7S Chapter 6 Exercises

exercise

Exercises from 7 Sketches, Chapter 6. Solutions: 7S Chapter 6 Solutions. Index: Map of Content.

Exercise 6.3

Consider the set $A=\{a,b\}$. Find a Preorder relation $\le$ on $A$ such that $(A,\le )$ 1. has no Initial Object; 2. has exactly one initial object; 3. has two initial objects.

Sources: 7 Sketches, Exercise 6.3 and Solution A.6.

Solution: Solution 6.3

Exercise 6.6

For each of the graphs below, consider the Free Category on that graph and say whether it has an Initial Object.

1. a single vertex $a$; 2. $a\to b\to c$; 3. two vertices $a$, $b$ and no edges; 4. a vertex $a$ with a loop.

Sources: 7 Sketches, Exercise 6.6 and Solution A.6.

Solution: Solution 6.6

Exercise 6.7

A Rig homomorphism $(R,0_R,{+}_R,1_R,{\ast }_R)\to (S,0_S,{+}_S,1_S,{\ast }_S)$ is a function $f$ with $f(0_R)=0_S$, $f(r_1{+}_R{r}_2)=f(r_1){+}_{S}f(r_2)$, etc.

1. Guess the remaining conditions.
2. Let $\mathbf{Rig}$ be the category of rigs and rig homomorphisms. It has an Initial Object. What is it?

Sources: 7 Sketches, Exercise 6.7 and Solution A.6.

Solution: Solution 6.7

Exercise 6.8

Explain the statement “the hallmark of universality is the existence of a unique map to any other comparable object” in the context of Definition 6.1 (Initial Object). What is being universal, and which is the “comparable object”?

Sources: 7 Sketches, Exercise 6.8 and Solution A.6.

Solution: Solution 6.8

Exercise 6.10

Let $\mathcal{C}$ be a category and $c_1,c_2$ two Initial Objects. Find an Isomorphism between them using only the universal property.

Sources: 7 Sketches, Exercise 6.10 and Solution A.6.

Solution: Solution 6.10

Exercise 6.13

Explain why, in a Preorder, Coproducts are the same as Joins.

Sources: 7 Sketches, Exercise 6.13 and Solution A.6.

Solution: Solution 6.13

Exercise 6.16

Let $T=\{a,\dots ,z\}$, $A=\{\mathrm{apple},\mathrm{banana},\mathrm{pear},\mathrm{cherry},\mathrm{orange}\}$, $B=\{\mathrm{apple},\mathrm{tomato},\mathrm{mango}\}$. Let $f:A\to T$ send each element to its first letter and $g:B\to T$ send each element to its last letter. Write down the copairing $[f,g]:A\bigsqcup B\to T$ on all eight elements.

Sources: 7 Sketches, Exercise 6.16 and Solution A.6.

Solution: Solution 6.16

Exercise 6.17

Let $f:A\to C$, $g:B\to C$, $h:C\to D$ in a category with Coproducts. Show

1. ${\iota }_A;[f,g]=f$;
2. ${\iota }_B;[f,g]=g$;
3. $[f,g];h=[f;h,g;h]$;
4. $[{\iota }_A,{\iota }_B]={\mathrm{id}}_{A+B}$.

Sources: 7 Sketches, Exercise 6.17 and Solution A.6.

Solution: Solution 6.17

Exercise 6.18

Suppose $\mathcal{C}$ has Coproducts $+$ and an Initial Object $\varnothing$. Then $(\mathcal{C},+,\varnothing )$ is a Symmetric Monoidal Category. Develop the data:

1. Show $+$ extends to a functor $\mathcal{C}\times \mathcal{C}\to \mathcal{C}$; how does it act on morphisms?
2. Show there are isomorphisms $A+\varnothing \to A$ and $\varnothing +A\to A$.
3. Write down morphisms (a) $(A+B)+C\to A+(B+C)$, (b) $A+B\to B+A$.

Sources: 7 Sketches, Exercise 6.18 and Solution A.6.

Solution: Solution 6.18

Exercise 6.24

For any set $S$ consider the Discrete Category ${\mathbf{Disc}}_S$.

1. Show that all Pushouts exist in ${\mathbf{Disc}}_S$.
2. For which sets $S$ does ${\mathbf{Disc}}_S$ have an Initial Object?

