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

On the Goodwillie Derivatives of the Identity in Structured Ring Spectra

Duncan Clark

Ohio State University

GOATS 2, June 6th, 2020

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 1 / 14

Main idea

Guiding principle

The Goodwillie derivatives of the identity functor in a suitably nice model category C (denoted ∂∗ IdC) should come equipped with a canonical

• perad structure.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 2 / 14

Main idea

Guiding principle

The Goodwillie derivatives of the identity functor in a suitably nice model category C (denoted ∂∗ IdC) should come equipped with a canonical

• perad structure.

Examples

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 2 / 14

Main idea

Guiding principle

The Goodwillie derivatives of the identity functor in a suitably nice model category C (denoted ∂∗ IdC) should come equipped with a canonical

• perad structure.

Examples

For C = S∗, the category of based spaces, Ching shows that ∂∗ IdS∗ is an operad

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 2 / 14

Main idea

Guiding principle

The Goodwillie derivatives of the identity functor in a suitably nice model category C (denoted ∂∗ IdC) should come equipped with a canonical

• perad structure.

Examples

For C = S∗, the category of based spaces, Ching shows that ∂∗ IdS∗ is an operad If O is a reduced operad of spectra, then ∂∗ IdAlgO is a “highly homotopy coherent” operad which is equivalent to O [C.]

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 2 / 14

Main idea

Guiding principle

The Goodwillie derivatives of the identity functor in a suitably nice model category C (denoted ∂∗ IdC) should come equipped with a canonical

• perad structure.

Examples

For C = S∗, the category of based spaces, Ching shows that ∂∗ IdS∗ is an operad If O is a reduced operad of spectra, then ∂∗ IdAlgO is a “highly homotopy coherent” operad which is equivalent to O [C.] General approach using ∞-categories [Ching, Lurie]

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 2 / 14

Main idea

Guiding principle

The Goodwillie derivatives of the identity functor in a suitably nice model category C (denoted ∂∗ IdC) should come equipped with a canonical

• perad structure.

Examples

For C = S∗, the category of based spaces, Ching shows that ∂∗ IdS∗ is an operad If O is a reduced operad of spectra, then ∂∗ IdAlgO is a “highly homotopy coherent” operad which is equivalent to O [C.] General approach using ∞-categories [Ching, Lurie] First, we’ll recall some necessary background on functor calculus and

• perads.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 2 / 14

Functor calculus

Let F : S∗ → S∗ be a homotopy functor (i.e. X ≃ Y = ⇒ F(X) ≃ F(Y ))

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(∗) ≃ ∗).

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 {PnF} and natural transformations of the following form

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 {PnF} and natural transformations of the following form P1F

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 {PnF} and natural transformations of the following form P2F

• P1F

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 {PnF} and natural transformations of the following form P3F

• P2F
• P1F

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• P1F

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F

P1F

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

Remarks

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

Remarks The functors PnF may be thought of as polynomials of degree n.

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

Remarks The functors PnF may be thought of as polynomials of degree n. P1F is a linear approximation to 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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

Remarks The functors PnF may be thought of as polynomials of degree n. P1F is a linear approximation to F. For nice (i.e. analytic) F and sufficiently connected spaces X, F(X) ≃ holimn PnF(X).

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

Remarks The functors PnF may be thought of as polynomials of degree n. P1F is a linear approximation to F. For nice (i.e. analytic) F and sufficiently connected spaces X, F(X) ≃ holimn PnF(X). X ≃ holimn Pn IdS∗(X), if X is 1-connected

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 {PnF} and natural transformations of the following form . . .

• P3F
• P2F
• F
• P1F

Remarks The functors PnF may be thought of as polynomials of degree n. P1F is a linear approximation to F. For nice (i.e. analytic) F and sufficiently connected spaces X, F(X) ≃ holimn PnF(X). X ≃ holimn Pn IdS∗(X), if X is 1-connected (i.e. IdS∗ is 1-analytic)

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
• C

D

(ho. push)

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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull)

