DaoFP Chapter 3 Exercises

exercise

Exercises from DaoFP, Chapter 3. Solutions: DaoFP Chapter 3 Solutions. Index: Map of Content.

Exercise 3.1.1

Argue that there is a bijection between arrows outgoing from two isomorphic objects.

Sources: DaoFP Exercise 3.1.1.

Solution: Solution 3.1.1

Exercise 3.1.2

Show every object is isomorphic to itself.

Sources: DaoFP Exercise 3.1.2.

Solution: Solution 3.1.2

Exercise 3.1.3

If there are two terminal objects, show they are isomorphic.

Sources: DaoFP Exercise 3.1.3.

Solution: Solution 3.1.3

Exercise 3.1.4

Show the isomorphism of the previous exercise is unique.

Sources: DaoFP Exercise 3.1.4.

Solution: Solution 3.1.4

Exercise 3.2.1

Show that both sides of the naturality condition $(-\circ f^{-1})\circ (g\circ -)=(g\circ -)\circ (-\circ f^{-1})$, acting on $h:a\to x$, reduce to $g\circ h\circ f^{-1}:b\to y$.

Sources: DaoFP Exercise 3.2.1.

Solution: Solution 3.2.1

Exercise 3.3.1

Given a natural family of bijections ${\beta }_x:\mathcal{C}(a,x)\to \mathcal{C}(b,x)$ (with $(g\circ -)\circ {\beta }_x={\beta }_y\circ (g\circ -)$), recover $f^{-1}$ using the identity trick.

Sources: DaoFP Exercise 3.3.1.

Solution: Solution 3.3.1

Exercise 3.3.2

Using $f^{-1}$ from the previous exercise, evaluate ${\beta }_y(g)$ for $g:a\to y$.

Sources: DaoFP Exercise 3.3.2.

Solution: Solution 3.3.2