7S Chapter 7 Exercises

exercise

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

Exercise 7.4

Prove Proposition 7.3 (the Pasting Lemma for Pullbacks) using the definition of Limit from Section 3.4.2.

Sources: 7 Sketches, Exercise 7.4 and Solution A.7.

Solution: Solution 7.4

Exercise 7.6

Show that in $\mathbf{Set}$, monomorphisms (defined via the Pullback square of Definition 7.5) are exactly the injections.

Sources: 7 Sketches, Exercise 7.6 and Solution A.7.

Solution: Solution 7.6

Exercise 7.7

1. Show that the Pullback of an Isomorphism $i:B^{\prime }\to B$ along any $f:A\to B$ is an isomorphism $i^{\prime }:A^{\prime }\to A$. 2. Show that for any $f:A\to B$, the square with $f$ on top and bottom and identities on the sides is a pullback.

Sources: 7 Sketches, Exercise 7.7 and Solution A.7.

Solution: Solution 7.7

Exercise 7.8

Suppose $A^{\prime }\to A$, $f^{\prime }:A^{\prime }\to B^{\prime }$, $f:A\to B$, $h:B^{\prime }\to B$ form a Pullback square. Use the Pasting Lemma for Pullbacks and 7S Exercise 7.7 to show that if $f$ is a Monomorphism then so is $f^{\prime }$.

Sources: 7 Sketches, Exercise 7.8 and Solution A.7.

Solution: Solution 7.8

Exercise 7.9

Factor the function $f:\underline{3}\to \underline{3}$ (with $f(1)=f(2)=1$, $f(3)=2$ in the picture) as an Epimorphism followed by a Monomorphism (Epi-Mono Factorization).

Sources: 7 Sketches, Exercise 7.9 and Solution A.7.

Solution: Solution 7.9

Exercise 7.11

Let $\mathcal{V}=(V,\le ,I,\otimes )$ be a Quantale with $v\le I$, $v\otimes w\le v$ and $v\otimes w\le w$, and ($x\le v$ and $x\le w$) $\Rightarrow x\le v\otimes w$. 1. Show $\mathcal{V}$ is a cartesian closed preorder. 2. Can every cartesian closed preorder be obtained this way?

Sources: 7 Sketches, Exercise 7.11 and Solution A.7.

Solution: Solution 7.11

Exercise 7.16

Let $m:\mathbb{N}\hookrightarrow \mathbb{Z}$ be the inclusion, with characteristic function $┌m┐:\mathbb{Z}\to \mathbb{B}$ (Subobject Classifier). 1. What is $┌m┐(-5)$? 2. What is $┌m┐(0)$?

Sources: 7 Sketches, Exercise 7.16 and Solution A.7.

Solution: Solution 7.16

Exercise 7.17

1. Describe the characteristic function $┌{\mathrm{id}}_{\mathbb{N}}┐:\mathbb{N}\to \mathbb{B}$ of the identity injection. 2. Describe $┌{!}_{\mathbb{N}}┐:\mathbb{N}\to \mathbb{B}$ for the unique function $\varnothing \to \mathbb{N}$.

Sources: 7 Sketches, Exercise 7.17 and Solution A.7.

Solution: Solution 7.17

Exercise 7.19

Negation $\neg :\mathbb{B}\to \mathbb{B}$ is the characteristic function of some thing $(\ast ?\ast )$. 1. What sort of thing is $(\ast ?\ast )$? 2. Which thing is it?

Sources: 7 Sketches, Exercise 7.19 and Solution A.7.

Solution: Solution 7.19

Exercise 7.20

Define $P\Rightarrow Q$ to mean $P=(P\wedge Q)$. 1. Write the truth table of $P=(P\wedge Q)$. 2. Does it agree with your idea of implication? 3. What is the characteristic function of $P\Rightarrow Q$? 4. Which subobject does it classify?

Sources: 7 Sketches, Exercise 7.20 and Solution A.7.

Solution: Solution 7.20

Exercise 7.21

Let $E$ = evens, $P$ = primes, $T=\{n\ge 10\}$, subsets of $\mathbb{N}$ with characteristic functions $┌E┐,┌P┐,┌T┐:\mathbb{N}\to \mathbb{B}$. 1. $┌E┐(17)$? 2. $┌P┐(17)$? 3. $┌T┐(17)$? 4. The smallest three elements of the set classified by $(┌E┐\wedge ┌P┐)\vee ┌T┐$?

Sources: 7 Sketches, Exercise 7.21 and Solution A.7.

Solution: Solution 7.21

Exercise 7.27

$\mathbb{R}$ with $d(x_1,x_2)=|x_1-x_2|$ is a Metric Space. 1. Define the $ϵ$-ball $B(x,ϵ)$. 2. When is $U\subseteq \mathbb{R}$ open? 3. Find opens $U_1,U_2$ covering an open $U$. 4. Find an infinite cover.

