On the Goodwillie Derivatives of the Identity in Structured Ring - PowerPoint PPT Presentation

� � � � � � Functor calculus Let F : S ∗ → S ∗ be a homotopy functor (i.e. X ≃ Y = ⇒ F ( X ) ≃ F ( Y )) and assume for simplicity that F is reduced (i.e. F ( ∗ ) ≃ ∗ ). Goodwillie constructs a Taylor tower of n-excisive approximations { P n F } and natural transformations of the following form . . Remarks . The functors P n F may be thought of as polynomials of degree n . P 1 F is a linear approximation to F . P 3 F For nice (i.e. analytic ) F and sufficiently connected spaces X , P 2 F F ( X ) ≃ holim n P n F ( X ) . X ≃ holim n P n Id S ∗ ( X ), if X is 1-connected F P 1 F Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 3 / 14

� � � � � � Functor calculus Let F : S ∗ → S ∗ be a homotopy functor (i.e. X ≃ Y = ⇒ F ( X ) ≃ F ( Y )) and assume for simplicity that F is reduced (i.e. F ( ∗ ) ≃ ∗ ). Goodwillie constructs a Taylor tower of n-excisive approximations { P n F } and natural transformations of the following form . . Remarks . The functors P n F may be thought of as polynomials of degree n . P 1 F is a linear approximation to F . P 3 F For nice (i.e. analytic ) F and sufficiently connected spaces X , P 2 F F ( X ) ≃ holim n P n F ( X ) . X ≃ holim n P n Id S ∗ ( X ), if X is 1-connected (i.e. Id S ∗ is 1-analytic) F P 1 F Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 3 / 14

Functor calculus (cont.) – Ex. linear functors Def. (linear functor) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) � D C Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Id S ∗ is not linear Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Id S ∗ is not linear (S ∗ is not stable). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Id S ∗ is not linear (S ∗ is not stable). In particular, P 1 Id S ∗ ≃ Ω ∞ Σ ∞ Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Id S ∗ is not linear (S ∗ is not stable). In particular, P 1 Id S ∗ ≃ Ω ∞ Σ ∞ If F ( ∗ ) ≃ ∗ , then P 1 F ( X ) ≃ Ω ∞ ( E ∧ Σ ∞ X ) for some E ∈ Spt Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Id S ∗ is not linear (S ∗ is not stable). In particular, P 1 Id S ∗ ≃ Ω ∞ Σ ∞ If F ( ∗ ) ≃ ∗ , then P 1 F ( X ) ≃ Ω ∞ ( E ∧ Σ ∞ X ) for some E ∈ Spt (note that E ∧ − : Spt → Spt is linear). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

