Kathryn Hess. Conference on Algebraic Topology, Group Theory and Representation Theory Isle of Skye 9 June 2009

Transcription

1 Institute of Geometry, lgebra and Topology Ecole Polytechnique Fédérale de Lausanne Conference on lgebraic Topology, Group Theory and Representation Theory Isle of Skye 9 June 2009

2 Outline

3 of rings: Data Commutative ring k Homomorphism of augmented k-algebras ϕ : k-bialgebra H, seen as a -algebra with trivial -action Coassociative, counital morphism ρ : k H of -algebras

4 of rings: Maps The Galois map β ϕ : ρ H µ H k H. k The corestriction map i ϕ : co H := H k = {a ρ(a) = a 1}

5 of rings: Definition The homomorphism ϕ : is an H- extension if the Galois map β ϕ : H k and the corestriction map are both isomorphisms. i ϕ : co H

6 of rings: Examples 1 Let G be a group. ny G-Galois extension ϕ : is a Hom(Z[G], Z)- extension. 2 If H is a Hopf algebra that is flat as a k-module and is a flat k-algebra, then H : a a 1 is an H- extension.

7 Generalization to ring spectra [Rognes] The unit map η : S MU is an S[U]- extension in a homotopical sense (i.e., replacing isomorphisms by weak equivalences), where the diagonal : U U U induces the comultiplication S[U] S[U] S[U]; the Thom diagonal MU MU U + gives rise to the coaction of S[U] on MU; and β η : MU MU MU S[U] is the Thom equivalence.

8 Goals Categorify the ring-level definition à la Morita, in a homotopical sense: isomorphisms of objects Quillen equivalences of model categories. Characterize homotopic in well-known monoidal model categories. Determine the role of homotopic in descent theory.

9 The framework Let M be a category endowed with a monoidal structure: : M M M and I Ob M such that ( ) C = ( C) and I = = I ; a model structure: a framework for defining homotopy relations on morphisms, involving distinguished classes of morphisms weak equivalences, fibrations and cofibrations satisfying axioms analogous to the properties of the continuous maps with the same names.

10 Definition Let be a monoid in M. n -co-ring is a comonoid in ( Mod, ), i.e., an -bimodule W endowed with a coassociative, counital comultiplication W W W that is a morphism of -bimodules. Example If = I, then Mod = M, and an -co-ring is just a comonoid in M. Example itself is an -co-ring, endowed with the comultiplication =.

11 Comodules over co-rings If W is an -co-ring, then M W is the category of W -comodules in the category of right -modules. n object of M W is thus a right -module M together with a coassociative, counital morphism of right -modules θ : M M W. Example If = I and C is a comonoid, then M C I category of right C-comodules. = Comod C, the Example M = Mod.

12 asic functors Let γ : W W be a morphism of -co-rings. The forgetful functor U W : M W Mod The cofree functor W : Mod M W The induced functor γ : M W MW Remark The functor U W is the left adjoint to W, and γ is a left adjoint if M W admits equalizers.

13 The trivial/coinvariants adjunction Let W be an -co-ring, endowed with a coaugmentation, i.e., a morphism of -co-rings η : W. The trivial W -comodule functor is Triv W = η : Mod M W. If Mod admits equalizers, then the W -coinvariants functor Coinv W : M W Mod is the right adjoint to Triv W, defined by Coinv(M, θ) = M co W = equal(m M η θ M W ).

14 Homotopy theory in M W Under certain reasonable conditions on the monoidal model category M and on W, the forgetful functor U W : M W Mod left-induces a model category structure on M W. Moreover, γ : M W MW is then the left member of a Quillen pair, for all morphisms γ : W W of nice enough -co-rings.

15 Example: The canonical co-ring I Let ϕ : be a morphism of monoids. The canonical co-ring on ϕ, denoted W ϕ, is, with comultiplication equal to the composite = ϕ = ( ) ( ). The morphism µ : induced by the multiplication map of is the counit.

16 Example: The canonical co-ring II For any morphism of monoids ϕ :, M Wϕ = D(ϕ), the descent category associated to ϕ. n object of D(ϕ) is a right -module M endowed with a morphism θ : M M such that the diagrams M θ M M θ M θ M M ϕ M θ = M r commute. The pair (M, θ) is a descent datum.

17 Canonical descent Let ϕ : be a morphism of monoids. The canonical descent functor Can : Mod D(ϕ) is defined on objects by Can(M) = (M, θ M ), with θ M = M ϕ. If M is a nice enough monoidal model category, then Can is the left member of a Quillen pair.

18 Example: Comodule algebras and co-rings Let H be any bimonoid in M. Let be an H-comodule algebra. H is naturally an -co-ring, with left -action and right -action H µ H H, H H ρ H H = H H µ µ H H and comultiplication H H H = ( H) ( H). Henceforth, we denote this co-ring W ρ.