Sources: 7 Sketches, Exercise 6.24 and Solution A.6.

Solution: Solution 6.24

Exercise 6.26

Compute the Pushout of $f:\underline{4}\to \underline{5}$ and $g:\underline{4}\to \underline{3}$ given by $f=(1,3,5,5)$, $g=(1^{\prime },1^{\prime },2^{\prime },3^{\prime })$ (the picture in the book), and check it against the description in Example 6.25 (Finite Colimits in Set).

Sources: 7 Sketches, Exercise 6.26 and Solution A.6; Kittenlab lecture on colimits (union-find).

Solution: Solution 6.26

Exercise 6.28

In Example 6.27 (Pushout over an Initial Object is a Coproduct) justify the three “why?“s.

Sources: 7 Sketches, Exercise 6.28 and Solution A.6.

Solution: Solution 6.28

Exercise 6.35

Check that the pushout of pushouts from Example 6.33 (three pushouts $Q=X{+}_{A}Y$, $R=Y{+}_{B}Z$, $S=Q{+}_{Y}R$) satisfies the universal property of the Colimit of the original diagram $X\leftarrow A\to Y\leftarrow B\to Z$.

Sources: 7 Sketches, Exercise 6.35 and Solution A.6.

Solution: Solution 6.35

Exercise 6.41

Use the formula of Theorem 6.37 (Finite Colimits in Set) to show that Pushouts $X\stackrel{f}{\leftarrow }N\stackrel{g}{\to }Y$ agree with the description of Example 6.25.

Sources: 7 Sketches, Exercise 6.41 and Solution A.6.

Solution: Solution 6.41

Exercise 6.48

Eq. (6.47) shows Cospans $A\to B$ and $B\to C$ in ${\mathbf{Cospan}}_{\mathbf{FinSet}}$. Draw their monoidal product as a morphism $A+B\to B+C$.

Sources: 7 Sketches, Exercise 6.48 and Solution A.6.

Solution: Solution 6.48

Exercise 6.49

Depicting the composite of the cospans in Eq. (6.47) with wire notation gives Eq. (6.50). Describe the composition rule in ${\mathbf{Cospan}}_{\mathbf{FinSet}}$ in terms of wires and connected components.

Sources: 7 Sketches, Exercise 6.49 and Solution A.6.

Solution: Solution 6.49

Exercise 6.57

Let $X$ carry a Frobenius structure (Frobenius Monoid). Which of the six depicted morphisms $X\otimes X\to X\otimes X\otimes X$ are necessarily equal?

Sources: 7 Sketches, Exercise 6.57 and Solution A.6.

Solution: Solution 6.57

Exercise 6.59

In the wiring diagram of Eq. (6.58) in a Hypergraph Category: 1. What label should be on the input to $h$? 2. On the output of $g$? 3. On the fourth output wire of the composite?

Sources: 7 Sketches, Exercise 6.59 and Solution A.6.

Solution: Solution 6.59

Exercise 6.62

${\mathbf{Cospan}}_{\mathbf{FinSet}}$ is a Hypergraph Category. Draw the Frobenius morphisms for the object $\underline{1}$ using both the function and wiring depictions.

Sources: 7 Sketches, Exercise 6.62 and Solution A.6.

Solution: Solution 6.62

Exercise 6.63

Show, using colimits, that the Frobenius maps on ${\mathbf{Cospan}}_{\mathbf{FinSet}}$ (Example 6.61) obey the special law $\delta ;\mu =\mathrm{id}$ (Frobenius Monoid).

Sources: 7 Sketches, Exercise 6.63 and Solution A.6.

Solution: Solution 6.63

Exercise 6.67

Fill in the missing diagram in the proof of Proposition 6.66 (every Hypergraph Category is a Compact Closed Category with every object self-dual) using the Frobenius equations (6.51), their opposites, and (6.53).

Sources: 7 Sketches, Exercise 6.67 and Solution A.6.

Solution: Solution 6.67

Exercise 6.70

Check that the maps ${\phi }_{S,T}:\mathcal{P}(S)\times \mathcal{P}(T)\to \mathcal{P}(S\times T)$, $(A,B)\mapsto A\times B$, of Example 6.69 are natural in $S$ and $T$: for $f:S\to S^{\prime }$, $g:T\to T^{\prime }$ the square with ${\mathrm{im}}_f\times {\mathrm{im}}_g$ and ${\mathrm{im}}_{f\times g}$ commutes. (This makes the Power Set functor a lax Monoidal Functor.)