� � � � � � Functor calculus (cont.) – Ex. linear functors Def. (linear functor) A functor F is linear (i.e. 1-excisive) if F takes homotopy pushout squares to homotopy pullback squares, i.e. A B (ho. push) F ( A ) F ( B ) (ho. pull) F − − → � D C � F ( D ) F ( C ) Remarks : The stabilization functor X �→ Ω ∞ Σ ∞ X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) Id S ∗ is not linear (S ∗ is not stable). In particular, P 1 Id S ∗ ≃ Ω ∞ Σ ∞ If F ( ∗ ) ≃ ∗ , then P 1 F ( X ) ≃ Ω ∞ ( E ∧ Σ ∞ X ) for some E ∈ Spt (note that E ∧ − : Spt → Spt is linear). We call E the first derivative of F . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 4 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. We can compute ∂ n F from D n F via cross-effects . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. We can compute ∂ n F from D n F via cross-effects . Ex. Derivatives of Id S ∗ Note, D 1 Id S ∗ ( X ) ≃ P 1 Id S ∗ ( X ) ≃ Ω ∞ Σ ∞ X and therefore ∂ 1 Id S ∗ ≃ S . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. We can compute ∂ n F from D n F via cross-effects . Ex. Derivatives of Id S ∗ Note, D 1 Id S ∗ ( X ) ≃ P 1 Id S ∗ ( X ) ≃ Ω ∞ Σ ∞ X and therefore ∂ 1 Id S ∗ ≃ S . For n ≥ 2, ∂ n Id S ∗ is related to the partition poset complex Par( n ) [Johnson, Arone-Mahowald]. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. We can compute ∂ n F from D n F via cross-effects . Ex. Derivatives of Id S ∗ Note, D 1 Id S ∗ ( X ) ≃ P 1 Id S ∗ ( X ) ≃ Ω ∞ Σ ∞ X and therefore ∂ 1 Id S ∗ ≃ S . For n ≥ 2, ∂ n Id S ∗ is related to the partition poset complex Par( n ) [Johnson, Arone-Mahowald]. In particular, ∂ 2 Id S ∗ ≃ Ω S with trivial Σ 2 action. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. We can compute ∂ n F from D n F via cross-effects . Ex. Derivatives of Id S ∗ Note, D 1 Id S ∗ ( X ) ≃ P 1 Id S ∗ ( X ) ≃ Ω ∞ Σ ∞ X and therefore ∂ 1 Id S ∗ ≃ S . For n ≥ 2, ∂ n Id S ∗ is related to the partition poset complex Par( n ) [Johnson, Arone-Mahowald]. In particular, ∂ 2 Id S ∗ ≃ Ω S with trivial Σ 2 action. The collection ∂ ∗ F forms a symmetric sequence of spectra. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives Set D n F to be the fiber D n F := hofib( P n F → P n − 1 F ). Thm. [Goodwillie] There is a unique (up to htpy.) spectrum ∂ n F with Σ n action such that D n F ( X ) ≃ Ω ∞ ( ∂ n F ∧ Σ n (Σ ∞ X ) ∧ n ). We call ∂ n F the n-th derivative of F . Remark : D n F ( X ) bears striking resemblance to ( f ( n ) (0) x n ) / n !. We can compute ∂ n F from D n F via cross-effects . Ex. Derivatives of Id S ∗ Note, D 1 Id S ∗ ( X ) ≃ P 1 Id S ∗ ( X ) ≃ Ω ∞ Σ ∞ X and therefore ∂ 1 Id S ∗ ≃ S . For n ≥ 2, ∂ n Id S ∗ is related to the partition poset complex Par( n ) [Johnson, Arone-Mahowald]. In particular, ∂ 2 Id S ∗ ≃ Ω S with trivial Σ 2 action. The collection ∂ ∗ F forms a symmetric sequence of spectra. We are interested in understanding what extra structure this sequence posses. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) unit map 1 → O [1] Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) unit map 1 → O [1] action maps O [ n ] ⊗ O [ k 1 ] ⊗ · · · ⊗ O [ k n ] → O [ k 1 + · · · + k n ] subject to equivariance, associativity and unitality conditions. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) unit map 1 → O [1] action maps O [ n ] ⊗ O [ k 1 ] ⊗ · · · ⊗ O [ k n ] → O [ k 1 + · · · + k n ] subject to equivariance, associativity and unitality conditions. Operads are precisely monoids with respect to the composition product ◦ for symmetric sequences Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) unit map 1 → O [1] action maps O [ n ] ⊗ O [ k 1 ] ⊗ · · · ⊗ O [ k n ] → O [ k 1 + · · · + k n ] subject to equivariance, associativity and unitality conditions. Operads are precisely monoids with respect to the composition product ◦ for symmetric sequences, i.e. there are associative and unital maps O ◦ O → O Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) unit map 1 → O [1] action maps O [ n ] ⊗ O [ k 1 ] ⊗ · · · ⊗ O [ k n ] → O [ k 1 + · · · + k n ] subject to equivariance, associativity and unitality conditions. Operads are precisely monoids with respect to the composition product ◦ for symmetric sequences, i.e. there are associative and unital maps O ◦ O → O and I → O Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads An operad may be thought of as a useful tool for describing spectra with extra algebraic structure, i.e. (commutative) ring spectra, A ∞ -ring spectra, or E n -ring spectra (1 ≤ n ≤ ∞ ). Def. [May, Boardman-Vogt] An operad O in a symmetric monoidal category (C , ⊗ , 1 ) consists of objects O [ n ] for n ≥ 0 with actions by Σ n (i.e. a symmetric sequence) unit map 1 → O [1] action maps O [ n ] ⊗ O [ k 1 ] ⊗ · · · ⊗ O [ k n ] → O [ k 1 + · · · + k n ] subject to equivariance, associativity and unitality conditions. Operads are precisely monoids with respect to the composition product ◦ for symmetric sequences, i.e. there are associative and unital maps O ◦ O → O and I → O (here I [1] = 1 and I [ k ] = ∗ for k � = 1). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 6 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt together with action maps O [ n ] ∧ Σ n X ∧ n → X for all n ≥ 0 Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt together with action maps O [ n ] ∧ Σ n X ∧ n → X for all n ≥ 0, subject to associativity and unitality conditions. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt together with action maps O [ n ] ∧ Σ n X ∧ n → X for all n ≥ 0, subject to associativity and unitality conditions. We think of the term O [ n ] as parametrizing the possible n -ary operations on X Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt together with action maps O [ n ] ∧ Σ n X ∧ n → X for all n ≥ 0, subject to associativity and unitality conditions. We think of the term O [ n ] as parametrizing the possible n -ary operations on X , e.g. if O [ n ] ≃ S (with free Σ n action), then O describes homotopy commutative (i.e. E ∞ -) monoids in Spt as its algebras. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt together with action maps O [ n ] ∧ Σ n X ∧ n → X for all n ≥ 0, subject to associativity and unitality conditions. We think of the term O [ n ] as parametrizing the possible n -ary operations on X , e.g. if O [ n ] ≃ S (with free Σ n action), then O describes homotopy commutative (i.e. E ∞ -) monoids in Spt as its algebras. We set Alg O to be the category of algebras over a given operad O together with structure preserving maps. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Operads (cont.) – Algebras We will focus on the category of symmetric spectra Spt = (Sp Σ , ∧ , S ). Def. (algebra over an operad) An algebra over an operad O in Spt is an object X ∈ Spt together with action maps O [ n ] ∧ Σ n X ∧ n → X for all n ≥ 0, subject to associativity and unitality conditions. We think of the term O [ n ] as parametrizing the possible n -ary operations on X , e.g. if O [ n ] ≃ S (with free Σ n action), then O describes homotopy commutative (i.e. E ∞ -) monoids in Spt as its algebras. We set Alg O to be the category of algebras over a given operad O together with structure preserving maps. Note, an algebra over X is equivalently an algebra over the assocaited monad on Spt � O [ n ] ∧ Σ n X ∧ n . X �→ O ◦ ( X ) = n ≥ 0 Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) � Mod O [1] Alg O U Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) � Mod O [1] Alg O U ∼ Let J be a factorization O ֒ → J − → τ 1 O . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� � Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) Q := J ◦ O ( − ) � Mod O [1] � Alg J ∼ Mod O [1] Alg O Alg O U U ∼ Let J be a factorization O ֒ → J − → τ 1 O . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� � Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) Q := J ◦ O ( − ) � Mod O [1] � Alg J ∼ Mod O [1] Alg O Alg O U U ∼ Let J be a factorization O ֒ → J − → τ 1 O . Set TQ := L Q (i.e. the left-derived functor of Q ). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� � Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) Q := J ◦ O ( − ) � Mod O [1] � Alg J ∼ Mod O [1] Alg O Alg O U U ∼ Let J be a factorization O ֒ → J − → τ 1 O . Set TQ := L Q (i.e. the left-derived functor of Q ). TQ( X ) is often called topological Quillen homology spectrum of X . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� � Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) Q := J ◦ O ( − ) � Mod O [1] � Alg J ∼ Mod O [1] Alg O Alg O U U ∼ Let J be a factorization O ֒ → J − → τ 1 O . Set TQ := L Q (i.e. the left-derived functor of Q ). TQ( X ) is often called topological Quillen homology spectrum of X . Basterra-Mandell show ( Q , U ) is equivalent to the stabilization adjunction for O -algebras Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