Sources: 7 Sketches, Exercise 7.27 and Solution A.7.

Solution: Solution 7.27

Exercise 7.29

1. Verify that the coarse topology $\{\varnothing ,X\}$ is a topology. 2. Verify that the fine topology $\mathcal{P}(X)$ is. 3. Show every function from a discrete space $(X,\mathcal{P}(X))$ to any space $Y$ is continuous.

Sources: 7 Sketches, Exercise 7.29 and Solution A.7.

Solution: Solution 7.29

Exercise 7.31

For the Sierpinski Space $(\{1,2\},{\mathrm{Op}}_1)$: 1. draw the Hasse Diagram of its preorder of opens; 2. write down all covers.

Sources: 7 Sketches, Exercise 7.31 and Solution A.7.

Solution: Solution 7.31

Exercise 7.32

For $Y\subseteq X$ with $(X,\mathrm{Op})$ a Topological Space, the subspace topology ${\mathrm{Op}}_{?\cap Y}$ declares $A\subseteq Y$ open iff $A=B\cap Y$ for some $B\in \mathrm{Op}$. 1. Show $Y$ is open. 2. Show it is a topology. 3. Show the inclusion $Y\hookrightarrow X$ is continuous.

Sources: 7 Sketches, Exercise 7.32 and Solution A.7.

Solution: Solution 7.32

Exercise 7.34

If $(X,\mathrm{Op})$ is a Topological Space and $\mathcal{V}=(\mathrm{Op},\subseteq ,X,\cap )$ the corresponding Quantale (Remark 7.33), how might we imagine a $\mathcal{V}$-category?

Sources: 7 Sketches, Exercise 7.34 and Solution A.7.

Solution: Solution 7.34

Exercise 7.38

For $f:X\to Y$ of Eq. (7.37): 1. the Fiber over $a$? 2. over $c$? 3. over $d$? 4. Give an $f^{\prime }:X\to Y$ all of whose fibers have one or two elements.

Sources: 7 Sketches, Exercise 7.38 and Solution A.7.

Solution: Solution 7.38

Exercise 7.40

For the Sheaf of Sections ${\mathrm{Sec}}_f$ of Eq. (7.37): 1. draw all sections over $V_1=\{a,b,c\}$; 2. over $V_2=\{a,b,c,d\}$; 3. how many sections over $V_3=\{a,b,d,e\}$?

Sources: 7 Sketches, Exercise 7.40 and Solution A.7.

Solution: Solution 7.40

Exercise 7.42

1. Write out ${\mathrm{Sec}}_f(\{a,b,c\})$ and ${\mathrm{Sec}}_f(\{a,c\})$. 2. Indicate the restriction map (Sheaf of Sections).

Sources: 7 Sketches, Exercise 7.42 and Solution A.7.

Solution: Solution 7.42

Exercise 7.44

With $U_1=\{a,b\}$, $U_2=\{b,e\}$, overlap $\{b\}$, in the Sheaf of Sections ${\mathrm{Sec}}_f$: 1. find $s_1\in {\mathrm{Sec}}_f(U_1)$, $s_2\in {\mathrm{Sec}}_f(U_2)$ not agreeing on the overlap; 2. can they be glued? 3. find $h_1,h_2$ that agree but differ from Eq. (7.43); 4. can they be glued?

Sources: 7 Sketches, Exercise 7.44 and Solution A.7.

Solution: Solution 7.44

Exercise 7.47

Is there a one-to-one correspondence between sheaves on the sphere $M$ and vector fields on $M$? If not, how are they related?

Sources: 7 Sketches, Exercise 7.47 and Solution A.7.

Solution: Solution 7.47

Exercise 7.49

For the Sierpinski Space: 1. What is the category $\mathrm{Op}$? 2. What is a Presheaf on it? 3. What is the sheaf condition? 4. How do we identify a sheaf with a function?

Sources: 7 Sketches, Exercise 7.49 and Solution A.7.

Solution: Solution 7.49

Exercise 7.52

The Subobject Classifier of $\mathbf{Set}$ is $\mathbb{B}$; how does that align with $\Omega (U)=\{U^{\prime }\in \mathrm{Op}\mid U^{\prime }\subseteq U\}$ on the one-point space $X=\{1\}$?

Sources: 7 Sketches, Exercise 7.52 and Solution A.7.

Solution: Solution 7.52

Exercise 7.53

1. Show that $\Omega (U)=\{U^{\prime }\subseteq U\text{ open}\}$ with restriction $U^{\prime }\mapsto U^{\prime }\cap V$ is functorial. 2. Is that all that is needed for $\Omega$ to be a Presheaf?

