7S Chapter 4 Exercises

exercise

Exercises from 7 Sketches, Chapter 4. Solutions: 7S Chapter 4 Solutions. Index: Map of Content.

Exercise 4.4

Let $X=\{\mathsf{monoid},\mathsf{preorder}\le \mathsf{category}\}$ and $Y=\{\mathsf{nothing}\le \mathsf{this}\mathsf{book}\}$. 1. Draw the Hasse diagram of $X^{\mathrm{op}}\times Y$. 2. Give a profunctor $\Lambda :X\nrightarrow Y$, reading $\Lambda (x,y)=\mathsf{true}$ as “my aunt can explain an $x$ given $y$”, and interpret the fact that ${\Lambda }^{-1}(\mathsf{true})$ is an Upper Set.

Sources: 7 Sketches, Exercise 4.4 and Solution A.4.

Solution: Solution 4.4

Exercise 4.7

Show that $\Rightarrow$ (with $\mathsf{true}\Rightarrow \mathsf{false}=\mathsf{false}$ and $\mathsf{true}$ otherwise) satisfies $b\wedge c\le d$ iff $b\le (c\Rightarrow d)$.

Sources: 7 Sketches, Exercise 4.7 and Solution A.4.

Solution: Solution 4.7

Exercise 4.9

Show that a $\mathcal{V}$-Profunctor $\Phi :{\mathcal{X}}^{\mathrm{op}}\times \mathcal{Y}\to \mathcal{V}$ is the same as a function $\Phi :\mathrm{Ob}(\mathcal{X})\times \mathrm{Ob}(\mathcal{Y})\to V$ with $\mathcal{X}(x^{\prime },x)\otimes \Phi (x,y)\otimes \mathcal{Y}(y,y^{\prime })\le \Phi (x^{\prime },y^{\prime })$.

Sources: 7 Sketches, Exercise 4.9 and Solution A.4.

Solution: Solution 4.9

Exercise 4.10

Is a $\mathbf{Bool}$-profunctor exactly a Feasibility Relation?

Sources: 7 Sketches, Exercise 4.10 and Solution A.4.

Solution: Solution 4.10

Exercise 4.12

Fill in the feasibility matrix of the bridge profunctor $\Phi :X\nrightarrow Y$ of Example 4.11.

Sources: 7 Sketches, Exercise 4.12 and Solution A.4.

Solution: Solution 4.12

Exercise 4.15

Fill in the Cost-matrix of the bridge profunctor of Example 4.13.

Sources: 7 Sketches, Exercise 4.15 and Solution A.4.

Solution: Solution 4.15

Exercise 4.17

Compute $M_X^3\ast M_{\Phi }\ast M_Y^2$ with min-plus multiplication and compare with 7S Exercise 4.15.

Sources: 7 Sketches, Exercise 4.17 and Solution A.4.

Solution: Solution 4.17

Exercise 4.18

The node $(g/n,\mathsf{funny})$ has no bridge out of it. Valid? Meaning?

Sources: 7 Sketches, Exercise 4.18 and Solution A.4.

Solution: Solution 4.18

Exercise 4.22

Fill in the composite $\Phi ;\Psi :X\nrightarrow Z$ of the two Cost-profunctors shown.

Sources: 7 Sketches, Exercise 4.22 and Solution A.4.

Solution: Solution 4.22

Exercise 4.26

Draw a bridge diagram for the unit profunctor $U_X$ of a Cost-category $X$.

Sources: 7 Sketches, Exercise 4.26 and Solution A.4.

Solution: Solution 4.26

Exercise 4.30

Justify the steps in Eqs. (4.28)–(4.29) of Lemma 4.27, and show they are equalities when $\mathcal{V}=\mathbf{Bool}$.

Sources: 7 Sketches, Exercise 4.30 and Solution A.4.

Solution: Solution 4.30

Exercise 4.32

Prove associativity of profunctor composition (Lemma 4.31).

Sources: 7 Sketches, Exercise 4.32 and Solution A.4.

Solution: Solution 4.32

Exercise 4.36

Check that the companion of $\mathrm{id}:P\to P$ is the unit profunctor.

Sources: 7 Sketches, Exercise 4.36 and Solution A.4.

Solution: Solution 4.36

Exercise 4.38

What is the conjoint $\check{+}$ of $+:{\mathbb{R}}^3\to \mathbb{R}$?

Sources: 7 Sketches, Exercise 4.38 and Solution A.4.

Solution: Solution 4.38

Exercise 4.41

For a skeletal quantale $\mathcal{V}$: 1. show $\mathcal{V}$-functors $F,G$ are $\mathcal{V}$-adjoint iff $\hat{F}=\check{G}$; 2. deduce $\widehat{\mathrm{id}}=\check{\mathrm{id}}$.

Sources: 7 Sketches, Exercise 4.41 and Solution A.4.

Solution: Solution 4.41

Exercise 4.44

Draw the Hasse diagram of the Collage of the Cost-profunctor of Example 4.13.

Sources: 7 Sketches, Exercise 4.44 and Solution A.4.

Solution: Solution 4.44

Exercise 4.48

Check that a Symmetric Monoidal Preorder is a Monoidal Category with at most one morphism between any two objects.

Sources: 7 Sketches, Exercise 4.48 and Solution A.4.

Solution: Solution 4.48

Exercise 4.50

In $(\mathbf{Set},1,\times )$ with $A=B=C=D=F=G=\mathbb{Z}$, $E=\mathbb{B}$, $f_C(a)=|a|$, $f_D(a)=5a$, $g_E(d,b)=[d\le b]$, $g_F(d,b)=d-b$, $h(c,e)=$ if $e$ then $c$ else $1-c$: compute the listed values and the composite $q:A\times B\to G\times F$.

Sources: 7 Sketches, Exercise 4.50 and Solution A.4.

Solution: Solution 4.50

Exercise 4.52

Does Rough Definition 4.51 with $\mathcal{V}=(\mathbf{Set},1,\times )$ agree with the definition of Category?

Sources: 7 Sketches, Exercise 4.52 and Solution A.4.

Solution: Solution 4.52

Exercise 4.54

What are identity elements in Cost-categories?

Sources: 7 Sketches, Exercise 4.54 and Solution A.4.

Solution: Solution 4.54

Exercise 4.62

For $\underline{3}$ in $\mathbf{Corel}$: draw the unit $\varnothing \to \underline{3}\bigsqcup \underline{3}$, the counit $\underline{3}\bigsqcup \underline{3}\to \varnothing$, and check a snake equation.

Sources: 7 Sketches, Exercise 4.62 and Solution A.4.

Solution: Solution 4.62

Exercise 4.64

Interpret monoidal products in $\mathbf{Feas}$ in terms of feasibility.

Sources: 7 Sketches, Exercise 4.64 and Solution A.4.

Solution: Solution 4.64

Exercise 4.65

What are the isomorphisms $X\times 1\cong X$ and $1\times X\cong X$ in ${\mathbf{Prof}}_{\mathcal{V}}$?

Sources: 7 Sketches, Exercise 4.65 and Solution A.4.

Solution: Solution 4.65

Exercise 4.66

Check the snake equations for ${\eta }_X(1,x,x^{\prime })=X(x,x^{\prime })$ and ${\epsilon }_X(x,x^{\prime },1)=X(x,x^{\prime })$ in ${\mathbf{Prof}}_{\mathcal{V}}$.

Sources: 7 Sketches, Exercise 4.66 and Solution A.4.

Solution: Solution 4.66