Algebraic Factorization for Chain Algebras

Jonathan ScottCleveland State University

Joint work with K. Hess and P.-E. Parent

Operads and Higher Structures in Algebraic Topology andCategory Theory

U Ottawa, August 2 2019

Lifting Functorially

Let C be a category with two classes of morphisms, C and F . Alifting problem with respect to C and F is a commutative square,

A C

B D

f g

where f ∈ C and g ∈ F . A solution to the given lifting problem isa dotted arrow that makes the diagram commute.

I When can the solution be provided in a functorial manner?

I We are interested C = cofibrations, F = acyclic fibrations, inDGA.

Factorization Systems

I Let n be the poset category 1→ 2→ · · · → n.

I If C is any category, then C2 is the category of arrows in Cand commutative squares. Similarly, C3 is the category ofcomposable pairs of morphisms. Composition defines afunctor C3 → C2.

I 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 ELϕ

ϕ

Rϕ

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 naturaltransformations,

ε : L⇒ I and η : I ⇒ R

where I is the identity functor on C2.

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.

R-Algebras, continued

A Gϕ A

E E E

ϕ

Lϕ m1

Rϕ ϕ

m2

In particular,

I ϕ is a retract of Rϕ.

I m2 = 1E , so we abuse notation and refer to m by its toparrow.

I m is a retraction of Lϕ.

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

I θ is a retract of Lθ;

I c is a section of Rϕ.

A Solution

The lifting problem with respect to L-coalgebras and R-algebrashas a functorial solution. Let θ : A→ B be an L-coalgebra andϕ : C → D be an R-algebra.

A C

Gθ Gϕ

B D

Lθ

θ

Lϕ

ϕ

Rθ

Rϕ

m

c

Putting the “A” in WAF

I Problem: If ϕ is an R-algebra, then Rϕ is not necessarily. Thisis an issue if we want to make this useful for model categories.

I (Riehl 2011) If R is a monad with structure µ : R2 ⇒ R thatis unital w.r.t. η, and L is a comonad with ∆ : L⇒ L2

co-unital w.r.t ε, then “everything works”.

I Every cofibrantly generated model category has a weakalgebraic factorization system for the (cofibration, acyclicfibration) and (acyclic cofibration, fibration) factorizations.

Our Project

Find comonad L : DGA2 → DGA2 and monadR : DGA2 → DGA2 such that

I (L,R) forms a factorization system;

I The cofibrations and acyclic fibrations in DGA are preciselythe L-coalgebras and R-algebras, respectively.

The Factorization

We make use of the obvious mapping-cylinder factorization inDGC and the bar-cobar adjunction.

TheoremLet 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 factorizationextends to M2 →M3.

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

Our Case

We let Q = ΩB. Let ϕ : A→ E be a morphism in DGA.

I There is a functorial cylinder object, IBA, on BA.

I The mapping cylinder on Bϕ : BA→ BE is given by thepushout

BA BE

IBA MBϕ

Bϕ

i1 jBϕ

`Bϕ

I Set λBϕ = `Bϕi0 : BA→ MBϕ

I ρBϕ : MBϕ → BE such that ρBϕλBϕ = Bϕ obtained bypush-out

I Apply cobar to get the required factorization of ΩBϕ.

The Comultiplication ∆ : L⇒ L2

Let ϕ : A→ E be a morphism of algebras.To construct the component ∆ϕ : L⇒ L2, we must construct anatural lift in the diagram

A GLϕ

Gϕ Gϕ

L2ϕ

Lϕ RLϕ∼∆ϕ

The Construction

We exploit the fact that Gϕ is a pushout.

ΩBA ΩMBLϕ

ΩMBϕ

Gϕ

A GLϕ

ΩλBLϕ

εA

ΩλBϕ

σLϕ

tϕ

σϕ

∆ϕLϕ

L2ϕ

First need tϕ!

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

`Bϕ

`BLϕ

?

And we run into trouble – upper cell only commutes up to DGChomotopy.

SHC Homotopies to the Rescue

LemmaThe diagram

BA BE MBϕ

BGϕ

MBLϕ

Bϕ

λBLϕ

jBϕ

σ]ϕ

jBLϕ

commutes up to natural SHC homotopy (i.e., once the cobarconstruction is applied).

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 thethe pushout,

ΩBA ΩBE

ΩIBA ΩMBϕ

ΩMBLϕ.

ΩBϕ

Ωi1 ΩjBϕ Ω(jBLϕσ]ϕjBϕ)

Ω`Bϕ

h

tϕ

L is a comonad for cofibrations

TheoremWith the above structure, L is a comonad, and the L-coalgebrasare precisely the cofibrations in DGA.

Proof.

I If θ is an L-coalgebra, then it is a retract of Lθ. Since Lθ is acofibration, so too is θ.

I If θ is a cofibration, then since Rθ is an acyclic fibration, canconstruct a lift

· ·

·, ·

Lθ

θ Rϕ∼δ

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 timein the diagram

Gϕ Gϕ

GRϕ E

LRϕ Rϕ∼

R2ϕ

µϕ

Exploit the fact that GRϕ is a pushout.

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, weobtain an SDR pair of DGCs,

BGϕ BE .

BRϕ

H

T

So we have

I BRϕ T = 1BE ,

I H : IBGϕ → BGϕ, H : 1BGϕ ' T BRϕ.

Putting it Together

Putting this data into a pushout diagram,

BGϕ BE

IBGϕ MBRϕ

BGϕ

BRϕ

i1 jBRϕT

`BRϕ

H

S

The Monad Structure

Define µϕ : GRϕ → Gϕ as the unique morphism that makes thepushout diagram of chain algebras,

ΩBGϕ ΩMBRϕ

Gϕ GRϕ

Gϕ

ΩλBRϕ

εGϕ σRϕS]

LRϕ

µϕ

commute.

Extending to Algebras over Koszul Operads

I Suppose we have an operad P (in chain complexes), acooperad Q, and a Koszul twisting cochain τ : Q → P.

I There are associated bar and cobar constructions that providea cofibrant replacement comonad.

I Need a homological perturbation machine. Berglund (2009)might be a good place to start.