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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) Remarks: The stabilization functor X → Ω∞Σ∞X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) IdS∗ 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) Remarks: The stabilization functor X → Ω∞Σ∞X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) IdS∗ 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) Remarks: The stabilization functor X → Ω∞Σ∞X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) IdS∗ is not linear (S∗ is not stable). In particular, P1 IdS∗ ≃ 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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) Remarks: The stabilization functor X → Ω∞Σ∞X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) IdS∗ is not linear (S∗ is not stable). In particular, P1 IdS∗ ≃ Ω∞Σ∞ If F(∗) ≃ ∗, then P1F(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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) Remarks: The stabilization functor X → Ω∞Σ∞X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) IdS∗ is not linear (S∗ is not stable). In particular, P1 IdS∗ ≃ Ω∞Σ∞ If F(∗) ≃ ∗, then P1F(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
• C

D

(ho. push)

F

− − → F(A)

• F(B)
• F(C)

F(D)

(ho. pull) Remarks: The stabilization functor X → Ω∞Σ∞X is 1-excisive Any homology theory is 1-excisive (i.e. satisfies Mayer-Vietoris) IdS∗ is not linear (S∗ is not stable). In particular, P1 IdS∗ ≃ Ω∞Σ∞ If F(∗) ≃ ∗, then P1F(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 DnF to be the fiber DnF := hofib(PnF → Pn−1F).

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives

Set DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n).

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives

Set DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF 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 DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives

Set DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!. We can compute ∂nF from DnF via cross-effects.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives

Set DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!. We can compute ∂nF from DnF via cross-effects.

• Ex. Derivatives of IdS∗

Note, D1 IdS∗(X) ≃ P1 IdS∗(X) ≃ Ω∞Σ∞X and therefore ∂1 IdS∗ ≃ S.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives

Set DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!. We can compute ∂nF from DnF via cross-effects.

• Ex. Derivatives of IdS∗

Note, D1 IdS∗(X) ≃ P1 IdS∗(X) ≃ Ω∞Σ∞X and therefore ∂1 IdS∗ ≃ S. For n ≥ 2, ∂n IdS∗ 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 DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!. We can compute ∂nF from DnF via cross-effects.

• Ex. Derivatives of IdS∗

Note, D1 IdS∗(X) ≃ P1 IdS∗(X) ≃ Ω∞Σ∞X and therefore ∂1 IdS∗ ≃ S. For n ≥ 2, ∂n IdS∗ is related to the partition poset complex Par(n) [Johnson, Arone-Mahowald]. In particular, ∂2 IdS∗ ≃ ΩS with trivial Σ2 action.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 5 / 14

Functor calculus (cont.) – Derivatives

Set DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!. We can compute ∂nF from DnF via cross-effects.

• Ex. Derivatives of IdS∗

Note, D1 IdS∗(X) ≃ P1 IdS∗(X) ≃ Ω∞Σ∞X and therefore ∂1 IdS∗ ≃ S. For n ≥ 2, ∂n IdS∗ is related to the partition poset complex Par(n) [Johnson, Arone-Mahowald]. In particular, ∂2 IdS∗ ≃ ΩS with trivial Σ2 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 DnF to be the fiber DnF := hofib(PnF → Pn−1F).

• Thm. [Goodwillie]

There is a unique (up to htpy.) spectrum ∂nF with Σn action such that DnF(X) ≃ Ω∞(∂nF ∧Σn (Σ∞X)∧n). We call ∂nF the n-th derivative of F. Remark: DnF(X) bears striking resemblance to (f (n)(0)xn)/n!. We can compute ∂nF from DnF via cross-effects.

• Ex. Derivatives of IdS∗

Note, D1 IdS∗(X) ≃ P1 IdS∗(X) ≃ Ω∞Σ∞X and therefore ∂1 IdS∗ ≃ S. For n ≥ 2, ∂n IdS∗ is related to the partition poset complex Par(n) [Johnson, Arone-Mahowald]. In particular, ∂2 IdS∗ ≃ ΩS with trivial Σ2 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 En-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 En-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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn

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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (i.e. a symmetric sequence)

unit map 1 → O[1] action maps O[n] ⊗ O[k1] ⊗ · · · ⊗ O[kn] → O[k1 + · · · + kn] 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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (i.e. a symmetric sequence)

unit map 1 → O[1] action maps O[n] ⊗ O[k1] ⊗ · · · ⊗ O[kn] → O[k1 + · · · + kn] 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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (i.e. a symmetric sequence)

unit map 1 → O[1] action maps O[n] ⊗ O[k1] ⊗ · · · ⊗ O[kn] → O[k1 + · · · + kn] 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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (i.e. a symmetric sequence)

unit map 1 → O[1] action maps O[n] ⊗ O[k1] ⊗ · · · ⊗ O[kn] → O[k1 + · · · + kn] 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 En-ring spectra (1 ≤ n ≤ ∞).

