Homotopical Adjoint Lifting Theorem David White Denison University - - PowerPoint PPT Presentation

Homotopical Adjoint Lifting Theorem

David White

Denison University

August 1, 2019 / Ottawa ATCT Conference

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 1 / 19

Lifting Quillen equivalences to algebra categories

Joint work with Donald Yau (Ohio State Newark). Alg(O)

L

• U
• Alg(P)

R

• U
• perad algebra categories

forget

• M

L

• O○−
• N

R

• P○−
• monoidal model categories

free

• L ⊣ R Quillen equivalence. R lax symmetric monoidal functor.

O, P are operads on M, N. f ∶ O → RP operad map with f ∶ LO

∼ P entrywise weak equivalence.

RU = UR. L(O ○ −) = (P ○ −)L by Adjoint Lifting Theorem.

Theorem (W.-Yau; 2016)

With some cofibrancy assumptions, L ⊣ R is a Quillen equivalence.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 2 / 19

Operads and algebras

Definition

An operad O = ({On}n≥0,γ,1) on a symmetric monoidal category (M,⊗,✶) : Right Σn-action on On, Operadic composition On ⊗ Ok1 ⊗ ⋯ ⊗ Okn

γ

Ok1+⋯+kn , Operadic unit ✶

1

O1 satisfying unity, associativity, and equivariance axioms.

Definition

An O-algebra is an object X ∈ M with structure maps On ⊗ X ⊗n

λ

X , satisfying unity, associativity, and equivariance axioms. Category of O-algebras = Alg(O) On parametrizes n-ary operations.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 3 / 19

Examples

Endomorphism operad End(X)n = [X ⊗n,X] As-algebras are monoids. Asn = ∐Σn ✶ Com-algebras are commutative monoids. Comn = ✶ Lie-algebras are Lie algebras in dg/simplicial modules. Op-algebras are operads. L∞

∼

Lie cofibrant resolution. A∞

∼

As cofibrant resolution. E∞

∼

Com cofibrant resolution.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 4 / 19

Monoidal Model Categories

A model category is a bicomplete category M and classes of maps W,F,Q (= weak equivalences, fibrations, cofibrations) satisfying axioms to behave like Top, e.g. lifting, factorization, 2 out of 3, retracts. Ho(M) = M[W −1]. Examples: Top, sSet, Ch(R), stable module cat, (G-)spectra, motivic spectra,

• perads, categories, graphs, flows, ...

Assume (M,⊗,1) is closed symmetric monoidal. For maps A1

f

B1 and A2

g

B2 , the pushout product is the map (A1 ⊗ B2)∐A1⊗B1(A2 ⊗ B1)

f◻g

B1 ⊗ B2 M is a monoidal model category if it satisfies the pushout product axiom: If f,g ∈ Q then f ◻ g ∈ Q, and if either f or g in W ∩ Q then so is f ◻ g. A Quillen equivalence M ⇆ N induces Ho(M) ≃ Ho(N)

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 5 / 19

Model structure on algebra categories

For j ∶ A → B, j◻n is the corner map from colimit of punctured n-cube (with vertices = words in A and B) to B⊗n.

Theorem (W.-Yau; 2014)

Suppose M is a strongly cofibrantly generated monoidal model category. For each n ≥ 1 and X ∈ MΣop

n , X ⊗Σn (−)◻n preserves acyclic cofibrations.

Then for each operad O, Alg(O) admits a projective model structure. strongly : domains of generating (acyclic) cofibrations are small. projective : weak equivalences and fibrations are defined in M. Ex : Ch(❦)(≥0), SSet(∗), SpΣ (positive (flat) stable), StMod(❦[G]) There are variations that assume less on M and more on O: if the condition only holds for X ∈ MΣop

n cofibrant in M, then get a semi-model

structure on O-algebras for objectwise cofibrant O. Always have a semi-model structure on O-algebras for Σ-cofibrant O.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 6 / 19

Semi-model categories (Spitzweck; 2001)

(M,W,Q,F) satisfies all model category axioms except 2 axioms: A

≃

• X
• K
• ≃
• L

B

• Y

& D

• nly hold if A and K are cofibrant. Still have cofibrant replacement.

All model category results have semi-model category analogues (often cofibrantly replace first): Ken Brown lemma, cylinders and path objects, cube lemma, Quillen equivalences, Reedy model structures, (co)simplicial frames, homotopy (co)limits, simplicial mapping spaces, Bousfield localization, etc. A combinatorial semi-model category has a Quillen equivalent model

• structure. The ∞-cat of Alg(O) agrees with AlgM∞(O⊗), for Σ-cofibrant O.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 7 / 19

Definition : Weak monoidal Quillen equivalence (Schwede-Shipley; 2003)

M

L

N

R

• Quillen equivalence between monoidal model categories, and

1

R is lax symmetric monoidal.

2

For cofibrant X,Y ∈ M, L(X ⊗ Y)

L2 ∼

LX ⊗ LY is a weak equivalence. L2 is adjoint to X ⊗ Y RLX ⊗ RLY R(LX ⊗ LY) .

