7S Chapter 2 Exercises

exercise

Exercises from 7 Sketches, Chapter 2. Solutions: 7S Chapter 2 Solutions. Index: Map of Content.

Exercise 2.5

On $(\mathbb{R},\le )$ someone proposes unit $1$ and product $\ast$. An expert says “that won’t work.” Why?

See Symmetric Monoidal Preorder.

Sources: 7 Sketches, Exercise 2.5 and Solution A.2.

Solution: Solution 2.5

Exercise 2.8

Is it really easy to check that a commutative Monoid $(M,\ast ,e)$ makes $({\mathrm{Disc}}_M,=,e,\ast )$ a Symmetric Monoidal Preorder?

Sources: 7 Sketches, Exercise 2.8 and Solution A.2.

Solution: Solution 2.8

Exercise 2.20

1. Prove formally from the axioms of a Symmetric Monoidal Preorder that the assertions $t\le v+w$, $w+u\le x+z$, $v+x\le y$ imply $t+u\le y+z$. 2. Where are reflexivity and transitivity used? 3. Why is symmetry not needed for diagram (2.12)?

See Wiring Diagrams for Monoidal Preorders.

Sources: 7 Sketches, Exercise 2.20 and Solution A.2.

Solution: Solution 2.20

Exercise 2.21

Check that $(\mathrm{Mat},\to ,0,+)$ (chemical materials and reactions) satisfies (a)–(d) of Definition 2.2.

See Resource Theory.

Sources: 7 Sketches, Exercise 2.21 and Solution A.2.

Solution: Solution 2.21

Exercise 2.29

On $(\mathbb{B},\le )$ with product $\vee$ (OR), what must the unit be, and do the other conditions hold?

Sources: 7 Sketches, Exercise 2.29 and Solution A.2.

Solution: Solution 2.29

Exercise 2.31

Show there is a monoidal structure on $(\mathbb{N},\le )$ with product $\ast$. What is the unit?

Sources: 7 Sketches, Exercise 2.31 and Solution A.2.

Solution: Solution 2.31

Exercise 2.33

On the Divisibility Order $(\mathbb{N},\mid )$ someone proposes unit $0$ and product $+$. Does it work?

Sources: 7 Sketches, Exercise 2.33 and Solution A.2.

Solution: Solution 2.33

Exercise 2.34

On $\mathsf{no}\le \mathsf{maybe}\le \mathsf{yes}$ with unit $\mathsf{yes}$ and product $\min$: 1. fill in the table for $\min$; 2. check the axioms for $\mathbf{NMY}$.

Sources: 7 Sketches, Exercise 2.34 and Solution A.2.

Solution: Solution 2.34

Exercise 2.35

Is $(\mathcal{P}(S),\subseteq ,S,\cap )$ a Symmetric Monoidal Preorder?

Sources: 7 Sketches, Exercise 2.35 and Solution A.2.

Solution: Solution 2.35

Exercise 2.36

${\mathrm{Prop}}_{\mathbb{N}}$ is the set of statements about a natural number, with $P\le Q$ iff $P(n)\Rightarrow Q(n)$ for all $n$. Define a monoidal unit and product satisfying Definition 2.2.

Sources: 7 Sketches, Exercise 2.36 and Solution A.2.

Solution: Solution 2.36

Exercise 2.39

Complete the proof of Proposition 2.38 (Opposite Monoidal Preorder): unitality, associativity and symmetry hold in $X^{\mathrm{op}}$.

Sources: 7 Sketches, Exercise 2.39 and Solution A.2.

Solution: Solution 2.39

Exercise 2.40

Describe ${\mathbf{Cost}}^{\mathrm{op}}$: its preorder, unit, and product.

Sources: 7 Sketches, Exercise 2.40 and Solution A.2.

Solution: Solution 2.40

Exercise 2.43

Check that $g:\mathbf{Bool}\to \mathbf{Cost}$, $g(\mathsf{false})=\infty$, $g(\mathsf{true})=0$, is a Monoidal Monotone Map; is it strict?

Sources: 7 Sketches, Exercise 2.43 and Solution A.2.

Solution: Solution 2.43

Exercise 2.44

Consider $d,u:[0,\infty ]\to \mathbb{B}$ with $d(x)=[x=0]$ and $u(x)=[x<\infty ]$. Are they monotone, monoidal monotone, strict?

Sources: 7 Sketches, Exercise 2.44 and Solution A.2.

Solution: Solution 2.44

Exercise 2.45

1. Is $(\mathbb{N},\le ,1,\ast )$ a monoidal preorder? 2. Is there a Monoidal Monotone Map $(\mathbb{N},\le ,0,+)\to (\mathbb{N},\le ,1,\ast )$? 3. Is $(\mathbb{Z},\le ,1,\ast )$ a monoidal preorder?

Sources: 7 Sketches, Exercise 2.45 and Solution A.2.

Solution: Solution 2.45

Exercise 2.50

Show the constructions of Theorem 2.49 (Preorders are Bool-Categories) are mutually inverse.

Sources: 7 Sketches, Exercise 2.50 and Solution A.2.

Solution: Solution 2.50

Exercise 2.52

With the “worst-case” regional distance, which is bigger: $d(\mathrm{Spain},\mathrm{US})$ or $d(\mathrm{US},\mathrm{Spain})$?

Sources: 7 Sketches, Exercise 2.52 and Solution A.2.

Solution: Solution 2.52

Exercise 2.55

How does a Lawvere Metric Space differ from an $({\mathbb{R}}_{\ge 0},\ge ,0,+)$-category?

Sources: 7 Sketches, Exercise 2.55 and Solution A.2.

