DaoFP Chapter 20 Exercises

exercise

Exercises from DaoFP, Chapter 20. Solutions: DaoFP Chapter 20 Solutions. Index: Map of Content.

Exercise 20.1.1

Define composition and unit in the $\mathcal{V}$-category ${\mathcal{C}}^{\mathrm{op}}$ (Opposite Enriched Category).

Sources: DaoFP Exercise 20.1.1; 7 Sketches Exercise 2.63.

Solution: Solution 20.1.1

Exercise 20.1.2

Show that every $\mathcal{V}$-category $\mathcal{C}$ has an underlying ordinary category ${\mathcal{C}}_0$ with hom-sets $\mathcal{V}(I,\mathcal{C}(a,b))$ (Enriched Category).

Sources: DaoFP Exercise 20.1.2; 7 Sketches §2.3.

Solution: Solution 20.1.2

Exercise 20.2.1

What is a $\mathcal{V}$-functor between two preorders ($\mathcal{V}=\mathbf{Bool}$)? (Enriched Functor)

Sources: DaoFP Exercise 20.2.1; 7 Sketches §2.4.2.

Solution: Solution 20.2.1

Exercise 20.2.2

Show that if $\mathcal{V}$ is monoidal closed, the tensor $\otimes :\mathcal{V}\times \mathcal{V}\to \mathcal{V}$ is a $\mathcal{V}$-functor (Enriched Functor, Monoidal Closed Category).

Sources: DaoFP Exercise 20.2.2.

Solution: Solution 20.2.2

Exercise 20.6.1

Show that for ordinary categories the weighted colimit definition of a coend, ${\mathrm{colim}}^{{\mathrm{Hom}}_{{\mathcal{C}}^{\mathrm{op}}}}P$, reproduces ${\int }^{c}P⟨c,c⟩$ (Weighted Limit, Coend).

Sources: DaoFP Exercise 20.6.1.

Solution: Solution 20.6.1

Exercise 20.6.2

Show ${\lim }^{W}D\cong {\int }_{j}Wj⋔Dj$ and ${\mathrm{colim}}^{W}D\cong {\int }^{j}Wj\cdot Dj$ in ordinary categories (Weighted Limit).

Sources: DaoFP Exercise 20.6.2.

Solution: Solution 20.6.2

Exercise 20.7.1

Derive $({\mathrm{Lan}}_{P}F)e={\mathrm{colim}}^{\mathcal{B}(P-,e)}F$ for ordinary categories (Kan Extension, Weighted Limit).

Sources: DaoFP Exercise 20.7.1.

Solution: Solution 20.7.1