7S Chapter 3 Exercises

exercise

Exercises from 7 Sketches, Chapter 3. Solutions: 7S Chapter 3 Solutions. Index: Map of Content.

Exercise 3.3

Count the non-ID columns in the Employee/Department database (3.1) and the arrows in its schema (3.2). Is it a coincidence that they agree?

Sources: 7 Sketches, Exercise 3.3 and Solution A.3.

Solution: Solution 3.3

Exercise 3.9

Check that $\mathrm{Free}(G)$ satisfies unitality and associativity.

Sources: 7 Sketches, Exercise 3.9 and Solution A.3.

Solution: Solution 3.9

Exercise 3.10

Name the six morphisms of $\underline{3}=\mathrm{Free}(v_1\stackrel{f_1}{\to }{v}_2\stackrel{f_2}{\to }{v}_3)$ and write the composition table.

Sources: 7 Sketches, Exercise 3.10 and Solution A.3.

Solution: Solution 3.10

Exercise 3.12

What are the categories $\underline{1}$ and $\underline{0}$? How many morphisms has $\underline{\mathbf{n}}$?

Sources: 7 Sketches, Exercise 3.12 and Solution A.3.

Solution: Solution 3.12

Exercise 3.15

Identifying paths in the one-loop graph with natural numbers, what does concatenation correspond to?

Sources: 7 Sketches, Exercise 3.15 and Solution A.3.

Solution: Solution 3.15

Exercise 3.16

Write the ten paths of the free square; name two parallel paths and two non-parallel ones.

Sources: 7 Sketches, Exercise 3.16 and Solution A.3.

Solution: Solution 3.16

Exercise 3.17

List the morphisms of the square with a diagonal $j:A\to D$ and equations $f;h=j=g;i$.

Sources: 7 Sketches, Exercise 3.17 and Solution A.3.

Solution: Solution 3.17

Exercise 3.19

List the morphisms of the category $\mathcal{D}$ presented by one loop $s$ with $s^4=s^2$.

Sources: 7 Sketches, Exercise 3.19 and Solution A.3.

Solution: Solution 3.19

Exercise 3.21

Which equations turn the graphs $G_1$ (two parallel arrows $f,g$), $G_2$ (a loop $f$), $G_3$ (a square $f,g,h,i$), $G_4$ (two arrows out of one vertex) into presentations of the associated preorders?

Sources: 7 Sketches, Exercise 3.21 and Solution A.3.

Solution: Solution 3.21

Exercise 3.22

What is the Preorder Reflection of the one-object category $\mathbb{N}$?

Sources: 7 Sketches, Exercise 3.22 and Solution A.3.

Solution: Solution 3.22

Exercise 3.25

List the nine elements of $\mathbf{Set}(\underline{2},\underline{3})$.

Sources: 7 Sketches, Exercise 3.25 and Solution A.3.

Solution: Solution 3.25

Exercise 3.30

For $f:\{a,b,c\}\to \underline{3}$, $f(a)=2,f(b)=1,f(c)=3$: what is $f^{-1}$, and how many isomorphisms $\{a,b,c\}\to \underline{3}$ are there?

Sources: 7 Sketches, Exercise 3.30 and Solution A.3.

Solution: Solution 3.30

Exercise 3.31

Show that every identity ${\mathrm{id}}_c$ is an Isomorphism.

Sources: 7 Sketches, Exercise 3.31 and Solution A.3.

Solution: Solution 3.31

Exercise 3.32

A monoid in which every morphism is an isomorphism is a Group. Is $\mathbb{N}$ (Example 3.13) a group? Is $\mathcal{C}$ from Example 3.18 ($s;s=z$)?

Sources: 7 Sketches, Exercise 3.32 and Solution A.3.

Solution: Solution 3.32

Exercise 3.33

Someone claims the only isomorphisms in a Free Category $\mathrm{Free}(G)$ are identities. Correct?

Sources: 7 Sketches, Exercise 3.33 and Solution A.3.

Solution: Solution 3.33

Exercise 3.37

Find the remaining functors $\underline{2}\to \underline{3}$ beyond the three of Example 3.36.

Sources: 7 Sketches, Exercise 3.37 and Solution A.3.

Solution: Solution 3.37

Exercise 3.39

Where does the functor $F$ from the free square $\mathcal{F}$ to the commutative square $\mathcal{C}$ (matching objects) send each of the ten morphisms?

Sources: 7 Sketches, Exercise 3.39 and Solution A.3.

Solution: Solution 3.39

Exercise 3.40

Give two functors $F,G:(a\stackrel{f}{\to }b)\to (a^{\prime }\underset{f_2}{\stackrel{f_1}{\rightrightarrows }}{b}^{\prime })$ that agree on objects but differ on morphisms.

Sources: 7 Sketches, Exercise 3.40 and Solution A.3.

Solution: Solution 3.40

Exercise 3.43

Construct the category $\mathbf{Cat}$: identity functors, composition of functors, and the category axioms.

Sources: 7 Sketches, Exercise 3.43 and Solution A.3.

Solution: Solution 3.43

Exercise 3.45

For any set $S$, give a functor $F_S:\underline{1}\to \mathbf{Set}$ with $F_S(1)=S$.