Solution: Solution 2.55

Exercise 2.58

Fill in the distance table for the Weighted Graph $X$ of Eq. (2.56) ($A\stackrel{3}{\to }C$, $B\stackrel{2}{\to }A$, $B\stackrel{5}{\to }D$, $C\stackrel{3}{\to }B$, $D\stackrel{6}{\to }C$).

Sources: 7 Sketches, Exercise 2.58 and Solution A.2.

Solution: Solution 2.58

Exercise 2.60

Fill out the graph matrix $M_X$ for the graph $X$ of Eq. (2.56).

Sources: 7 Sketches, Exercise 2.60 and Solution A.2.

Solution: Solution 2.60

Exercise 2.61

Interpret an $\mathbf{NMY}$-category ($\mathbf{NMY}$ from 7S Exercise 2.34).

Sources: 7 Sketches, Exercise 2.61 and Solution A.2.

Solution: Solution 2.61

Exercise 2.62

Let $\mathcal{M}=(\mathcal{P}(M),\subseteq ,M,\cap )$ with $M=\{\mathsf{car},\mathsf{boat},\mathsf{foot}\}$ (“modes of transportation”). 1. Draw a graph with four vertices labeled by subsets. 2. Build the $\mathcal{M}$-category (union over paths of the intersection along each path) and write its matrix. 3. Is the interpretation “the hom-object is the set of modes that get you from $a$ to $b$” right?

Sources: 7 Sketches, Exercise 2.62 and Solution A.2.

Solution: Solution 2.62

Exercise 2.63

For $\mathbf{W}=(\mathbb{N}\cup \{\infty \},\le ,\infty ,\min )$: draw a labeled graph, compute the matrix of max-over-paths of min-edge-label, show it is a $\mathbf{W}$-category, and interpret.

Sources: 7 Sketches, Exercise 2.63 and Solution A.2.

Solution: Solution 2.63

Exercise 2.67

Apply the Change of Base $f(x)=[x=0]$ to the regions Boston, US, Spain and draw the resulting preorder. Interpretation?

Sources: 7 Sketches, Exercise 2.67 and Solution A.2.

Solution: Solution 2.67

Exercise 2.68

1. Find another Monoidal Monotone Map $\mathbf{Cost}\to \mathbf{Bool}$. 2. Find a Lawvere Metric Space on which it and $f(x)=[x=0]$ give different preorders.

Sources: 7 Sketches, Exercise 2.68 and Solution A.2.

Solution: Solution 2.68

Exercise 2.73

1. Show that a skeletal dagger Cost-category is an extended Metric Space. 2. Make sense of “preorders are to sets as Lawvere metric spaces are to extended metric spaces.”

See Opposite Enriched Category.

Sources: 7 Sketches, Exercise 2.73 and Solution A.2.

Solution: Solution 2.73

Exercise 2.75

Verify that the $\mathcal{V}$-product $\mathcal{X}\times \mathcal{Y}$ is a $\mathcal{V}$-category, and point out where symmetry is used.

Sources: 7 Sketches, Exercise 2.75 and Solution A.2.

Solution: Solution 2.75

Exercise 2.78

In the Cost-product $\mathbb{R}\times \mathbb{R}$, what is the distance from $(5,6)$ to $(-1,4)$?

Sources: 7 Sketches, Exercise 2.78 and Solution A.2.

Solution: Solution 2.78

Exercise 2.82

Prove that a monoidal preorder is closed iff $(-\otimes v)$ has a right adjoint $(v⊸-)$ for every $v$: 1. $(-\otimes v)$ is monotone; 2. if closed, $(v⊸w)\otimes v\le w$; 3. $(v⊸-)$ is monotone; 4. conclude.

Sources: 7 Sketches, Exercise 2.82 and Solution A.2.

Solution: Solution 2.82

Exercise 2.84

Show that $\mathbf{Bool}=(\mathbb{B},\le ,\mathsf{true},\wedge )$ is monoidal closed.

Sources: 7 Sketches, Exercise 2.84 and Solution A.2.

Solution: Solution 2.84

Exercise 2.92

1. What is $\bigvee \varnothing$ ("$0$") in $\mathbf{Bool}$ and in $\mathbf{Cost}$? 2. What is $x\vee y$ in each?

Sources: 7 Sketches, Exercise 2.92 and Solution A.2.

Solution: Solution 2.92

Exercise 2.93

Show $\mathbf{Bool}$ is a Quantale.

Sources: 7 Sketches, Exercise 2.93 and Solution A.2.

Solution: Solution 2.93

Exercise 2.94

Is $(\mathcal{P}(S),\subseteq ,S,\cap )$ a Quantale?

Sources: 7 Sketches, Exercise 2.94 and Solution A.2.

Solution: Solution 2.94

Exercise 2.103

Write the $2\times 2$ identity $\mathcal{V}$-matrix for $(\mathbb{N},\le ,1,\ast )$, $\mathbf{Bool}$, $\mathbf{Cost}$.

Sources: 7 Sketches, Exercise 2.103 and Solution A.2.

Solution: Solution 2.103

Exercise 2.104

In a Quantale, prove 1. $I_X\ast M=M$; 2. $(M\ast N)\ast P=M\ast (N\ast P)$.

Sources: 7 Sketches, Exercise 2.104 and Solution A.2.

Solution: Solution 2.104

Exercise 2.105

Compute $M_X^2$, $M_X^3$, $M_X^4$ for the matrix of 7S Exercise 2.60 and compare with 7S Exercise 2.58.

Sources: 7 Sketches, Exercise 2.105 and Solution A.2.

Solution: Solution 2.105