CTfS Chapter 5 Exercises

exercise

Exercises from Category Theory for Scientists (CTfS), Chapter 5 (“Categories at work”). CTfS gives no solutions; the solutions here are the wiki’s own. This is a selection: the exercises referenced from concept notes. Solutions: CTfS Chapter 5 Solutions. Index: Map of Content.

Exercise 5.1.1.3

Let $X=\{a,b,c\}$, $M=(\mathbb{N},1,\ast )$ the multiplicative monoid of natural numbers, and $f:X\to \mathbb{N}$ with $f(a)=7$, $f(b)=2$, $f(c)=2$. Let ${\beta }_{X,M}:{\mathrm{Hom}}_{\mathbf{Set}}(X,U(M))\to {\mathrm{Hom}}_{\mathbf{Mon}}(\mathrm{List}(X),M)$ be the adjunction bijection. What is ${\beta }_{X,M}(f)([b,b,a,c])$?

CTfS §5.1.1; context: Free-Forgetful Adjunction, Map of Content.

Sources: CTfS, Exercise 5.1.1.3.

Solution: Solution 5.1.1.3

Exercise 5.1.1.6

Let $U:\mathbf{Grph}\to \mathbf{Set}$ send a graph to its set of vertices. This functor has both a left and a right adjoint. What are they?

CTfS §5.1.1; context: Adjunction, Map of Content.

Sources: CTfS, Exercise 5.1.1.6.

Solution: Solution 5.1.1.6

Exercise 5.1.1.9

The discrete category functor $\mathrm{Disc}:\mathbf{Set}\to \mathbf{Cat}$ has a left adjoint $p:\mathbf{Cat}\to \mathbf{Set}$. Describe it. (Hint: look at the mate isomorphism at $[1]\in \mathbf{Cat}$ and $\underline{2}\in \mathbf{Set}$.)

CTfS §5.1.1; context: Adjunction, Map of Content.

Sources: CTfS, Exercise 5.1.1.9.

Solution: Solution 5.1.1.9

Exercise 5.1.4.5

Let $[1]=(0\stackrel{f}{\to }1)$, $[2]=(0\stackrel{g}{\to }1\stackrel{h}{\to }2)$ and $F:[1]\to [2]$ with $0\mapsto 0$, $1\mapsto 2$.

• a. How many possibilities are there for $F(f)$?
• b. Let $I:[2]\to \mathbf{Set}$ be the instance with table 0 (word ↦ $g$): Am ↦ To be verb, Baltimore ↦ Place, Carla ↦ Person, Develop ↦ Action verb, Edward ↦ Person, Foolish ↦ Adjective, Green ↦ Adjective; table 1 ($h$): Action verb ↦ Verb, Adjective ↦ Adjective, Place ↦ Noun, Person ↦ Noun, To be verb ↦ Verb; table 2: Adjective, Noun, Verb. Write out the two tables of ${\Delta }_F(I)$.

CTfS §5.1.4; context: Data Migration Functor, Map of Content.

Sources: CTfS, Exercise 5.1.4.5.

Solution: Solution 5.1.4.5

Exercise 5.1.4.8

Let $F:\underline{3}\to \underline{2}$ (discrete categories) send $1\mapsto 1$, $2\mapsto 2$, $3\mapsto 2$.

• a. Write down an instance $I:\underline{3}\to \mathbf{Set}$.
• b. Given that ”${\Sigma }_F$ performs a parameterized colimit”, guess ${\Sigma }_F(I)$ as two sets made from the three sets you wrote down.

CTfS §5.1.4; context: Data Migration Functor, Map of Content.

Sources: CTfS, Exercise 5.1.4.8.

Solution: Solution 5.1.4.8

Exercise 5.1.4.11

With $F:\underline{3}\to \underline{2}$ as in Exercise 5.1.4.8:

• a. Write down an instance $I:\underline{3}\to \mathbf{Set}$.
• b. Given that ”${\Pi }_F$ performs a parameterized limit”, guess ${\Pi }_F(I)$ as two sets made from the three sets.

CTfS §5.1.4; context: Data Migration Functor, Map of Content.

Sources: CTfS, Exercise 5.1.4.11.

Solution: Solution 5.1.4.11

Exercise 5.2.3.3

• a. Come up with 4 overlapping open subsets covering the square $X=[0,3]\times [0,3]$; label each open set and each overlap, and draw the preorder $\mathrm{Op}(X)$ of these regions under inclusion. Make up formulas $R_1,R_2:X\to \mathbb{R}$ with $R_1\le R_2$, a range of temperatures that can exist at each point.
• b. Define a presheaf $\mathcal{O}$ by $\mathcal{O}(A)=\{f:A\to \mathbb{R}\mid R_1(a)\le f(a)\le R_2(a)\}$. What are the restriction maps? Do you like the name “value-assignment throughout $A$”?
• c. Define ${\mathcal{O}}^{\prime }(A)=\{f\in \mathcal{O}(A)\mid f\text{ continuous}\}$. Is there a morphism of presheaves ${\mathcal{O}}^{\prime }\to \mathcal{O}$?

CTfS §5.2.3; context: Map of Content, Sheaf.

Sources: CTfS, Exercise 5.2.3.3.

Solution: Solution 5.2.3.3

Exercise 5.3.2.5

Let $E$ be a set of “exceptions” (like “overflow!”, “division by zero!”). Let $T:\mathbf{Set}\to \mathbf{Set}$ be $X\mapsto X\bigsqcup E$. Following Example 5.3.2.4 (the Maybe monad), find a unit $\eta$ and multiplication $\mu$ making $(T,\eta ,\mu )$ a monad.

CTfS §5.3.2; context: Map of Content, Maybe Monad.

Sources: CTfS, Exercise 5.3.2.5.

Solution: Solution 5.3.2.5

Exercise 5.3.3.5

Let $J=(\mathcal{P},\eta ,\mu )$ be the power set monad.

• a. Given a morphism $f:A\to B$ in $\mathrm{Kls}(J)$ (a function $A\to \mathcal{P}(B)$), is there a natural way to associate a relation $R\subseteq A\times B$?
• b. How does composition in $\mathrm{Kls}(J)$ relate to composition of relations?

CTfS §5.3.3; context: Map of Content, Power Set Monad.

Sources: CTfS, Exercise 5.3.3.5.

Solution: Solution 5.3.3.5

Exercise 5.3.3.6

Let $J$ be the power set monad. $\mathrm{Kls}(J)$ has binary products. What is the product of $A=\{1,2,3\}$ and $B=\{a,b\}$?

CTfS §5.3.3; context: Map of Content, Power Set Monad.

Sources: CTfS, Exercise 5.3.3.6.

Solution: Solution 5.3.3.6

Exercise 5.3.3.7

Let $J$ be the power set monad. $\mathrm{Kls}(J)$ has binary coproducts. What is the coproduct of $A=\{1,2,3\}$ and $B=\{a,b\}$?

CTfS §5.3.3; context: Map of Content, Power Set Monad.

Sources: CTfS, Exercise 5.3.3.7.

Solution: Solution 5.3.3.7

Exercise 5.4.1.4

Consider an operad $\mathcal{O}$ like the little squares operad but with three objects: square, circle, equilateral triangle. A morphism is a positioning of non-overlapping shapes inside a shape.

• a. Draw an example of a morphism $f$ from two circles and a square to a triangle.
• b. Find three other morphisms that compose into $f$, and draw the composite.

CTfS §5.4.1; context: Map of Content, Operad.

Sources: CTfS, Exercise 5.4.1.4.

Solution: Solution 5.4.1.4