Sources: 7 Sketches, Exercise 3.45 and Solution A.3.

Solution: Solution 3.45

Exercise 3.48

Give data that makes sense for the schemas 1. one loop $s$ with $s;s=z$ (identity); 2. $a\stackrel{f}{\to }b\underset{h}{\stackrel{g}{\rightrightarrows }}c$ with $f;g=f;h$.

Sources: 7 Sketches, Exercise 3.48 and Solution A.3.

Solution: Solution 3.48

Exercise 3.55

In the Functor Category ${\mathcal{D}}^{\mathcal{C}}$: 1. how do natural transformations compose? 2. what is the identity, and is it unital?

Sources: 7 Sketches, Exercise 3.55 and Solution A.3.

Solution: Solution 3.55

Exercise 3.58

Let $\mathcal{P}$ be a preorder. 1. Is there at most one natural transformation between any two functors $F,G:\mathcal{C}\to \mathcal{P}$? 2. Between $F,G:\mathcal{P}\to \mathcal{C}$?

Sources: 7 Sketches, Exercise 3.58 and Solution A.3.

Solution: Solution 3.58

Exercise 3.62

Write the graph easySchema (3.2) as a $\mathsf{Gr}$-instance.

Sources: 7 Sketches, Exercise 3.62 and Solution A.3.

Solution: Solution 3.62

Exercise 3.64

For $G=1\stackrel{a}{\to }2\stackrel{b}{\to }3$ and $H=4\stackrel{c,d}{\to }5\circlearrowleft e$, the unique Graph Homomorphism with ${\alpha }_{\mathrm{Arrow}}(a)=d$: find the other values and check naturality.

Sources: 7 Sketches, Exercise 3.64 and Solution A.3.

Solution: Solution 3.64

Exercise 3.67

Migrate the DDS instance (3.65) along $G:\mathsf{Gr}\to \mathsf{DDS}$ with $\mathsf{source}\mapsto \mathsf{next}$, $\mathsf{target}\mapsto \mathrm{id}$.

Sources: 7 Sketches, Exercise 3.67 and Solution A.3.

Solution: Solution 3.67

Exercise 3.73

For the currying adjunction $(-\times B)⊣(-{)}^B$: 1. what does $-\times B$ do to $f:X\to Y$? 2. what does $(-{)}^B$ do? 3. currying $+:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ gives $p:\mathbb{N}\to {\mathbb{N}}^{\mathbb{N}}$; what is $p(3)$?

Sources: 7 Sketches, Exercise 3.73 and Solution A.3.

Solution: Solution 3.73

Exercise 3.76

Describe the unique functor $!:\mathcal{C}\to \underline{1}$.

Sources: 7 Sketches, Exercise 3.76 and Solution A.3.

Solution: Solution 3.76

Exercise 3.78

Draw the email instance (3.77) as a graph.

Sources: 7 Sketches, Exercise 3.78 and Solution A.3.

Solution: Solution 3.78

Exercise 3.81

Show that $z$ is a Terminal Object in a preorder $P$ (as a category) iff it is a top element.

Sources: 7 Sketches, Exercise 3.81 and Solution A.3.

Solution: Solution 3.81

Exercise 3.82

Name a terminal object in $\mathbf{Cat}$.

Sources: 7 Sketches, Exercise 3.82 and Solution A.3.

Solution: Solution 3.82

Exercise 3.83

Find a category without a terminal object.

Sources: 7 Sketches, Exercise 3.83 and Solution A.3.

Solution: Solution 3.83

Exercise 3.88

Show that in a preorder $P$ viewed as a category, the Product $x\times y$ is the Meet $x\wedge y$.

Sources: 7 Sketches, Exercise 3.88 and Solution A.3.

Solution: Solution 3.88

Exercise 3.90

In a Product Category $\mathcal{C}\times \mathcal{D}$: 1. identities? 2. why associative? 3. what is $\underline{1}\times \underline{2}$? 4. what is $\mathcal{P}\times \mathcal{Q}$ for preorders?

Sources: 7 Sketches, Exercise 3.90 and Solution A.3.

Solution: Solution 3.90

Exercise 3.91

Check that a Product $X\leftarrow X\times Y\to Y$ is exactly a Terminal Object in $\mathrm{Cone}(X,Y)$.

Sources: 7 Sketches, Exercise 3.91 and Solution A.3.

Solution: Solution 3.91

Exercise 3.97

Show that the limit formula of Theorem 3.95 gives products.

Sources: 7 Sketches, Exercise 3.97 and Solution A.3.

Solution: Solution 3.97

Exercise 3.98

What is the limit of $D:\underline{1}\to \mathbf{Set}$?

Sources: 7 Sketches, Exercise 3.98 and Solution A.3.

Solution: Solution 3.98

Exercise 3.101

Define the opposite $F^{\mathrm{op}}:{\mathcal{C}}^{\mathrm{op}}\to {\mathcal{D}}^{\mathrm{op}}$ of a functor $F$.

Sources: 7 Sketches, Exercise 3.101 and Solution A.3.

Solution: Solution 3.101