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Algebraic Factorization for Chain Algebras Jonathan Scott Cleveland - - PowerPoint PPT Presentation
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
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Factorization Systems
◮ Let n be the poset category 1 → 2 → · · · → n. ◮ If C is any category, then C2 is the category of arrows in C and commutative squares. Similarly, C3 is the category of composable pairs of morphisms. Composition defines a functor C3 → C2. ◮ 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 : C2 → C2. A Gϕ E
Lϕ ϕ Rϕ
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The Beginnings of Structure
Given a factorization system (L, R), the (same) diagrams A A Gϕ E
Lϕ ϕ Rϕ
and A Gϕ E E
Lϕ ϕ Rϕ
provide the components of co-unit and unit natural transformations, ε : L ⇒ I and η : I ⇒ R where I is the identity functor on C2.
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R-Algebras
An R-algebra is a morphism ϕ along with a structure “morphism”: Gϕ A E E
m1 Rϕ
m
ϕ m2
that is unital, as expressed in the following diagram.
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R-Algebras, continued
A Gϕ A E E E
ϕ Lϕ m1 Rϕ ϕ m2
In particular, ◮ ϕ is a retract of Rϕ. ◮ m2 = 1E, so we abuse notation and refer to m by its top arrow. ◮ m is a retraction of Lϕ.
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L-Coalgebras
Dually, an L-coalgebra is a morphism θ with a structure “morphism”, A A E Gθ
θ Lθ c
that is co-unital with respect to εθ, so ◮ θ is a retract of Lθ; ◮ c is a section of Rϕ.
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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 Gθ Gϕ B D
Lθ θ Lϕ ϕ Rθ Rϕ m c
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Putting the “A” in WAF
◮ Problem: If ϕ is an R-algebra, then Rϕ 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 : R2 ⇒ R that is unital w.r.t. η, and L is a comonad with ∆ : L ⇒ L2 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.
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Our Project
Find comonad L : DGA2 → DGA2 and monad R : DGA2 → DGA2 such that ◮ (L, R) forms a factorization system; ◮ The cofibrations and acyclic fibrations in DGA are precisely the L-coalgebras and R-algebras, respectively.
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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(M2) → M3. Then the functorial cofibration-acyclic fibration factorization extends to M2 → M3.
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Proof by Diagram
Let ϕ : X → Y . Then Qϕ : QX → QY factors functorially as QX Nϕ QY .
λϕ ρϕ ∼
Form the diagram QX Nϕ QY X Gϕ Y .
λϕ ∼ σ ∼ ρϕ ∼ Lϕ ϕ Rϕ
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Our Case
We let Q = ΩB. Let ϕ : A → E be a morphism in DGA. ◮ There is a functorial cylinder object, IBA, on BA. ◮ The mapping cylinder on Bϕ : BA → BE is given by the pushout BA BE IBA MBϕ
Bϕ i1 jBϕ ℓBϕ
◮ Set λBϕ = ℓBϕi0 : BA → MBϕ ◮ ρBϕ : MBϕ → BE such that ρBϕλBϕ = Bϕ obtained by push-out ◮ Apply cobar to get the required factorization of ΩBϕ.
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The Comultiplication ∆ : L ⇒ L2
Let ϕ : A → E be a morphism of algebras. To construct the component ∆ϕ : L ⇒ L2, we must construct a natural lift in the diagram A GLϕ Gϕ Gϕ
L2ϕ Lϕ RLϕ ∼ ∆ϕ
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The Construction
We exploit the fact that Gϕ is a pushout. ΩBA ΩMBLϕ ΩMBϕ Gϕ A GLϕ
ΩλBLϕ εA ΩλBϕ σLϕ tϕ σϕ ∆ϕ Lϕ L2ϕ
First need tϕ!
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Construction of tϕ
We exploit the fact that MBϕ is a pushout – naively. BGϕ BA BE MBϕ IBA MBϕ MBLϕ
jBLϕ Bϕ i1 BLϕ jBϕ jBϕ σ♯
ϕ
ℓBϕ ℓBLϕ ?
And we run into trouble – upper cell only commutes up to DGC homotopy.
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SHC Homotopies to the Rescue
Lemma
The diagram BA BE MBϕ BGϕ MBLϕ
Bϕ λBLϕ jBϕ σ♯
ϕ
jBLϕ
commutes up to natural SHC homotopy (i.e., once the cobar construction is applied).
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The Missing Piece
In the proof of the lemma, we construct a natural DGA homotopy h : Ω(IBA) → ΩMBLϕ from ΩλBLϕ to Ω(jBLϕ ◦ σ♯
ϕ ◦ jBϕ ◦ Bϕ). Then tϕ is given by the
the pushout, ΩBA ΩBE ΩIBA ΩMBϕ ΩMBLϕ.
ΩBϕ Ωi1 ΩjBϕ Ω(jBLϕ◦σ♯
ϕ◦jBϕ)
ΩℓBϕ h tϕ
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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 Lθ. Since Lθ is a cofibration, so too is θ. ◮ If θ is a cofibration, then since Rθ is an acyclic fibration, can construct a lift · · ·, ·
Lθ θ Rϕ ∼ δ
whose top triangle is an L-coalgebra structure on θ.
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The Monad Structure of R
For DGC morphism ϕ : A → E, again want a natural lift, this time in the diagram Gϕ Gϕ GRϕ E
LRϕ Rϕ ∼ R2ϕ µϕ
Exploit the fact that GRϕ is a pushout.
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A Little Homological Perturbation
(In progress) There is an EZ-SDR pair of DGAs, Gϕ E
Rϕ h t
From the Perturbation Lemma of Gugenheim-Munkholm, we
- btain an SDR pair of DGCs,
BGϕ BE.
BRϕ H T
So we have ◮ BRϕ ◦ T = 1BE, ◮ H : IBGϕ → BGϕ, H : 1BGϕ ≃ T ◦ BRϕ.
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Putting it Together
Putting this data into a pushout diagram, BGϕ BE IBGϕ MBRϕ BGϕ
BRϕ i1 jBRϕ T ℓBRϕ H S
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The Monad Structure
Define µϕ : GRϕ → Gϕ as the unique morphism that makes the pushout diagram of chain algebras, ΩBGϕ ΩMBRϕ Gϕ GRϕ Gϕ
ΩλBRϕ εGϕ σRϕ S♯ LRϕ µϕ
commute.
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