M. Künzer A_{∞}-categories, WS 16/17

Sheet 8

Problem 19 Let Z be a grading category.

(1) Let Ṽ = (Ṽ , Δ̃) and V = (V, Δ) be coalgebras over Z. Let Ṽ ⟶^{f} V be a morphism of coalgebras. Suppose that f is piecewise bijective. Show that f is an isomorphism of coalgebras, i.e. that there exists a morphism of coalgebras Ṽ ⟵^{g} V such that fg = idṼ and gf = id_{V} . Then g is uniquely determined and written f^{−} := g.

(2) Let Ṽ = (Ṽ , Δ̃, δ̃) and V = (V, Δ, δ) be coalgebras with differential over Z. Let Ṽ ⟶^{f} V be a morphism of coalgebras with differential. Suppose that f is piecewise bijective. Show that f is an isomorphism of coalgebras with codifferential, i.e. that there exists a morphism of coalgebras with codifferential Ṽ ⟵^{g} V such that fg = idṼ and gf = id_{V} . Then g is uniquely determined and written f^{−} := g.

(3) Let Ṽ = (Ṽ , Δ̃) and V = (V, Δ) be coalgebras over Z.

Let Ṽ ⟶^{f} V be an isomorphism of coalgebras. Suppose given a codifferential δ on V. Show that fδf^{−} is a codifferential on Ṽ.

(4) Let V = (V, Δ) be a coalgebra over Z. Let λ : V \overline{}Ṙ be a shift-graded linear map of degree 1 ; cf. Problem 7.(3). Recall that Ṙ ⊗ V = V = V ⊗ Ṙ by identification. Let δ_{λ} := Δ(id ⊗λ) − Δ(λ ⊗ id). Show that δ_{λ} is a coderivation. Coderivations of this form are called inner.

Problem 20 Let Z be a grading category. Let I be a finite set. Let V_{i} be a Z-graded module for i ∈ I. Recall that the Z-graded module ⨁_{i∈I} V_{i} is defined by letting (⨁_{i∈I} V_{i})^{z} = ⨁_{i∈I} V_{i}^{z} for z ∈ Mor(Z).

(1) Given j ∈ I, construct a shift-graded linear inclusion map ι_{j} : V_{j} → ⨁_{i∈I} V_{i} of degree 0 and a shift-graded linear projection map π_{j} : ⨁_{i∈I} V_{i} → V_{j} of degree 0.

(2) Suppose given a Z-graded module S. Suppose given d ∈ Z. Suppose given a shift-graded linear map s_{j} : S → V_{j} of degree d for j ∈ I. Show that there exists a unique shift-graded linear map s : S → ⨁_{i∈I} V_{i} of degree d such that sπ_{j} = s_{j} for j ∈ I.

(3) Suppose given a Z-graded module T. Suppose given d ∈ Z. Suppose given shift-graded linear maps t_{j} : V_{j} → T of degree d for j ∈ I. Show that there exists a unique shift-graded linear map t : ⨁_{i∈I} V_{i} → T of degree d such that ι_{j}t = t_{j} for j ∈ I.

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