Eilenberg MacLane mapping algebras and higher distributivity up to homotopy

Transcription

1 Eilenberg McLne mpping lgebrs nd higher distributivity up to homotopy Hns-Jochim Bues 1 Mrtin Frnklnd* 2 1 Mx-Plnck-Institut für Mthemtik 2 Universität Osnbrück Homotopy Theory: Tools nd Applictions University of Illinois t Urbn-Chmpign July 19, 2017 Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

2 Outline 1 Bckground 2 Mpping theories 3 Higher distributivity 4 Min results 5 Exmples in mod 2 cohomology Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

3 Some history: A -spces Stsheff (1963): Higher ssocitivity vi ssocihedr. Existence Loop spces (group-like) A -spces Recognition Homotopy invrince: Assume X Y. Then X dmits n A n -structure if nd only if Y does. Strictifiction: An A -spce is wekly equivlent to topologicl monoid. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

4 Stble cohomology opertions Slogn: Higher distributivity vi cubes. Let X be spectrum. Cohomology H n (X; F p ) = [X, Σ n HF p ] is given by homotopy clsses of mps to Eilenberg McLne spectr. Primry stble cohomology opertions re given by homotopy clsses of mps between Eilenberg McLne spectr. The mod p Steenrod lgebr A is given by For exmple, Sq k : HF 2 Σ k HF 2. A k = [HF p, Σ k HF p ]. More generlly, consider mps between finite products A = Σ n 1 HF p... Σ n k HF p. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

5 Higher order opertions Higher order cohomology opertions re encoded by the mpping spces between Eilenberg McLne spectr. Exmple The 3-fold Tod brckets b,, x [X, ΩC] define secondry cohomology opertion b,,. α 0 X x A B b C β 0 Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

6 Distributivity up to homotopy In the homotopy ctegory of spectr, composition is biliner. This does not hold in Top -enriched ctegory of spectr. The eqution X x,x A, B ( + )x = x + x holds strictly in mp(x, B), becuse of pointwise ddition. Tht is, left linerity holds. The eqution (x + x ) x + x holds up to coherent homotopy in mp(x, B). Gol Describe the higher distributivity lws stisfied by mps between Eilenberg McLne spectr. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

7 Mps between Eilenberg McLne spectr Work in simplicil model ctegory of spectr Sp (e.g. Bousfield Friedlnder). Importnt ingredient: A model of the Eilenberg McLne spectrum HF p which is n belin group object, fibrnt, nd cofibrnt. (Ht tip: Mrc Stephn.) Ech mpping spce mp(x, A) is topologicl belin group. Nottion Let EM denote the full Top -enriched ctegory of Sp consisting of the finite products A = Σ n 1 HF p... Σ n k HF p. Note tht EM is smll ctegory. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

8 Left liner mpping theories Slient fetures of EM: 1 Top -enriched. 2 Hs finite products (i.e., is theory). 3 Ech mpping spce EM(A, B) is n topologicl belin group, with bsepoint 0: A B. 4 Composition is strictly left liner: ( + )x = x + x. Definition A left liner mpping theory T is defined by (1) (4). Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

9 Exmples Exmple EM is left liner mpping theory, clled the Eilenberg McLne mpping theory. Exmple Consider models of Eilenberg McLne spces K (F p, n) s topologicl belin groups. Let EM unstble be the full subctegory of Top consisting of finite products K (F p, n 1 )... K (F p, n k ) with n i 1. Then EM unstble is left liner mpping theory. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

10 Enriched cohomology For spectrum X, the functor mp(x, ): EM Top preserves products strictly, i.e., is model of EM. It is clled the (stble) Eilenberg McLne mpping lgebr of X. For our purposes: It suffices to focus on EM itself. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

11 Wht bout right linerity? Stbly, finite products become coproducts, more precisely in the homotopy ctegory of spectr: A B A B. In spectr, finite products re wek coproducts: A B A B. Definition A mpping theory T is wekly biliner if it is left liner nd moreover for ll objects A, B, Z of T, the mp T (A B, Z ) (i A,i B ) T (A, Z ) T (B, Z ) is trivil Serre fibrtion. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

12 Exmples Exmple The mpping theory EM is wekly biliner. Exmple The mpping theory EM unstble is left liner, but not wekly biliner. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

13 1-distributivity Definition A left liner Top -enriched ctegory is 1-distributive if for ll, x, y T, there is pth (x + y) ϕ x,y x + y in T. In other words, T is right liner up to homotopy. A choice of such pths is denoted ϕ 1 = { ϕ x,y, x, y T } nd is clled 1-distributor for T. Also, ϕ 1 is required to be continuous in the inputs, x, y. More precisely, for ll objects X, A, B of T, the following mp is continuous: T (A, B) T (X, A) 2 ϕ 1 T (X, B) I (, x, y) ϕ x,y. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

14 2-distributivity Definition T is clled 2-distributive if it dmits 1-distributor ϕ 1 such tht for ll, x, y, z T, the mp I 2 T defined by (x + y) + z ϕ x,y + z x + y + z ϕ x+y,z ϕ x,y,z x + ϕ y,z (x + y + z) ϕ x,y+z x + (y + z). dmits n extension ϕ x,y,z : I 2 T. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