� � Functor calculus in Alg O – Stabilization Fix an operad O in Spt with O [0] = ∗ and assume for simplicity that O [1] ∼ = S . We define the n-th truncation of O , τ n O , by � O [ k ] k ≤ n τ n O [ k ] := ∗ k > n There is a tower O → · · · → τ 3 O → τ 2 O → τ 1 O of operads and the bottom map O → τ 1 O induces a change of operads adjunction τ 1 O◦ O ( − ) Q := J ◦ O ( − ) � Mod O [1] � Alg J ∼ Mod O [1] Alg O Alg O U U ∼ Let J be a factorization O ֒ → J − → τ 1 O . Set TQ := L Q (i.e. the left-derived functor of Q ). TQ( X ) is often called topological Quillen homology spectrum of X . Basterra-Mandell show ( Q , U ) is equivalent to the stabilization adjunction for O -algebras, i.e. TQ( X ) ≃ Σ ∞ X . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

� � � � � � Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form . . . τ 3 O ◦ O ( − ) τ 2 O ◦ O ( − ) Id τ 1 O ◦ O ( − ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

� � � � � � Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form In particular, there are equivalences . . . D n Id( X ) ≃ U ( O [ n ] ∧ Σ n TQ( X ) ∧ n ) τ 3 O ◦ O ( − ) τ 2 O ◦ O ( − ) Id τ 1 O ◦ O ( − ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

� � � � � � Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form In particular, there are equivalences . . . D n Id( X ) ≃ U ( O [ n ] ∧ Σ n TQ( X ) ∧ n ) ∂ n Id ≃ O [ n ] (as Σ n -objects in Spt) τ 3 O ◦ O ( − ) τ 2 O ◦ O ( − ) Id τ 1 O ◦ O ( − ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

� � � � � � Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form In particular, there are equivalences . . . D n Id( X ) ≃ U ( O [ n ] ∧ Σ n TQ( X ) ∧ n ) ∂ n Id ≃ O [ n ] (as Σ n -objects in Spt) τ 3 O ◦ O ( − ) Thus, ∂ ∗ Id ≃ O as symmetric sequences . τ 2 O ◦ O ( − ) Id τ 1 O ◦ O ( − ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

� � � � � � Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form In particular, there are equivalences . . . D n Id( X ) ≃ U ( O [ n ] ∧ Σ n TQ( X ) ∧ n ) ∂ n Id ≃ O [ n ] (as Σ n -objects in Spt) τ 3 O ◦ O ( − ) Thus, ∂ ∗ Id ≃ O as symmetric sequences . It has been a long standing conjecture τ 2 O ◦ O ( − ) that ∂ ∗ Id ≃ O as operads Id τ 1 O ◦ O ( − ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

� � � � � � Functor calculus in Alg O (cont.) – Taylor tower of Id Harper-Hess and Pereira show that the Taylor tower of the identity in Alg O takes the following form In particular, there are equivalences . . . D n Id( X ) ≃ U ( O [ n ] ∧ Σ n TQ( X ) ∧ n ) ∂ n Id ≃ O [ n ] (as Σ n -objects in Spt) τ 3 O ◦ O ( − ) Thus, ∂ ∗ Id ≃ O as symmetric sequences . It has been a long standing conjecture τ 2 O ◦ O ( − ) that ∂ ∗ Id ≃ O as operads , the missing piece being the lack of an intrinsic operad structure on ∂ ∗ Id with which to Id τ 1 O ◦ O ( − ) compare to O . Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Example: Set ℓ ( X ) = Cobar( ∗ , X , ∗ ) w.r.t. the diagonal map on X Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

���� Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Example: Set ℓ ( X ) = Cobar( ∗ , X , ∗ ) w.r.t. the diagonal map on X , i.e. ��� X × 2 �� X X × 3 · · · ℓ ( X ) = ∗ Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

���� Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Example: Set ℓ ( X ) = Cobar( ∗ , X , ∗ ) w.r.t. the diagonal map on X , i.e. ��� X × 2 �� X X × 3 · · · ℓ ( X ) = ∗ Then, there is a monoidal pairing ℓ ( X ) � ℓ ( X ) → ℓ ( X ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

���� Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Example: Set ℓ ( X ) = Cobar( ∗ , X , ∗ ) w.r.t. the diagonal map on X , i.e. ��� X × 2 �� X X × 3 · · · ℓ ( X ) = ∗ Then, there is a monoidal pairing ℓ ( X ) � ℓ ( X ) → ℓ ( X ) induced by X × p × X × q ∼ = X × p + q Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

���� Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Example: Set ℓ ( X ) = Cobar( ∗ , X , ∗ ) w.r.t. the diagonal map on X , i.e. ��� X × 2 �� X X × 3 · · · ℓ ( X ) = ∗ Then, there is a monoidal pairing ℓ ( X ) � ℓ ( X ) → ℓ ( X ) induced by X × p × X × q ∼ = X × p + q , and Tot ℓ ( X ) ≃ Ω X as A ∞ -monoids Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

���� Main theorem Thm. [C] The derivatives of the identity in Alg O posses an intrinsic “homotopy coherent” operad structure with respect to which ∂ ∗ Id ≃ O as operads. Idea of proof: Our method is to adapt a technique of McClure-Smith that if Y • is a cosimplicial space which is a monoid with respect to the box product � [Batanin], then Tot Y • is an A ∞ -monoid in spaces. Example: Set ℓ ( X ) = Cobar( ∗ , X , ∗ ) w.r.t. the diagonal map on X , i.e. ��� X × 2 �� X X × 3 · · · ℓ ( X ) = ∗ Then, there is a monoidal pairing ℓ ( X ) � ℓ ( X ) → ℓ ( X ) induced by X × p × X × q ∼ = X × p + q , and Tot ℓ ( X ) ≃ Ω X as A ∞ -monoids (note that Ω X is an A ∞ -monoid under composition of loops). Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 10 / 14

Proof sketch of main thm. We use the Bousfield-Kan cosimplicial resolution of Id with respect to the stabilization adjunction ( Q , U ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 11 / 14

Proof sketch of main thm. We use the Bousfield-Kan cosimplicial resolution of Id with respect to the stabilization adjunction ( Q , U ), i.e. Id Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 11 / 14

Proof sketch of main thm. We use the Bousfield-Kan cosimplicial resolution of Id with respect to the stabilization adjunction ( Q , U ), i.e. ��� UQUQUQ · · · � �� UQUQ � Id → UQ Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 11 / 14

Proof sketch of main thm. We use the Bousfield-Kan cosimplicial resolution of Id with respect to the stabilization adjunction ( Q , U ), i.e. ��� UQUQUQ · · · � �� UQUQ � Id → UQ ��� J ◦ O J ◦ O J ◦ O ( − ) · · · � �� J ◦ O J ◦ O ( − ) � ≃ J ◦ O ( − ) Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 11 / 14

Proof sketch of main thm. We use the Bousfield-Kan cosimplicial resolution of Id with respect to the stabilization adjunction ( Q , U ), i.e. ��� UQUQUQ · · · � �� UQUQ � Id → UQ ��� J ◦ O J ◦ O J ◦ O ( − ) · · · � �� J ◦ O J ◦ O ( − ) � ≃ J ◦ O ( − ) Blomquist shows the maps Id → holim ∆ ≤ k − 1 C ( − ) are sufficiently connected to induce equivalences on derivatives Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 11 / 14