M. Künzer A_{∞}-categories, WS 16/17

Sheet 10

Problem 23 Let n ∈ [1, ∞]. Let Z be a grading category. Let (A, (m_{ℓ})_{ℓ}), (A′, (m′_{ℓ})_{ℓ}), (A′′, (m′′_{ℓ})_{ℓ}), (A′′′, (m′′′_{ℓ})_{ℓ}) be A_{n}-algebras over Z. Let f = (f_{ℓ})_{ℓ} : A → A′, f′ = (f′_{ℓ})_{ℓ} : A′ → A′′, f′′ = (f′′_{ℓ})_{ℓ} : A′′ → A′′′ be A_{n}-morphisms. Write ^{ω}f := (^{ω}f_{ℓ})_{ℓ}. Define

f · f′ := ^{ω^{−}}(((^{ω}f)β · (^{ω}f′)β)α) .

(1) Show that f · f′ is a morphism of A_{n}-algebras from A to A′′.

(2) Write f · f′ in terms of (f_{ℓ})_{ℓ} and (f′_{ℓ})_{ℓ}. What is the entry of f · f′ at ℓ = 1 ?

(3) Show that (f · f′) · f′′ = f · (f′ · f′′).

(4) Suppose given shift-graded linear maps g : A → A′ and g′ : A′ → A′′ of degree 0. Define strict_{n}(g) := (g, 0, 0, …). When is strict_{n}(g) : A → A′ a morphism of A_{n}-algebras? Is strict_{n}(id_{A}) a morphism of A_{n}-algebras? If strict_{n}(g) and strict_{n}(g′) are morphisms of A_{n}-algebras, show that strict_{n}(gg′) = strict_{n}(g) · strict_{n}(g′).

(5) Show that f · strict_{n}(id_{A′}) = f and that strict_{n}(id_{A′}) · f′ = f′.

(6) Define the category A_{n}-Z-alg of A_{n}-algebras over Z and A_{n}-morphisms.

Therein, define the subcategory strict-A_{n}-Z-alg of A_{n}-algebras over Z and strict A_{n}- morphisms.

(7) Show that H is a functor from A_{n}-Z-alg to Z-grad_{0} .

Problem 24 Let n ∈ [1, ∞]. Let Z ⟶^{F} Z̃ be a morphism of grading categories; cf. Problem 5. Let (A, (m_{k})_{k}) be an A_{n}-algebra over Z.

(1) Show that F_{&}A = (F_{&}A, (σ^{−} · F_{&}m_{ℓ})_{ℓ}) is an A_{n}-algebra over Z̃, where σ = σ_{(A,...,A)} ; cf. Problem 10.

(2) Consider the case n = ∞, u ∈ Z_{⩾1} , Z = Z × [1, u]^{×2}, Z̃ = Z and P being the projection, mapping a morphism (j, (s, t)) to j.

Given a unital Z × [1, u]^{×2}-algebra A, i.e. an A_{∞}-category with set of objects [1, u], show that its total A_{∞}-algebra P_{&}A is unital.

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