Picard Cohomology Michael Horst The Ohio State University - - PowerPoint PPT Presentation

Picard Cohomology

Michael Horst

The Ohio State University horst.59@osu.edu https://u.osu.edu/horst.59/

12 July 2019

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

Pic(R) := R-Mod

∼ = inv, for R ∈ CRing

Note: π0(Pic(R)) = pic(R)

Π1X for X ∈ Ω3Top 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

Free Picard Categories

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, pseudonatural in G and A .

Michael Horst OSU

Tensoring over Grpd

Corollary (H) Pic is tensored over Grpd Specifically: For G ∈ Grpd and A ∈ Pic, there exists A [G ] ∈ Pic so that for all B ∈ Pic, Pic(A [G ], B) ≃ Grpd(G , Pic(A , B)) pseudonaturally.

Michael Horst OSU

Picard Cohomology

Definition: Category of Modules over G ∈ Pic For G ∈ Pic, G -Mod := PsFunk (ΣG , Pic) Definition: Picard Cohomology For G ∈ Pic and M ∈ G -Mod, define H n (G ; M ) := Rn G -Mod (Ztriv, )

• (M )

i.e. H n (G ; M ) = Extn

G-Mod (Ztriv, M )

i.e. H n (G ; M ) = Ln G -Mod ( , M )

• (Ztriv)

Michael Horst OSU

A first computation: H m (Z/n; Z)

Proposition (H) The chain complex . . .

N

− → Z[Z/n]

x−1

− − → Z[Z/n]

N

− → Z[Z/n]

x−1

− − → Z[Z/n] − → 0 − → . . . provides a projective resolution of . . . − → 0 − → Ztriv − → 0 − → . . . Lemma (H) Applying Z/n-Mod ( , Ztriv) to the above yields − → 0 − → Z − → Z

n

− → Z − → Z

n

− → . . .

Michael Horst OSU

A first computation: H m (Z/n; Z)

Theorem (H) For n even, H m (Z/n; Z) =          Z ⊕ Z/2 m = 0 Z/2 ⊕ Σ(Z × Z/2) m = 1 Z/n ⊕ Z/2 m > 1 is even Z/2 ⊕ Σ(Z/n × Z/2) m > 1 is odd For n odd, H m (Z/n; Z) =          Z m = 0 Σ(Z) m = 1 Z/n m > 1 is even Σ(Z/n) m > 1 is odd

Michael Horst OSU

A second computation: H m (Z/n; Z)

Lemma (H) Applying Z/n-Mod ( , Ztriv) to the previous resolution yields − → 0 − → Z − → Z

n

− → Z − → Z

n

− → . . . Theorem (H) For all n, H m (Z/n; Z) =          Z m = 0 Σ(Z) m = 1 Z/n m > 1 is even Σ(Z/n) m > 1 is odd

Michael Horst OSU

Chain complexes of Picard categories

Chain complex of Picard categories . . . An−2 An−1 An An+1 An+2 . . .

dn−2 ∂n−2 dn−1 ∂n−1 dn dn+1 ∂n

Compute cohomology using relative kernel and cokernel Ker (F, ϕ) A B C

k κ F G ϕ

If H n(A•) ≃ 0, call A• relative exact at An

Michael Horst OSU

Projective Picard categories

Definition P ∈ Pic is projective if for all P B C

H G

∼ = with G essentially surjective, such a lift exists. Problems:

No homological rephrasing P is projective if and only if P = Z⊕κ = Z[G ] for discrete G

Michael Horst OSU

Relative projective Picard categories

Definition (H) P ∈ Pic is relative projective if for all P A B C

H F ϕ G

∼ = with G essentially surjective and ϕ-full, such a lift exists. “ϕ-full” is a generalization of full

Michael Horst OSU

Relative projective Picard categories

Definition, rephrased P ∈ Pic is relative projective if for all P A B C

H F ϕ G

∼ = with row relative exact at C , such a lift exists.

Michael Horst OSU

Relative projective Picard categories

Theorem (H) P ∈ Pic is relative projective if and only if Pic (P, ) is exact. But not all free Picard categories are relative projective! Proposition (H) Z[G ] is relative projective if and only if for all G ∈ G , End(G) is free. Proposition (H) For free A ∈ Ab, ΣA ∈ Pic is relative projective but not free.

Michael Horst OSU

Enough relative projectives

Note: If G is ϕ-full, then it is κG-full A B C Ker(G) B C

F ϕ G kG κG G

Definition Say Pic has enough relative projectives if for all C ∈ Pic, there is a relative projective P and G : P → C that is essentially surjective and κG-full. Corollary (H) Pic has enough relative projectives.

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

• M. Dupont, Abelian Categories in dimension 2., Ph.D. thesis, Universit´

e catholique de Louvain, 2008. arXiv:0809.1760

• 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