Sources: 7 Sketches, Exercise 6.70 and Solution A.6.

Solution: Solution 6.70

Exercise 6.78

The notation ${\mathbf{Cospan}}_{\mathcal{C}}$ (cospans in $\mathcal{C}$) looks like ${\mathbf{Cospan}}_F$ (Decorated Cospans), though they are different. An expert says “one is a special case of the other: just use the constant functor $F(c):=\{\ast \}$.” What does the expert mean?

Sources: 7 Sketches, Exercise 6.78 and Solution A.6.

Solution: Solution 6.78

Exercise 6.79

Write a tuple $(V,A,s,t,\ell )$ representing the circuit of Eq. (6.71) (a square with resistors on three sides, a capacitor on the top, and an inductor on the diagonal).

Sources: 7 Sketches, Exercise 6.79 and Solution A.6.

Solution: Solution 6.79

Exercise 6.80

Let $c\in \mathrm{Circ}(\underline{4})$ be the circuit $1-23\stackrel{3\Omega }{\to }4$ (a bare wire from $1$ to $2$ and a $3\Omega$ resistor from $3$ to $4$), and let $f:\underline{4}\to \underline{3}$ identify $2$ and $3$. What is $\mathrm{Circ}(f)(c)$?

Sources: 7 Sketches, Exercise 6.80 and Solution A.6.

Solution: Solution 6.80

Exercise 6.82

Given circuits $b$ (a battery) and $s$ (a switch) in $\mathrm{Circ}(\underline{2})$, use the definition of the laxator ${\psi }_{V,V^{\prime }}:\mathrm{Circ}(V)\times \mathrm{Circ}(V^{\prime })\to \mathrm{Circ}(V+V^{\prime })$ to compute ${\psi }_{2,2}(b,s)\in \mathrm{Circ}(\underline{4})$.

Sources: 7 Sketches, Exercise 6.82 and Solution A.6.

Solution: Solution 6.82

Exercise 6.84

Eq. (6.83) depicts a morphism of ${\mathbf{Cospan}}_{\mathrm{Circ}}$: a cospan with a decoration in $\mathrm{Circ}(\underline{2})$. What are they?

Sources: 7 Sketches, Exercise 6.84 and Solution A.6.

Solution: Solution 6.84

Exercise 6.86

Express the two circuits of Eq. (6.73) as morphisms in ${\mathbf{Cospan}}_{\mathrm{Circ}}$ and compute their composite. Does it match Eq. (6.74)?

Sources: 7 Sketches, Exercise 6.86 and Solution A.6.

Solution: Solution 6.86

Exercise 6.88

Let $x$ be the open circuit of Eq. (6.87) (a morphism $\underline{2}\to \underline{2}$ in ${\mathbf{Cospan}}_{\mathrm{Circ}}$). Define $\eta :\underline{0}\to \underline{2}$ as the cospan $\underline{0}\to \underline{1}\leftarrow \underline{2}$ and $ϵ:\underline{2}\to \underline{0}$ as $\underline{2}\to \underline{1}\leftarrow \underline{0}$, both decorated by the empty circuit $(\underline{1},\varnothing ,!,!,!)$. Compute $\eta ;x;ϵ$, a closed circuit $\underline{0}\to \underline{0}$.

Sources: 7 Sketches, Exercise 6.88 and Solution A.6.

Solution: Solution 6.88

Exercise 6.96

In the Operad $\mathbf{Cospan}$ (Undirected Wiring Diagrams):

1. Draw the cospan $f\in \mathbf{Cospan}(2,2;2)$, $\underline{2}+\underline{2}\to \underline{3}\leftarrow \underline{2}$, as a wiring diagram with two inner circles.
2. Draw the cospan $g\in \mathbf{Cospan}(2,2,2;0)$.
3. Compute $g{\circ }_{1}f$ and its arity.
4. Draw $g{\circ }_{1}f$. Do you see it as substitution?

Sources: 7 Sketches, Exercise 6.96 and Solution A.6.

Solution: Solution 6.96