19 data a bimonoid H a monoid an H-comodule algebra, with coaction ρ : H ϕ : Triv H () a morphism of H-comodule algebras ssume that M and (H,,, ϕ) are nice enough to ensure the existence of all the necessary model category structures.

20 The Galois functor The Galois map β ϕ, which is equal to the composite ρ H µ H H, underlies a morphism of -co-rings, from W ϕ to W ρ. The Galois map therefore induces (β ϕ ) : D(ϕ) M Wρ, the Galois functor associated to ϕ, which is the left member of a Quillen pair, under reasonable conditions.

21 The corestriction functor Let j : be a fibrant replacement in the category of H-comodule algebras. model for the homotopy coinvariants of the H-coaction on is then hco H := ( ) co H. The homotopy corestriction map i ϕ : hco H is equal to the composite = ( Triv H () ) co H ϕ co H co H jco H ( ) co H = hco H and induces a functor i ϕ : Mod hco H Mod.

22 Reminder of the ring case Recall that......a homomorphism of rings ϕ : is an H- extension if the Galois map β ϕ : H k and the corestriction map are both isomorphisms. i ϕ : co H

23 Definition The morphism ϕ : Triv H () of H-comodule algebras is a homotopic H- extension if the Galois functor (β ϕ ) : D(ϕ) M Wρ and the corestriction functor i ϕ : Mod hco H Mod are both Quillen equvalences.

24 Trivial I bimonoid H is a Hopf monoid if the Galois functor (β η ) : D(η) M W H associated to the H-comodule algebra map η : Triv(I) H is a Quillen equivalence. Examples The monoid of Moore loops on a topological space is a Hopf monoid in Top. ny bialgebra in the category of chain complexes over a commutative ring is a Hopf monoid.

25 Trivial II Proposition If H is a Hopf monoid and is a fibrant monoid such that preserves weak equivalences, then η H is a homotopic H- extension.

26 Example: Simplicial monoids Let H be a simplicial monoid, seen as a simplicial bimonoid, via the diagonal map. Let be a fibrant H-comodule algebra, i.e., a simplicial monoid endowed with a simplicial homomorphism p : H that is a Kan fibration. Let be a simplicial monoid, and let ϕ : Triv H () be a morphism of H-comodule algebras. Proposition ϕ is a homotopic H- extension iff it is homotopy equivalent to a principal fibration of simplicial monoids.

27 Example: Chain algebras I Let H be a 1-connected bialgebra in the category of finite-type chain complexes of k-vector spaces. Let be a connected H-comodule algebra, with H-coaction ρ. Proposition The algebra map Ω(; H; H) : a a i 1 h i, where ρ(a) = a i h i, is a fibrant replacement of as an H-comodule algebra. [H.-Levi] Ω(; H; H) admits an algebra structure. Fibrancy of Ω(; H; H) proved by showing that it is the limit of a Postnikov tower.

28 Example: Chain algebras II Example The algebra map induced by the unit of H ι : Ω(; H; k) Ω(; H; H) is a homotopic H- extension. Remark and Ω(; H; H) Ω(;H;k) Ω(; H; H) = Ω(; H; H) H Ω(; H; H) hco H = Ω(; H; H) co H = Ω(; H; k), since Ω(; H; H) is fibrant. Thus, both i ϕ and β ϕ are actually isomorphisms in this case.

29 Goal To prove a structure theorem relating the notions of homotopic, homotopical faithful flatness, and descent, analogous to a well-known and important theorem in the ring case, due to Schneider.

30 Induction Let H be a bimonoid, and let be an H-comodule algebra with H-coaction map ρ. The ρ-induction functor Ind ρ : Mod co H M Wρ is defined on objects by Ind ρ (M) = (M, M ρ). co H co H If M is a nice enough monoidal model category, then Ind ρ is the left member of a Quillen pair.

31 Schneider s structure theorem Theorem Let k be a commutative ring, and let H be a k-flat Hopf algebra. The following are equivalent for any H-comodule algebra, with coinvariant algebra = co H. The inclusion is an H- extension, and is a faithfully flat -module. The functor Ind ρ : Mod M Wρ is an equivalence, where ρ denotes the H-coaction on. ( is faithfully flat over if is flat over and M = 0 M = 0.)

32 Characterizing faithful flatness Theorem Let ϕ : be an inclusion of rings. The canonical descent functor Can : Mod D(ϕ) is an equivalence of categories if and only if is faithfully flat as a -module.

33 al faithful flatness Let ϕ : be a morphism of monoids in M. The monoid is homotopically faithfully flat over if Can : Mod D(ϕ) is the left member of a Quillen equivalence.

34 The homotopical structure theorem Theorem Let M be a monoidal model category. Let (H,, Triv H () ϕ ) be data. Under reasonable conditions on M and on the data, the following conditions are equivalent. The monoid map ϕ is a homotopic H- extension, and is homotopically faithfully flat over. The functor Ind ρ ( co H ) : Mod M Wρ is a Quillen equivalence.