• Def. [May, Boardman-Vogt]

An operad O in a symmetric monoidal category (C, ⊗, 1) consists of

• bjects O[n] for n ≥ 0 with actions by Σn (i.e. a symmetric sequence)

unit map 1 → O[1] action maps O[n] ⊗ O[k1] ⊗ · · · ⊗ O[kn] → O[k1 + · · · + kn] 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 = (SpΣ, ∧, 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 = (SpΣ, ∧, 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 = (SpΣ, ∧, 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] ∧Σn 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 = (SpΣ, ∧, 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] ∧Σn 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 = (SpΣ, ∧, 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] ∧Σn 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

• n 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 = (SpΣ, ∧, 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] ∧Σn 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

• n X, e.g. if O[n] ≃ S (with free Σn 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 = (SpΣ, ∧, 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] ∧Σn 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

• n X, e.g. if O[n] ≃ S (with free Σn action), then O describes homotopy

commutative (i.e. E∞-) monoids in Spt as its algebras. We set AlgO 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 = (SpΣ, ∧, 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] ∧Σn 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

• n X, e.g. if O[n] ≃ S (with free Σn action), then O describes homotopy

commutative (i.e. E∞-) monoids in Spt as its algebras. We set AlgO 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 X → O ◦ (X) =

• n≥0

O[n] ∧Σn X ∧n.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 7 / 14

Functor calculus in AlgO – 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 AlgO – 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 AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O 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 AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• Duncan Clark (Ohio State University)

Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• Let J be a factorization O ֒

→ J

∼

− → τ1O.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• AlgO

Q:=J◦O(−)

AlgJ ∼ ModO[1]

U

• Let J be a factorization O ֒

→ J

∼

− → τ1O.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 8 / 14

Functor calculus in AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• AlgO

Q:=J◦O(−)

AlgJ ∼ ModO[1]

U

• Let J be a factorization O ֒

→ J

∼

− → τ1O. Set TQ := LQ (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 AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• AlgO

Q:=J◦O(−)

AlgJ ∼ ModO[1]

U

• Let J be a factorization O ֒

→ J

∼

− → τ1O. Set TQ := LQ (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 AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• AlgO

Q:=J◦O(−)

AlgJ ∼ ModO[1]

U

• Let J be a factorization O ֒

→ J

∼

− → τ1O. Set TQ := LQ (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 AlgO – 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, τnO, by τnO[k] :=

• O[k]

k ≤ n ∗ k > n There is a tower O → · · · → τ3O → τ2O → τ1O of operads and the bottom map O → τ1O induces a change of operads adjunction AlgO

τ1O◦O(−)

ModO[1]

U

• AlgO

Q:=J◦O(−)

AlgJ ∼ ModO[1]

U

• Let J be a factorization O ֒

→ J

∼

− → τ1O. Set TQ := LQ (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 AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Functor calculus in AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form . . .

• τ3O ◦O (−)
• τ2O ◦O (−)
• Id
• τ1O ◦O (−)

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Functor calculus in AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form . . .

• τ3O ◦O (−)
• τ2O ◦O (−)
• Id
• τ1O ◦O (−)

In particular, there are equivalences Dn Id(X) ≃ U(O[n] ∧Σn TQ(X)∧n)

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Functor calculus in AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form . . .

• τ3O ◦O (−)
• τ2O ◦O (−)
• Id
• τ1O ◦O (−)

In particular, there are equivalences Dn Id(X) ≃ U(O[n] ∧Σn TQ(X)∧n) ∂n Id ≃ O[n] (as Σn-objects in Spt)

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Functor calculus in AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form . . .

• τ3O ◦O (−)
• τ2O ◦O (−)
• Id
• τ1O ◦O (−)

In particular, there are equivalences Dn Id(X) ≃ U(O[n] ∧Σn TQ(X)∧n) ∂n Id ≃ O[n] (as Σn-objects in Spt) Thus, ∂∗ Id ≃ O as symmetric sequences.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Functor calculus in AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form . . .

