A Categorification of Group Cohomology Michael Horst The Ohio State University horst.59@osu.edu https://www.asc.ohio-state.edu/horst.59/ 27 October 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: 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 ∼ = Pic ( R ) := R - Mod inv , for R ∈ CRing Note: π 0 ( Pic ( R )) = pic ( R ) Π 1 X for X ∈ Ω 3 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
Free Picard Categories Theorem (H) The forgetful functor U : Pic → Grpd has a left biadjoint 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. Proposition (H) Pic is cotensored over Grpd Michael Horst OSU
Picard Cohomology Definition: Category of Modules over G ∈ Pic For G ∈ Pic , G - Mod := PsFunk (Σ G , Pic ) Definition: Picard Cohomology For G ∈ Pic and M ∈ G - Mod , define H n ( G ; M ) := R n � � G - Mod ( Z triv , ) ( M ) i.e. H n ( G ; M ) = Ext n G - Mod ( Z triv , M ) i.e. H n ( G ; M ) = L n � � G - Mod ( , M ) ( Z triv ) Michael Horst OSU
A first computation: H m ( Z / n ; Z ) Proposition (H) The chain complex N x − 1 N x − 1 0 0 . . . − → Z [ Z / n ] − − → Z [ Z / n ] − → Z [ Z / n ] − − → Z [ Z / n ] − → 0 − → . . . provides a projective resolution of 0 0 0 0 . . . − → 0 → Z triv − − → 0 − → . . . Lemma (H) Applying Z / n- Mod ( , Z triv ) to the above yields 0 0 0 n 0 n 0 → 0 − → Z − → Z − − → Z − → Z − → . . . Michael Horst OSU
A first computation: H m ( Z / n ; Z ) Theorem (H) For n even, Z ⊕ Z / 2 m = 0 Z / 2 ⊕ Σ( Z × Z / 2) m = 1 H m ( Z / n ; Z ) = Z / n ⊕ Z / 2 m > 1 is even Z / 2 ⊕ Σ( Z / n × Z / 2) m > 1 is odd For n odd, m = 0 Z Σ( Z ) m = 1 H m ( Z / n ; Z ) = 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 ( , Z triv ) to the previous resolution yields 0 0 0 n 0 n 0 → 0 − − → Z − → Z − → Z − → Z → . . . − Theorem (H) For all n, m = 0 Z Σ( Z ) m = 1 H m ( Z / n ; Z ) = 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 0 0 ∂ n − 2 ∂ n d n − 2 d n − 1 d n d n +1 . . . A n − 2 A n − 1 A n A n +1 A n +2 . . . ∂ n − 1 0 Compute cohomology using relative kernel and cokernel 0 ϕ k F G Ker ( F , ϕ ) A B C κ 0 If H n ( A • ) ≃ 0, call A • relative exact at A n Michael Horst OSU
G -modules vs. Picard Categories Theorem (H) Let A , B , and D be bicategories, and assume that A has all bicategorical limits (dually, colimits) of shape D . Then for any F : D → PsFunk ( B , A ) , Lim F exists and all its data are computed objectwise. Dually, the same holds for Colim F. Proposition Any biadjunction F ⊣ G between bicategories A and B yields for any bicategory D a biadjunction F D ⊣ G D between PsFunk ( D , A ) and PsFunk ( D , B ) . Michael Horst OSU
Projective Picard categories Definition P ∈ Pic is projective if for all P H ∼ = B C 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 H ∼ = A B C F G ϕ 0 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 H ∼ = A B C 0 0 0 0 F G ϕ 0 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 relative 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 , Σ A ∈ Pic is relative projective but not free. 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
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