∈ Grpd(G , A )

uF(G) = F(1.G) uF(g : G → G ′) = F(11.g) Michael Horst OSU

Proof Highlights

Pic(Z[G ], A ) ∋ F → uF ∈ Grpd(G , A )

uF(G) = F(1.G) uF(g : G → G ′) = F(11.g) u(F) = F Michael Horst OSU

Proof Highlights

Pic(Z[G ], A ) ∋ F → uF ∈ Grpd(G , A )

uF(G) = F(1.G) uF(g : G → G ′) = F(11.g) u(F) = F

F(n.G) ∼ =

F(δ) F

• |n| sgn(n).G
• Michael Horst

OSU

Proof Highlights

Pic(Z[G ], A ) ∋ F → uF ∈ Grpd(G , A )

uF(G) = F(1.G) uF(g : G → G ′) = F(11.g) u(F) = F

F(n.G) ∼ =

F(δ) F

• |n| sgn(n).G
• ∼

=

|n| F(sgn(n).G)

∼ =

|n| sgn(n)F(1.G) = uF(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 Pic CMon(Pic, ∗) Ab CMon(Ab, ⊗)

Z[ ] π0 π0 Z[ ]

⊣ ⊣

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic)

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Conjecture If M ∈ G -Mod, then G -Mod (Z[G ], M ) ≃ M

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Conjecture If M ∈ G -Mod, then G -Mod (Z[G ], M ) ≃ M Example use case:

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Conjecture If M ∈ G -Mod, then G -Mod (Z[G ], M ) ≃ M Example use case: Hn(G ; M ) := RnG -Mod (Ztriv, ) (M )

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Conjecture If M ∈ G -Mod, then G -Mod (Z[G ], M ) ≃ M Example use case: Hn(G ; M ) := RnG -Mod (Ztriv, ) (M ) Compute Hn(Z/2; Z) via resolving Z:

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Conjecture If M ∈ G -Mod, then G -Mod (Z[G ], M ) ≃ M Example use case: Hn(G ; M ) := RnG -Mod (Ztriv, ) (M ) Compute Hn(Z/2; Z) via resolving Z: . . . → Z[Z/2] → Z[Z/2] → Z[Z/2] → Z

Michael Horst OSU

Group rings and the free module

Definition For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Conjecture If M ∈ G -Mod, then G -Mod (Z[G ], M ) ≃ M Example use case: Hn(G ; M ) := RnG -Mod (Ztriv, ) (M ) Compute Hn(Z/2; Z) via resolving Z: . . . → Z[Z/2] → Z[Z/2] → Z[Z/2] → Z and take cohomology of Z/2-Mod (Z[Z/2], Z) → Z/2-Mod (Z[Z/2], Z) → . . .

Michael Horst OSU

Thank you∼!

Michael Horst OSU

References

• J. C. Baez and A. D. Lauda, Higher-dimensional algebra. v: 2-groups., Theory

and Applications of Categories [electronic only] 12 (2004), 423–491.

• A. del R´

ıo, J. Mart´ ınez-Moreno, and E. Vitale, Chain complexes of symmetric categorical groups., J. Pure Appl. Algebra 196 (2005), no. 2-3, 279–312. doi:10.1016/j.jpaa.2004.08.029

• N. Gurski, N. Johnson, and A. M. Osorno, Star product on Picard Categories.,

2018, personal correspondence.

• N. Johnson and A. M. Osorno, Modeling stable one-types., Theory Appl. Categ.

26 (2012), 520–537.

• M. Kapranov, Supergeometry in mathematics and physics., 2015.

arXiv:1512.07042

• T. Pirashvili, On Abelian 2-categories and derived 2-functors., 2010.

arXiv:1007.4138 , Projective and injective symmetric categorical groups and duality.,

• Proc. Am. Math. Soc. 143 (2015), no. 3, 1315–1323.

doi:10.1090/S0002-9939-2014-12354-9

Michael Horst OSU