• τ3O ◦O (−)
• τ2O ◦O (−)
• Id
• τ1O ◦O (−)

In particular, there are equivalences Dn Id(X) ≃ U(O[n] ∧Σn TQ(X)∧n) ∂n Id ≃ O[n] (as Σn-objects in Spt) Thus, ∂∗ Id ≃ O as symmetric sequences. It has been a long standing conjecture that ∂∗ Id ≃ O as operads

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 9 / 14

Functor calculus in AlgO (cont.) – Taylor tower of Id

Harper-Hess and Pereira show that the Taylor tower of the identity in AlgO takes the following form . . .

• τ3O ◦O (−)
• τ2O ◦O (−)
• Id
• τ1O ◦O (−)

In particular, there are equivalences Dn Id(X) ≃ U(O[n] ∧Σn TQ(X)∧n) ∂n Id ≃ O[n] (as Σn-objects in Spt) Thus, ∂∗ Id ≃ O as symmetric sequences. It has been a long standing conjecture that ∂∗ Id ≃ O as operads, the missing piece being the lack of an intrinsic

• perad structure on ∂∗ Id with which to

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 AlgO 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 AlgO 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 AlgO 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 AlgO 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. ℓ(X) = ∗

X X ×2

• X ×3 · · ·

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 AlgO 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. ℓ(X) = ∗

X X ×2

• X ×3 · · ·

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 AlgO 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. ℓ(X) = ∗

X X ×2

• X ×3 · · ·

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 AlgO 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. ℓ(X) = ∗

X X ×2

• X ×3 · · ·

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) ≃ ΩX 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 AlgO 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. ℓ(X) = ∗

X X ×2

• X ×3 · · ·

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) ≃ ΩX as A∞-monoids (note that ΩX 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. Id →

• UQ

UQUQ UQUQUQ · · ·

• 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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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. Id →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting):

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n where

B(O) := | Bar(J, O, J)| ≃ J ◦O J.

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n where

B(O) := | Bar(J, O, J)| ≃ J ◦O J. Corollary: ∂n(QU) ≃ B(O)[n]

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n where

B(O) := | Bar(J, O, J)| ≃ J ◦O J. Corollary: ∂n(QU) ≃ B(O)[n] and therefore ∂n(U(QU)kQ)

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n where

B(O) := | Bar(J, O, J)| ≃ J ◦O J. Corollary: ∂n(QU) ≃ B(O)[n] and therefore ∂n(U(QU)kQ) ≃ ∂n(QU)k

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n where

B(O) := | Bar(J, O, J)| ≃ J ◦O J. Corollary: ∂n(QU) ≃ B(O)[n] and therefore ∂n(U(QU)kQ) ≃ ∂n(QU)k ≃ (B(O) ◦ · · · ◦ B(O))[n]

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 →

• UQ

UQUQ UQUQUQ · · ·

• ≃
• 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 ∂n Id ∼ − → holim∆

• ∂n(UQ)

∂n(UQUQ) ∂n(UQUQUQ) · · ·

• .

Lemma (AlgO Snaith splitting): QU(Y ) ≃

n≥1 B(O)[n] ∧Σn Y ∧n where

B(O) := | Bar(J, O, J)| ≃ J ◦O J. Corollary: ∂n(QU) ≃ B(O)[n] and therefore ∂n(U(QU)kQ) ≃ ∂n(QU)k ≃ (B(O) ◦ · · · ◦ B(O))[n] ≃ (J ◦O · · · ◦O J)[n]

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 11 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O).

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• Duncan Clark (Ohio State University)

Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J ∼ = J at the j-th position.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J ∼ = J at the j-th position. Then, (after applying a suitable fibrant replacement to C(O))

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J ∼ = J at the j-th position. Then, (after applying a suitable fibrant replacement to C(O)) C(O) is an (oplax) -monoid

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J ∼ = J at the j-th position. Then, (after applying a suitable fibrant replacement to C(O)) C(O) is an (oplax) -monoid (roughly, C(O) = Cobar(J, J ◦O J, J))

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J ∼ = J at the j-th position. Then, (after applying a suitable fibrant replacement to C(O)) C(O) is an (oplax) -monoid (roughly, C(O) = Cobar(J, J ◦O J, J)) ∂∗ Id ≃ Tot C(O) inherits an A∞-monoid structure with respect to the composition product ◦ of symmetric sequences

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Proof sketch of main thm. (cont.)

We obtain the model ∂∗ Id ≃ holim∆ C(O). Here, C(O) ∼ =

• J

J ◦O J J ◦O J ◦O J · · ·

• with coface map di given by inserting O → J at the i-th position, i.e.

