DaoFP Chapter 17 Exercises

exercise

Exercises from DaoFP, Chapter 17. Solutions: DaoFP Chapter 17 Solutions. Index: Map of Content.

Exercise 17.1.1

Show that there is a functor from a Collage of two categories to the Walking Arrow category $0\to 1$.

Sources: DaoFP Exercise 17.1.1.

Solution: Solution 17.1.1

Exercise 17.1.2

Show that if there is a functor $F:\mathcal{E}\to 2$ (the Walking Arrow) then $\mathcal{E}$ splits into a Collage of two categories.

Sources: DaoFP Exercise 17.1.2.

Solution: Solution 17.1.2

Exercise 17.2.1

Verify that for an extranatural transformation $P\to {\Delta }_d$ the first extranaturality diamond is the cowedge condition and the second is trivial (Coend).

Sources: DaoFP Exercise 17.2.1.

Solution: Solution 17.2.1

Exercise 17.2.2

Define a Profunctor instance for newtype ProPair q p a b x y = ProPair (q a y, p x b) keeping a b fixed (Coend).

Sources: DaoFP Exercise 17.2.2.

Solution: Solution 17.2.2

Exercise 17.2.3

Define a Profunctor instance for newtype CoEndCompose p q a b = CoEndCompose (Coend (ProPair q p a b)) (Coend, profunctor composition).

Sources: DaoFP Exercise 17.2.3.

Solution: Solution 17.2.3

Exercise 17.3.1

Show that a Product (a limit over the two-object discrete category $2$) can be defined as an End.

Sources: DaoFP Exercise 17.3.1.

Solution: Solution 17.3.1

Exercise 17.6.1

Prove the contravariant co-Yoneda lemma ${\int }^x\mathcal{C}(a,x)\times Gx\cong Ga$ for a presheaf $G$ (Ninja Yoneda Lemma).

Sources: DaoFP Exercise 17.6.1.

Solution: Solution 17.6.1

Exercise 17.7.1

Define the Functor instance for Day.

Sources: DaoFP Exercise 17.7.1.

Solution: Solution 17.7.1

Exercise 17.7.2

Implement the associator assoc :: Day f (Day g h) x -> Day (Day f g) h x (Day Convolution).

Sources: DaoFP Exercise 17.7.2.

Solution: Solution 17.7.2

Exercise 17.7.3

Define the Functor instance for the free applicative FreeA (Day Convolution).

Sources: DaoFP Exercise 17.7.3.

Solution: Solution 17.7.3

Exercise 17.9.1

Show that $\mathcal{C}(s,y\times a)\times \mathcal{C}(x\times b,t)$ is a Profunctor in $⟨x,y⟩$ (Existential Lens).

Sources: DaoFP Exercise 17.9.1.

Solution: Solution 17.9.1