Algebraic Factorization for Chain Algebras Jonathan Scott Cleveland - PowerPoint PPT Presentation

Algebraic Factorization for Chain Algebras Jonathan Scott Cleveland State University Joint work with K. Hess and P.-E. Parent Operads and Higher Structures in Algebraic Topology and Category Theory U Ottawa, August 2 2019

Lifting Functorially Let C be a category with two classes of morphisms, C and F . A lifting problem with respect to C and F is a commutative square, A C g f B D where f ∈ C and g ∈ F . A solution to the given lifting problem is a dotted arrow that makes the diagram commute. ◮ When can the solution be provided in a functorial manner? ◮ We are interested C = cofibrations, F = acyclic fibrations, in DGA .

Factorization Systems ◮ Let n be the poset category 1 → 2 → · · · → n . ◮ If C is any category, then C 2 is the category of arrows in C and commutative squares. Similarly, C 3 is the category of composable pairs of morphisms. Composition defines a functor C 3 → C 2 . ◮ A factorization system is a section of the composition functor. Composing with the “first arrow” and “second arrow” functors leads to two functors L , R : C 2 → C 2 . L ϕ R ϕ A G ϕ E ϕ

The Beginnings of Structure Given a factorization system ( L , R ), the (same) diagrams L ϕ A A A G ϕ and ϕ L ϕ ϕ R ϕ G ϕ E E E R ϕ provide the components of co-unit and unit natural transformations, ε : L ⇒ I and η : I ⇒ R where I is the identity functor on C 2 .

R -Algebras An R - algebra is a morphism ϕ along with a structure “morphism”: m 1 G ϕ A ϕ m R ϕ E E m 2 that is unital, as expressed in the following diagram.

R -Algebras, continued A G ϕ A m 1 L ϕ ϕ ϕ R ϕ m 2 E E E In particular, ◮ ϕ is a retract of R ϕ . ◮ m 2 = 1 E , so we abuse notation and refer to m by its top arrow. ◮ m is a retraction of L ϕ .

L -Coalgebras Dually, an L - coalgebra is a morphism θ with a structure “morphism”, A A θ L θ c E G θ that is co-unital with respect to ε θ , so ◮ θ is a retract of L θ ; ◮ c is a section of R ϕ .

A Solution The lifting problem with respect to L -coalgebras and R -algebras has a functorial solution. Let θ : A → B be an L -coalgebra and ϕ : C → D be an R -algebra. A C L ϕ L θ m G θ G ϕ ϕ θ R θ R ϕ c B D

Putting the “A” in WAF ◮ Problem: If ϕ is an R -algebra, then R ϕ is not necessarily. This is an issue if we want to make this useful for model categories. ◮ (Riehl 2011) If R is a monad with structure µ : R 2 ⇒ R that is unital w.r.t. η , and L is a comonad with ∆ : L ⇒ L 2 co-unital w.r.t ε , then “everything works”. ◮ Every cofibrantly generated model category has a weak algebraic factorization system for the (cofibration, acyclic fibration) and (acyclic cofibration, fibration) factorizations.

Our Project Find comonad L : DGA 2 → DGA 2 and monad R : DGA 2 → DGA 2 such that ◮ ( L , R ) forms a factorization system; ◮ The cofibrations and acyclic fibrations in DGA are precisely the L -coalgebras and R -algebras, respectively.

The Factorization We make use of the obvious mapping-cylinder factorization in DGC and the bar-cobar adjunction. Theorem Let M be a left-proper model category that satisfies 1. If gf is a fibration, then g is a fibration; 2. there is a functorial cofibrant replacement functor Q on M ; 3. there is a functorial (cofibration, acyclic fibration) factorization Q ( M 2 ) → M 3 . Then the functorial cofibration-acyclic fibration factorization extends to M 2 → M 3 .

Proof by Diagram Let ϕ : X → Y . Then Q ϕ : QX → QY factors functorially as λ ϕ ρ ϕ QX N ϕ QY . ∼ Form the diagram λ ϕ ρ ϕ QX N ϕ QY ∼ σ ∼ L ϕ X G ϕ ∼ R ϕ Y . ϕ

Our Case We let Q = Ω B . Let ϕ : A → E be a morphism in DGA . ◮ There is a functorial cylinder object, IBA , on BA . ◮ The mapping cylinder on B ϕ : BA → BE is given by the pushout B ϕ BA BE i 1 j B ϕ ℓ B ϕ IBA M B ϕ ◮ Set λ B ϕ = ℓ B ϕ i 0 : BA → M B ϕ ◮ ρ B ϕ : M B ϕ → BE such that ρ B ϕ λ B ϕ = B ϕ obtained by push-out ◮ Apply cobar to get the required factorization of Ω B ϕ .