J ◦O · · · ◦O J ∼ = J ◦O · · · ◦O O ◦O · · · ◦O J → J ◦O · · · ◦O J ◦O · · · ◦O J and codegeneracy sj given by J ◦O J → J ◦J J ∼ = J at the j-th position. Then, (after applying a suitable fibrant replacement to C(O)) C(O) is an (oplax) -monoid (roughly, C(O) = Cobar(J, J ◦O J, J)) ∂∗ Id ≃ Tot C(O) inherits an A∞-monoid structure with respect to the composition product ◦ of symmetric sequences The coaugmentation O → C(O) which yields an equivalence of (homotopy coherent) operads O ≃ ∂∗ Id.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 12 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra. Can show that a 0-connected O-algebra X is naturally equivalent to a ∂∗ Id-algebra by

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra. Can show that a 0-connected O-algebra X is naturally equivalent to a ∂∗ Id-algebra by first replacing X by its TQ-completion, i.e. X ∧

TQ := holim∆ C(X)

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra. Can show that a 0-connected O-algebra X is naturally equivalent to a ∂∗ Id-algebra by first replacing X by its TQ-completion, i.e. X ∧

TQ := holim∆ C(X)

If X is 0-connected then X ≃ X ∧

TQ [Ching-Harper]

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra. Can show that a 0-connected O-algebra X is naturally equivalent to a ∂∗ Id-algebra by first replacing X by its TQ-completion, i.e. X ∧

TQ := holim∆ C(X)

If X is 0-connected then X ≃ X ∧

TQ [Ching-Harper] and there is a

natural pairing C(O)C(X) → C(X) which induces the desired algebra structure.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra. Can show that a 0-connected O-algebra X is naturally equivalent to a ∂∗ Id-algebra by first replacing X by its TQ-completion, i.e. X ∧

TQ := holim∆ C(X)

If X is 0-connected then X ≃ X ∧

TQ [Ching-Harper] and there is a

natural pairing C(O)C(X) → C(X) which induces the desired algebra structure. Similar box product pairings can be used to provide a new description for the operad structure on ∂∗ IdS∗.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Applications

Can use similar box product pairing to induce a “highly homotopy coherent” chain rule, i.e. comparison map ∂∗F ◦ ∂∗G → ∂∗(FG) for functors of structured ring spectra. Can show that a 0-connected O-algebra X is naturally equivalent to a ∂∗ Id-algebra by first replacing X by its TQ-completion, i.e. X ∧

TQ := holim∆ C(X)

If X is 0-connected then X ≃ X ∧

TQ [Ching-Harper] and there is a

natural pairing C(O)C(X) → C(X) which induces the desired algebra structure. Similar box product pairings can be used to provide a new description for the operad structure on ∂∗ IdS∗. Hope to extend this technique to

• ther suitable model categories C to attack the “guiding principle”

and better understand the relation between C to Alg∂∗ IdC

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 13 / 14

Selected references

Gregory Arone and Michael Ching. Operads and Chain Rules for the Calculus of

• Functors. Number 338 in Ast´
• erisque. Soci´

et´ e Math´ ematique de France, 2011. Michael Ching. Bar constructions for topological operads and the Goodwillie derivatives of the identity. Geometry and Topology, 9(2):833–934, 2005. Michael Ching and John E. Harper. Derived Koszul duality and TQ-homology completion of structured ring spectra. Advances in Math., 341:118–187, 2019. Thomas G. Goodwillie. Calculus III: Taylor series. Geometry and Topology, 7(2):645–711, 2003. John E. Harper. Homotopy theory of modules over operads in symmetric spectra. Algebraic and Geometric Topology, 9(3):1637–1680, Aug 2009. John E. Harper and Kathryn Hess. Homotopy completion and topological Quillen homology of structured ring spectra. Geometry and Topology, 17(3):1325–1416, Jun 2013. James E. McClure and Jeffrey H. Smith. Cosimplical objects and little n-cubes, i. American Journal of Mathematics, 126(5):1109–1153, 2004. Lu´ ıs Pereira. Goodwillie calculus and algebras over a spectral operad. PhD thesis, Massachusetts Institute of Technology, 2013.

Duncan Clark (Ohio State University) Derivatives of the identity GOATS 2 (6/6/20) 14 / 14