M. Künzer

A_{∞}-categories, WS 16/17

Sheet 2

Problem 4 Let Z = (Z, S, deg) be a grading category. Show.

(1) The shift S is an automorphism of Z-grad_{0} .

(2) Z-grad is a category.

(3) By S(f, k) := (Sf, k) for (f, k) ∈ Mor(Z-grad), an automorphism S on Z-grad is defined.

(4) Z-grad_{0} is additive.

(5) Z-grad_{0} is isomorphic to a subcategory of Z-grad.

Is this subcategory full? Does Z-grad have a zero object?

Problem 5 Let Z = (Z, S, deg) and Z̃ = (Z̃, S̃, d̃eg) be grading categories. A (1-)morphism of grading categories from Z to Z̃ is a functor F : Z → Z̃ such that

\begin{aligned}F (zS) & = & (F z)S̃ \\ (z) deg & = & (F z)d̃eg \end{aligned}

for z ∈ Mor(Z).

(1) Show that grading categories, together with morphisms of such, form a category Grad.

(2) Show that (Z, S^{−}, − deg) is a grading category, where (S^{−})_{X,Y} := (S_{X,Y})^{−} for X, Y ∈ Ob(Z) and z(− deg) := −(z deg) for z ∈ Mor(Z). Construct an automorphism of order 2 on the category of grading categories.

(3) Show that (id_{X}) deg = 0 for X ∈ Ob(Z).

(4) Show that there exists exactly one morphism of grading categories from Z to Z, i.e. that Z is the terminal grading category.

(5) Show that there is a bijection from the set of morphisms of grading categories from Z to Z to the set of endomorphisms of Z of degree 0.

(6) Suppose given a morphism of grading categories Z ⟶^{F} Z̃.

Show that there exist functors

Z-grad [figure: F_{&}; F^{&}] Z̃-grad

having (F^{&}M̃)^{z} = M̃^{F z} for M̃ ∈ Ob(Z̃-grad) and z ∈ Mor(Z), having

(F_{&}M)z̃ = ⨁_{\begin{matrix}z∈Mor(Z) \\ F z=z̃ \end{matrix}}M^{z} for M ∈ Ob(Z-grad) and z̃ ∈ Mor(Z̃) and having F_{&} ⊣ F^{&}.

Problem 6 Let Z = (Z, S, deg) be a grading category. Define a category (Z-grad)^{×n,±} such that we have a functor

(Z-grad)^{×n,±} \underrightarrow{⨂_{i∈[1,n]}} Z-grad
(L_{i} ⟶^{(f_{i},k_{i})} M_{i})_{i∈[1,n]} ⟼ (⨂_{i∈[1,n]} L_{i} \underrightarrow{⨂_{i∈[1,n]}(f_{i},k_{i})} ⨂_{i∈[1,n]} M_{i}) .

w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/ai16/