The Comultiplication ∆ : L ⇒ L 2 Let ϕ : A → E be a morphism of algebras. To construct the component ∆ ϕ : L ⇒ L 2 , we must construct a natural lift in the diagram L 2 ϕ A G L ϕ ∆ ϕ L ϕ RL ϕ ∼ G ϕ G ϕ

The Construction We exploit the fact that G ϕ is a pushout. Ω λ BL ϕ Ω BA Ω M BL ϕ Ω λ B ϕ t ϕ Ω M B ϕ ε A σ ϕ σ L ϕ G ϕ ∆ ϕ L ϕ A G L ϕ L 2 ϕ First need t ϕ !

Construction of t ϕ We exploit the fact that M B ϕ is a pushout – naively. BL ϕ BG ϕ σ ♯ ϕ j B ϕ B ϕ BA BE M B ϕ j B ϕ i 1 j BL ϕ ℓ B ϕ IBA M B ϕ ? M BL ϕ ℓ BL ϕ And we run into trouble – upper cell only commutes up to DGC homotopy.

SHC Homotopies to the Rescue Lemma The diagram j B ϕ B ϕ BA BE M B ϕ σ ♯ ϕ BG ϕ λ BL ϕ j BL ϕ M BL ϕ commutes up to natural SHC homotopy (i.e., once the cobar construction is applied).

The Missing Piece In the proof of the lemma, we construct a natural DGA homotopy h : Ω( IBA ) → Ω M BL ϕ from Ω λ BL ϕ to Ω( j BL ϕ ◦ σ ♯ ϕ ◦ j B ϕ ◦ B ϕ ). Then t ϕ is given by the the pushout, Ω B ϕ Ω BA Ω BE Ω i 1 Ω j B ϕ Ω( j BL ϕ ◦ σ ♯ ϕ ◦ j B ϕ ) Ω ℓ B ϕ Ω IBA Ω M B ϕ t ϕ Ω M BL ϕ . h

L is a comonad for cofibrations Theorem With the above structure, L is a comonad, and the L-coalgebras are precisely the cofibrations in DGA . Proof. ◮ If θ is an L -coalgebra, then it is a retract of L θ . Since L θ is a cofibration, so too is θ . ◮ If θ is a cofibration, then since R θ is an acyclic fibration, can construct a lift L θ · · δ R ϕ θ ∼ · , · whose top triangle is an L -coalgebra structure on θ .

The Monad Structure of R For DGC morphism ϕ : A → E , again want a natural lift, this time in the diagram G ϕ G ϕ µ ϕ LR ϕ R ϕ ∼ G R ϕ E R 2 ϕ Exploit the fact that G R ϕ is a pushout.

A Little Homological Perturbation (In progress) There is an EZ-SDR pair of DGAs, t G ϕ E h R ϕ From the Perturbation Lemma of Gugenheim-Munkholm, we obtain an SDR pair of DGCs, T BG ϕ BE . H BR ϕ So we have ◮ BR ϕ ◦ T = 1 BE , ◮ H : IBG ϕ → BG ϕ , H : 1 BG ϕ ≃ T ◦ BR ϕ .

Putting it Together Putting this data into a pushout diagram, BR ϕ BG ϕ BE j BR ϕ i 1 T ℓ BR ϕ IBG ϕ M BR ϕ S BG ϕ H

The Monad Structure Define µ ϕ : G R ϕ → G ϕ as the unique morphism that makes the pushout diagram of chain algebras, Ω λ BR ϕ Ω BG ϕ Ω M BR ϕ ε G ϕ σ R ϕ S ♯ LR ϕ G ϕ G R ϕ µ ϕ G ϕ commute.

Extending to Algebras over Koszul Operads ◮ Suppose we have an operad P (in chain complexes), a cooperad Q , and a Koszul twisting cochain τ : Q → P . ◮ There are associated bar and cobar constructions that provide a cofibrant replacement comonad. ◮ Need a homological perturbation machine. Berglund (2009) might be a good place to start.