M. Künzer

A_{∞}-categories, WS 16/17

Sheet 1

Problem 1 Consider the commutative ring Z. Consider the Z-algebra Z. Determine the isoclasses of the Z-modules M that have a chain of submodules

M = M_{0} ⊇ M_{1} ⊇ M_{2} ⊇ M_{3} = 0

such that

\begin{aligned}M_{0}/M_{1} & ≃ & Z/(2) \\ M_{1}/M_{2} & ≃ & Z/(4) \\ M_{2}/M_{3} & ≃ & Z/(2) . \end{aligned}

Problem 2 Let Cat denote the (1-)category of categories, (1-)morphisms being functors. Let Set denote the category of sets, morphisms being maps.

(1) Given a set X, how many isoclasses does the pair category X^{×2} have?

(2) Construct a full and faithful functor P : Set → Cat sending X to X^{×2}.

(3) Show that the functor Ob : Cat → Set has P as a right adjoint, i.e. Ob ⊣ P.

(4) Determine unit and counit of the adjunction in (3). Problem 3 Let Poset denote the category of posets and monotone maps.

(1) Suppose given a poset X. Show that we have a subcategory CX of the pair category X^{×2} with Ob(CX) = X and Mor(CX) = { (x, y) ∈ X^{×2} : x ⩽ y }.

(2) Construct a functor C : Poset \overline{}Cat.

(3) Given n ∈ Z_{⩾0} , we write Δ_{n} := C[0, n].

We have the monotone map ω : [0, 1] → [0, n] , 0 ↦ 0, 1 ↦ n.

Suppose given a category Z and z ∈ Mor(Z). Let F_{z} : Δ_{1} → Z, (0, 1) ↦ z.

Let n ⩾ 1. Show that fact_{n}(z) is in bijection to

{ Δ_{n} ⟶^{G} Z : G is a functor such that G ∘ (Cω) = F_{z} } . w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/ai16/