3

For some cofibrant replacement q ∶ Q✶M → ✶M, the map LQ✶M

Lq

L✶M ✶N is a weak equivalence in N. Example : M = N and L = R = Id Example : Dold-Kan K ∶ Ch≥0 (❦Mod)∆op ∶ N

• ver a field ❦ of char. 0

Example (Castiglioni-Cortiñas) : Monoidal dual Dold-Kan Q ∶ Ch≥0 (❦Mod)Fin ∶ P

• David White (Denison University)

Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 8 / 19

Definition : Nice Quillen equivalence

1

M

L

N

R

• weak monoidal Quillen eq., both cofibrantly generated

2

Every generating cofibration in M has cofibrant domain.

3

U

g ∼

V ∈ NΣop

n , X ∈ NΣn, U, V, X cofibrant in N

⇒ U ⊗Σn X

g⊗Σn X ∼

V ⊗Σn X is a weak equivalence in N

4

In both M and N: For W ∈ MΣop

n , X ∈ MΣn cofibrant in M

coinvariants XΣn ∈ M is cofibrant [L(W ⊗ X)]Σn

(L2)Σn ∼

[LW ⊗ LX]Σn

W ⊗Σn (−)◻n preserves (acyclic) cofibrations

Ex: idM ⊣ idM, Dold-Kan, monoidal dual Dold-Kan, L strong sym. mon.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 9 / 19

Main Theorems : Lifting Quillen equivalences

Theorem (Entrywise cofibrant operads)

Suppose L ∶ M N ∶ R

• nice Quillen equivalence.

O, P are entrywise cofibrant operads on M, N. f ∶ O → RP operad map with f ∶ LO

∼ P entrywise weak equivalence.

Then L ⊣ R lifts to a Quillen equivalence Alg(O)

L ∼

Alg(P)

R

• between algebra categories.

Σ-cofibrant means cofibrant in ∏n≥0 MΣop

n .

Theorem (Σ-cofibrant operads)

If O, P are Σ-cofibrant, then we can replace nice Quillen eq with: L ⊣ R is a weak monoidal Quillen equivalence. Every generating cofibration in M has cofibrant domain.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 10 / 19

Proof outline

For cofibrant A ∈ Alg(O) and fibrant B ∈ Alg(P), need to show: (§) LA

ϕ ∼

B ∈ Alg(P) ⇐ ⇒ A

ϕ# ∼

RB ∈ Alg(O) Alg(O)

L

• U
• Alg(P)

R

• U
• M

L

• O○−
• N
• LUA

χA comparison map (Uϕ)χA

• ULA

Uϕ

• UB ∈ N

Key Lemma : For cofibrant A ∈ Alg(O), χA is a weak equivalence. Up to retract, ∅ = A0 → A1 → A2 → ⋯ → A, where for t ≥ 1 O ○ Xt

O○it

• pushout

At−1

• O ○ Yt

At for some generating cofibration it ∶ Xt → Yt in M. Use this filtration to successively approximate the comparison map χA.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 11 / 19

Special cases of the Main Theorems

Alg(O)

L

• U
• Alg(P)

R

• U
• perad algebra categories

forget

• M

L

• O○−
• N

R

• P○−
• monoidal model categories

free

• 1

(Rectification) Id ∶ M = N ∶ Id nice, O

∼ P of entrywise cofibrant operads

Alg(O)

∼

Alg(P)

• Similar rectification : Berger-Moerdijk, Elmendorf-Mandell, Harper, Muro,

Pavlov-Scholbach

2

Fixed operad O = P: AlgM(O)

∼

• AlgN(O)
• M

∼

• N
• For example, O = P = As (Schwede-Shipley), Com (Richter-Shipley), Op

(Berger-Moerdijk), Opnon−Σ (Muro), Σ-cofibrant operads (Fresse)

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 12 / 19

Rectification of ∞-algebras

M = N = Ch≥0(❦), ❦ a field of characteristic 0 Alg(O)

L ∼

• forget
• Alg(P)

R

• forget
• Ch(❦)≥0

free

• Ch(❦)≥0

free

• cofibrant replacement

Quillen equivalence O

∼ P

Alg(O)

∼

• Alg(P)

A∞

∼ As

A∞-algebras

∼

• DGA

E∞

∼ Com

E∞-algebras

∼

• CDGA

L∞

∼ Lie

L∞-algebras

∼

• Lie algebras

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 13 / 19

Commutative monoids / algebras and operads

Corollary

Suppose L ∶ M N ∶ R

• is a nice Quillen equivalence.

