Exercise sheet 6

Algorithmic homological algebra

Exercise sheet 6

Prof. Dr. Mohamed Barakat, Sebastian Posur

Exercise 1. (Functors and natural transformations)

LetA be a category. We define itsarrow category A^•→• as follows: Objects are given by triples (A, B, α), where A, B ∈ A and α ∈ HomA(A, B). A morphism between (A, B, α) and (A⁰, B⁰, α⁰) is given by a pair of morphisms µ : A → A⁰, ν : B → B⁰ such that µ·α⁰ =α·ν. Projection to the first component

Source :A^•→• → A: (A, B, α)7→A gives rise to a functor.

(a) Assume thatA has kernels. Show that there exists a unique functor with underlying function on objects

ker :A^•→• → A: (A, B, α)7→ker(α)

such that the kernel embeddings ker(α) ,→ A define the components of a natural transformation ker =⇒Source.

(b) State and prove a similar statement for pullbacks.

Exercise 2. (Yoneda lemma) Prove the Yoneda lemma:

LetA be a category. For each objectM ∈ A and each functor F :A →(Sets) the map [h^M, F]→F(M),(η :h^M →F)7→η_M(1_M)∈F(M)

is an isomorphism of sets. Likewise [h_N, G]∼=G(N) for a presheaf G:A^op →(Sets).

This exercise sheet will be discussed on 2.2.2017.