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). L Alg ( O ) Alg ( P ) operad algebra categories R forget O ○− U P ○− U free L M N monoidal model categories R L ⊣ R Quillen equivalence. R lax symmetric monoidal functor. O, P are operads on M, N. ∼ � P entrywise weak equivalence. f ∶ O � → R P operad map with f ∶ L O 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 = ({ O n } n ≥ 0 ,γ, 1 ) on a symmetric monoidal category ( M , ⊗ , ✶ ) : Right Σ n -action on O n , Operadic composition O n ⊗ O k 1 ⊗ ⋯ ⊗ O k n γ � O k 1 +⋯+ k n , 1 � O 1 Operadic unit ✶ satisfying unity, associativity, and equivariance axioms. Definition An O -algebra is an object X ∈ M with structure maps O n ⊗ X ⊗ n λ � X , satisfying unity, associativity, and equivariance axioms. Category of O-algebras = Alg ( O ) O n 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. As n = ∐ Σ n ✶ Com-algebras are commutative monoids. Com n = ✶ Lie-algebras are Lie algebras in dg/simplicial modules. Op-algebras are operads. ∼ � Lie cofibrant resolution. L ∞ ∼ � As cofibrant resolution. A ∞ ∼ � Com cofibrant resolution. E ∞ 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, operads, categories, graphs, flows, ... Assume ( M , ⊗ , 1 ) is closed symmetric monoidal. g f � B 1 and A 2 � B 2 , the pushout product is the map For maps A 1 f ◻ g � B 1 ⊗ B 2 ( A 1 ⊗ B 2 )∐ A 1 ⊗ B 1 ( A 2 ⊗ B 1 ) 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 → B , j ◻ n is the corner map from colimit of punctured n -cube (with For j ∶ A � 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 ∈ M Σ op n , X ⊗ Σ n (−) ◻ 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 (∗) , Sp Σ (positive (flat) stable), StMod ( ❦ [ G ]) There are variations that assume less on M and more on O: if the condition only holds for X ∈ M Σ op 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: � X A � � K L � � ≃ ≃ B Y & D only 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 Alg M ∞ ( 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) L � N M Quillen equivalence between monoidal model categories, and R R is lax symmetric monoidal. 1 L 2 � LX ⊗ LY is a weak equivalence. For cofibrant X , Y ∈ M, L ( X ⊗ Y ) 2 ∼ L 2 is adjoint to X ⊗ Y � RLX ⊗ RLY � R ( LX ⊗ LY ) . For some cofibrant replacement q ∶ Q ✶ M � → ✶ M , the map 3 Lq � L ✶ M � ✶ N LQ ✶ M is a weak equivalence in N. Example : M = N and L = R = Id � ( ❦ Mod ) ∆ op ∶ N Example : Dold-Kan K ∶ Ch ≥ 0 over a field ❦ of char. 0 Example (Castiglioni-Cortiñas) : Monoidal dual Dold-Kan � ( ❦ Mod ) Fin ∶ P Q ∶ Ch ≥ 0 David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 8 / 19

� Definition : Nice Quillen equivalence L � N M weak monoidal Quillen eq., both cofibrantly generated 1 R Every generating cofibration in M has cofibrant domain. 2 g � V ∈ N Σ op n , X ∈ N Σ n , U , V , X cofibrant in N U 3 ∼ g ⊗ Σ n X � V ⊗ Σ n X is a weak equivalence in N ⇒ U ⊗ Σ n X ∼ In both M and N: For W ∈ M Σ op n , X ∈ M Σ n cofibrant in M 4 coinvariants X Σ n ∈ M is cofibrant ( L 2 ) Σ n � [ LW ⊗ LX ] Σ n [ L ( W ⊗ X )] Σ n ∼ W ⊗ Σ n (−) ◻ n preserves (acyclic) cofibrations Ex: id M ⊣ id M , 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) � N ∶ R Suppose L ∶ M nice Quillen equivalence. O , P are entrywise cofibrant operads on M , N . ∼ � P entrywise weak equivalence. f ∶ O � → R P operad map with f ∶ L O Then L ⊣ R lifts to a Quillen equivalence L � Alg ( P ) Alg ( O ) ∼ R between algebra categories. Σ -cofibrant means cofibrant in ∏ n ≥ 0 M Σ op 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: ϕ # ϕ � B ∈ Alg ( P ) � RB ∈ Alg ( O ) ( § ) ⇐ ⇒ LA A ∼ ∼ L χ A Alg ( O ) Alg ( P ) LUA ULA comparison map � R O ○− U U ( U ϕ ) χ A U ϕ L UB ∈ N M N Key Lemma : For cofibrant A ∈ Alg ( O ) , χ A is a weak equivalence. Up to retract, ∅ = A 0 � → A 1 � → A 2 � → ⋯ � → A , where for t ≥ 1 O ○ X t A t − 1 O ○ i t � pushout O ○ Y t � A t for some generating cofibration i t ∶ X t � → Y t in M. Use this filtration to successively approximate the comparison map χ A . David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 11 / 19

� � � � � � � � � � � � � � � � � � � Special cases of the Main Theorems L Alg ( O ) Alg ( P ) operad algebra categories R forget O ○− U P ○− U free L M N monoidal model categories R (Rectification) Id ∶ M = N ∶ Id nice, O ∼ � P of entrywise cofibrant operads 1 � Alg ( P ) Alg ( O ) ∼ Similar rectification : Berger-Moerdijk, Elmendorf-Mandell, Harper, Muro, Pavlov-Scholbach Fixed operad O = P: Alg M ( O ) Alg N ( O ) 2 ∼ M N ∼ For example, O = P = As (Schwede-Shipley), Com (Richter-Shipley), Op (Berger-Moerdijk), Op non − 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 L Alg ( O ) Alg ( P ) ∼ R forget forget free free Ch ( ❦ ) ≥ 0 Ch ( ❦ ) ≥ 0 cofibrant replacement Quillen equivalence ∼ � P Alg ( O ) Alg ( P ) O ∼ ∼ � As A ∞ A ∞ -algebras DGA ∼ ∼ � Com E ∞ E ∞ -algebras CDGA ∼ ∼ � Lie L ∞ L ∞ -algebras Lie algebras ∼ David White (Denison University) Homotopical Adjoint Lifting Theorem August 1, 2019 / Ottawa ATCT Conference 13 / 19