1

If the monoidal units in M, N are cofibrant, then: commutative monoids CMonoid(M)

∼ CMonoid(N)

• perads

Operad(M)

∼ Operad(N)

• enriched categories, objects = C

CatC(M)

∼

CatC(N)

• generalized props (G = pasting sch)

PropG(M)

∼

PropG(N)

• 2

If L is lax symmetric monoidal, and T ∈ CMonoid(M) cofibrant in M, then: CAlg(T)

∼

CAlg(LT)

• CAlg(T) = commutative T-algebras = commutative monoids in Mod(T)

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 14 / 19

Quillen : Reduced dg Lie ≃ Reduced simplicial Lie

Recover Quillen’s Quillen equivalence in Rational homotopy theory: start with Dold-Kan between reduced dg/simplicial ❦-modules apply Main Theorem to O = P = Lie operad (⋯)r = reduced (DGLie)r

Quillen ∼

• forget
• (SLie)r
• forget
• Chr

≥0(❦) Dold-Kan ∼

• free
• SModr(❦)
• free
• Quillen’s original proof is a calculation of the normalization

DGLie SLie

N

• .

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 15 / 19

Shipley : HR-algebra spectra ≃ DGA

Over each commutative unital ring R, there are Quillen equivalences: Mod(HR)

Z ∼

SpΣ(SMod)

N

• U
• SpΣ(Ch≥0)

D ∼

• L

∼

• Ch

C

• (†)

HR : Eilenberg-Mac Lane spectrum of R SpΣ(⋯) : symmetric spectra in (⋯) U : induced by forgetful SMod → SSet∗ N : induced by normalization SMod → Ch≥0 C = {C0(? ⊗ R[m])}m≥0, C0 = connective cover Apply the Main Theorem to (†) and the associative operad As:

Theorem (Shipley; 2007)

Each Quillen equivalence in (†) lifts to monoids, so Alg(HR)

∼

Monoid(⋯)

• Monoid(⋯)

∼

• ∼
• DGA
• .

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 16 / 19

Commutative HR-algebra spectra ≃ CDGA

Applying the Main Theorem to Shipley’s Quillen equivalences Mod(HR)

Z ∼

SpΣ(SMod)

N

• U
• SpΣ(Ch≥0)

D ∼

• L

∼

• Ch

C

• and the commutative operad Com:

Corollary (W.-Yau; 2016)

If R contains the rationals, then there is a zig-zag of three Quillen equivalences: CAlg(HR)

∼ CMonoid(⋯)

• CMonoid(⋯)

∼ ∼

• CDGA
• The hypothesis on R is needed to make sure CDGA inherits a model

structure. Shipley suggested that this might be true, but did not prove it. This is an improvement of a recent result of Richter-Shipley.

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 17 / 19

Commutative HR-algebra spectra ≃ CDGA

Theorem (Richter-Shipley; 2017)

If char(R) = 0, then there is a zig-zag of six Quillen equivalences: CAlg(HR)

∼ C(SpΣ(SMod))

• C(SpΣ(Ch≥0))

∼

• ∼
• C(SpΣ(Ch))
• CDGA

E∞Ch

• ∼
• E∞(SpΣ(Ch))

∼

• ∼
• All six Quillen equivalences are consequences of the Main Theorem:
• uses Com.
• uses E∞-operad.
• uses E∞

∼ Com

However, our version

Corollary (W.-Yau; 2018)

If R contains the rationals, then there is a zig-zag of three Quillen equivalences: CAlg(HR)

∼ C(SpΣ(SMod))

• C(SpΣ(Ch≥0))

∼ ∼

• CDGA
• involves three instead of six Quillen equivalences

does not go through E∞-algebras every Quillen equivalence is an application of the same theorem

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 18 / 19

References

Spitzweck: Operads, Algebras and Modules in General Model Categories, arxiv:0101102 Schwede-Shipley: Equivalences of monoidal model categories, arxiv:0209342, AGT 2003. Elmendorf-Mandell: Rings, modules, and algebras in infinite loop space theory, 0403403, Advances 2006. Fresse: Modules over Operads and Functors, Springer, 2009. Muro: Homotopy theory of nonsymmetric operads, AGT 2011. Richter-Shipley: An algebraic model for commutative HZ-algebras, 1411.7238, AGT 2017. White: Model structures on commutative monoids in general model categories, 1403.6759, JPAA 2017. White: Monoidal Bousfield localization and algebras over operads, 1404.5197 White-Yau: Bousfield Localization and Algebras over Colored Operads, 1503.06720, ACS 2017. White-Yau: Right Bousfield Localization and Operadic Algebras, 1512.07570 Batanin-White: Baez-Dolan Stabilization via (semi-)model categories of operads, CRM Proceedings, Barcelona, 2015. Batanin-White: Bousfield Localization and Eilenberg-Moore Categories, 1606.01537, HHA 2019? White-Yau: Homotopical Adjoint Lifting Theorem, 1606.01803, ACS 2019 White-Yau: Right Bousfield Localization and Eilenberg-Moore Categories, 1609.03635 Gutierrez-White: Encoding Equivariant Commutativity via Operads, 1707.02130, AGT 2018. White-Yau: Arrow Categories of Monoidal Model Categories, 1703.05359, Math. Scan. 2018 White-Yau: Smith Ideals of Operadic Algebras in Monoidal Model Categories, 1703.05377 Chorny-White: Homotopy theory of homotopy presheaves, 1805.05378 White-Yau: Comonadic Coalgebras and Bousfield Localization, 1805.11536

David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 19 / 19