15 2-distributivity (cont d) Definition A choice of such 2-cubes is denoted ϕ 2 = { ϕ x,y,z, x, y, z T } nd is clled 2-distributor for T, bsed on the 1-distributor ϕ 1. As before, the 2-distributor ϕ 2 is required to be continuous in the inputs, x, y, z T. More precisely, for ll objects X, A, B of T, the following mp is continuous: T (A, B) T (X, A) 3 ϕ 2 T (X, B) I2 (, x, y, z) ϕ x,y,z. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

16 n-distributivity Definition T is clled n-distributive if there re collections of cubes ϕ 0, ϕ 1,..., ϕ n, where ϕ m = {ϕ x 0,...,x m, x 0,..., x m T } is collection of m-cubes ϕ x 0,...,x m : I m T, stisfying the following. ϕ 0 is 0-distributor, i.e., the collection of 0-cubes ϕ x = x. For 1 m n, the following boundry conditions hold: ϕ x 0,...,x m (t 1,..., {}}{ (t 1,..., ϕ x 0,...,x m t j {}}{ 0,..., t m ) = ϕ x 0,...,x j 1 +x j,...,x m (t 1,..., t j,..., t m ) t j 1,..., t m ) = ϕ x 0,...,x j 1 (t 1,...,..., t j 1 ) ϕ x j,...,x m (t j+1,..., t m ). Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

17 n-distributivity (cont d) Such collection ϕ n of n-cubes in T is clled n n-distributor for T, bsed on the (n 1)-distributor ϕ n 1. The n-distributor ϕ n is required to be continuous in the inputs, x 0,..., x n T. More precisely, for ll objects X, A, B of T, the following mp is continuous: T (A, B) T (X, A) n+1 ϕ n T (X, B) In (, x 0,..., x n ) ϕ x 0,...,x n. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

18 Exmple: 3-distributor (x 0 + x 1 ) + x 2 + x 3 ϕ x 0,x 1 + x 2 + x 3 x 0 + x 1 + x 2 + x 3 (x 0 + x 1 + x 2 ) + x 3 ϕ x 0+x 1,x 2 + x 3 x 0 + ϕ x 1,x 2 + x 3 ϕ x 0,x 1,x 2 + x 3 ϕ x 0,x 1 +x 2 ϕ x 0,x 1 + x 3 ϕ x 2,x 3 x 0 + (x 1 + x 2 ) + x 3 (x 0 + x 1 ) + ϕ x 2,x 3 ϕ x x 0 + x 1 + ϕ x 2,x 3 0+x 1 +x 2,x 3 x 0 + ϕ x 1+x 2,x 3 (x 0 + x 1 ) + (x 2 + x 3 ) x 0 + x 1 + (x 2 + x 3 ) ϕ x 0,x 1 + (x 2 + x 3 ) ϕ x 0 +x 1,x 2,x 3 ϕ x 0,x 1 +x 2,x 3 ϕ x 0+x 1,x 2 +x 3 ϕ x 0,x 1,x 2 +x 3 x 0 + ϕ x 1,x 2,x 3 x 0 + ϕ x 1,x 2 +x 3 (x 0 + x 1 + x 2 + x 3 ) ϕ x 0,x 1 +x 2 +x 3 x 0 + (x 1 + x 2 + x 3 ) Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

19 Existence Theorem (Bues,F.) Let T be wekly biliner mpping theory in which every mpping spce T (A, B) hs the homotopy type of CW complex. Then T is -distributive. Remrk In fct, T dmits good -distributor, for which ech distributor ϕ n is determined up to homotopy rel I n by the previous distributor ϕ n 1. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

20 Homotopy invrince Theorem (Bues,F.) Let F : S T be morphism of left liner Top -enriched ctegories which is moreover Dwyer Kn equivlence. Assume tht ll mpping spces in S nd in T hve the homotopy type of CW complex. Then for every n 1 (or n = ), S is n-distributive if nd only if T is n-distributive. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

21 The Kristensen derivtion Fix p = 2, nd let ϕ 1 be good 1-distributor for the mod 2 Eilenberg McLne mpping theory EM. For clss in the Steenrod lgebr A m, the loop defines clss 0 = 0 = (1 + 1) ϕ1, = + = 0 κ() π 1 EM(HF 2, Σ m HF 2 ) = [HF 2, Σ m 1 HF 2 ] = A m 1. Proposition (Bues 2006) The function κ: A A 1 is the Kristensen derivtion, i.e., the derivtion stisfying κ(sq m ) = Sq m 1. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

22 The 2-dimensionl nlogue Now let ϕ 2 be good 2-distributor for EM. Consider the 2-cube in EM(HF 2, Σ m HF 2 ): = (1 + 1) + 1 ϕ 1, = = ϕ 0,1 ϕ 1,1,1 + ϕ 1,1 = ( ) ϕ 1,0 = 1 + (1 + 1) =. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

23 A derivtion of degree 2 The 2-cube 0 ϕ 1,1 0 0 ɛ 0 0 ϕ 1,1,1 ϕ 1, defines clss λ() π 2 EM(HF 2, Σ m HF 2 ) = A m 2. Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

24 A derivtion of degree 2 (cont d) Proposition The function λ: A A 2 is derivtion. Question Is λ given by λ = κ 2? Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

25 Thnk you! Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31

26 Reference H.J. Bues nd M. Frnklnd. Eilenberg McLne mpping lgebrs nd higher distributivity up to homotopy. rxiv: Bues, Frnklnd (MPIM nd Osnbrück) Higher distributivity UIUC, July / 31