M. Künzer

A_{∞}-categories, WS 16/17

Sheet 9

Problem 21 Let Z be a grading category. Let n ∈ [1, ∞] . Let (A, (m_{ℓ})_{ℓ}) be a pre-A_{n}-algebra. Write 𝔪 := ((^{ω}m_{ℓ})_{ℓ∈[1,n]∩Z})β_{Coder,n,A^{[1]}} .

(1) Suppose n ∈ Z_{⩾1} . Suppose that (m_{ℓ})_{ℓ} satisfies the Stasheff equation at each k ∈ [1, n−1]. Suppose given z ∈ Mor(Z) and a ∈ ((A^{[1]})^{⊗n})^{z}. Consider the following assertions.

(i) We have

a ( ∑_{\begin{matrix}(r,s,t)⩾(0,1,0) \\ r+s+t = n \end{matrix}}(id^{⊗r} ⊗^{ω}m_{s} ⊗ id^{⊗t}) · ^{ω}m_{r+1+t}) = 0

(ii) We have a𝔪^{2} = 0. Show that (i) and (ii) are equivalent.

(2) Suppose given p ∈ [1, n]. Write 𝔪′ := ((^{ω}m_{ℓ})_{ℓ∈[1,p]∩Z})β_{Coder,p,A^{[1]}} .

Show that 𝔪′ = 𝔪|_{T_{⩽p}(A^{[1]})}^{T_{⩽p}(A^{[1]})} . Problem 22 Let Z be a grading category. Let n ∈ [1, ∞] . Let (Ã, (m̃_{ℓ})_{ℓ}) and (A, (m_{ℓ})_{ℓ}) be pre-A_{n}-algebras. Let f = (f_{ℓ})_{ℓ} be a pre-A_{n}-morphism from Ã to A. Write

𝔪̃ := ((^{ω}m̃_{ℓ})_{ℓ})β_{Coder,n,Ã^{[1]}}
𝔪 := ((^{ω}m_{ℓ})_{ℓ})β_{Coder,n,A^{[1]}}
𝔣 := ((^{ω}f_{ℓ})_{ℓ})β_{Coalg,n,Ã^{[1]},A^{[1]}} .

(1) Suppose n ∈ Z_{⩾1} . Suppose that (f_{ℓ})_{ℓ} satisfies the Stasheff equation for morphisms at each k ∈ [1, n − 1]. Suppose given z ∈ Mor(Z) and ã ∈ ((Ã^{[1]})^{⊗n})^{z}. Consider the following assertions.

(i) We have

ã ( ∑_{\begin{matrix}(r,s,t)⩾(0,1,0) \\ r+s+t = n \end{matrix}}(id^{⊗r} ⊗^{ω}m̃_{s} ⊗ id^{⊗t}) · ^{ω}f_{r+1+t}) = ã ( ∑_{r∈[1,n]} ∑_{\begin{matrix}(i_{j})_{j∈[1,r]}⩾(1)_{j} \\ ∑_{j} i_{j} = n \end{matrix}}( ⨂_{j∈[1,r]} ^{ω}f_{i_{j}}) · ^{ω}m_{r})

(ii) We have ã(𝔪̃𝔣 − 𝔣𝔪) = 0. Show that (i) and (ii) are equivalent.

(2) Suppose given p ∈ [1, n]. Write 𝔣′ := ((^{ω}f_{ℓ})_{ℓ∈[1,p]∩Z})β_{Coalg,p,Ã^{[1]},A^{[1]}} .

Show that 𝔣′ = 𝔣|_{T_{⩽p}(Ã^{[1]})}^{T_{⩽p}(A^{[1]})} .

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