Sources: 7 Sketches, Exercise 7.53 and Solution A.7.

Solution: Solution 7.53

Exercise 7.55

Let $G$ be the graph $A\stackrel{f}{\to }B$, $A\stackrel{g}{\to }B$, $B\stackrel{h}{\to }C$, $C\stackrel{i}{\to }D$ and $H\subseteq G$ the subgraph with vertices $A,B,C$ and the single arrow $f$. Find the classifying graph homomorphism $┌H┐:G\to {\Omega }_{\mathbf{Grph}}$ (Topos of Graphs).

Sources: 7 Sketches, Exercise 7.55 and Solution A.7.

Solution: Solution 7.55

Exercise 7.59

In $\mathbb{R}$ let $U=\mathbb{R}\setminus \{0\}$. 1. What is $\neg U$? 2. $\neg \neg U$? 3. Is $U\subseteq \neg \neg U$? 4. Is $\neg \neg U\subseteq U$? (Internal Logic of a Topos)

Sources: 7 Sketches, Exercise 7.59 and Solution A.7.

Solution: Solution 7.59

Exercise 7.60

In $\mathbf{Shv}(X)$: 1. which open set is $\mathsf{true}$, given $\mathsf{true}\wedge U=U$ for all $U$? 2. Check $\mathsf{true}\vee U=\mathsf{true}$, $U\Rightarrow \mathsf{true}=\mathsf{true}$, $\mathsf{true}\Rightarrow U=U$. 3. Which open set is $\mathsf{false}$, given $\mathsf{false}\vee U=U$? 4. Check $\mathsf{false}\wedge U=\mathsf{false}$ and $\mathsf{false}\Rightarrow U=\mathsf{true}$.

Sources: 7 Sketches, Exercise 7.60 and Solution A.7.

Solution: Solution 7.60

Exercise 7.62

A Predicate $p:S\to \Omega$ such as ”… likes the weather” also defines a subsheaf $\{S\mid p\}\subseteq S$. Describe its sections.

Sources: 7 Sketches, Exercise 7.62 and Solution A.7.

Solution: Solution 7.62

Exercise 7.64

Give a space $X$, a sheaf $S$, and predicates $p,q:S\to \Omega$ with $p(s){⊢}_{s:S}q(s)$.

Sources: 7 Sketches, Exercise 7.64 and Solution A.7.

Solution: Solution 7.64

Exercise 7.66

In $\mathbf{Set}$ let $p:\mathbb{N}\times \mathbb{Z}\to \mathbb{B}$, $p(n,z)=[n\le |z|]$. Find the sets where 1. $\forall (z:\mathbb{Z}).p(n,z)$; 2. $\exists (z:\mathbb{Z}).p(n,z)$; 3. $\forall (n:\mathbb{N}).p(n,z)$; 4. $\exists (n:\mathbb{N}).p(n,z)$ hold (Quantification).

Sources: 7 Sketches, Exercise 7.66 and Solution A.7.

Solution: Solution 7.66

Exercise 7.67

Apply the definition of universal Quantification to $p(s,t)$ = “person $s$ is worried about news $t$”, for $s$ alive throughout $U$. 1. What open subset of $U$ is $\forall (t:T).p(s,t)$? 2. Does it match the informal meaning?

Sources: 7 Sketches, Exercise 7.67 and Solution A.7.

Solution: Solution 7.67

Exercise 7.68

Apply the definition of existential Quantification to “person $s$ is worried about news $t$“. 1. What open set is $\exists (t:T).p(s,t)$? 2. Does it match expectations?

Sources: 7 Sketches, Exercise 7.68 and Solution A.7.

Solution: Solution 7.68

Exercise 7.70

Let $j:\Omega \to \Omega$ satisfy $p\le j(p)$ for all $p$. Show $j(j(q))\le j(q)$ iff $j(j(q))=j(q)$ (Modality).

Sources: 7 Sketches, Exercise 7.70 and Solution A.7.

Solution: Solution 7.70

Exercise 7.72

Let $S$ be the sheaf of people and $j$ = “assuming Bob is in San Diego, …” (Modality of type (a)). 1. Name a predicate $p$. 2. For a time interval $U$ and $s\in S(U)$, what is $p(s)$? 3. What is $j(p(s))$? 4. Is $p(s)\le j(p(s))$? 5. Is $j(j(p(s)))=j(p(s))$? 6. For another $q$, is $j(p\wedge q)=j(p)\wedge j(q)$?

Sources: 7 Sketches, Exercise 7.72 and Solution A.7.

Solution: Solution 7.72

Exercise 7.76

In the Interval Domain: 1. Why is $[2,6]\in o_{[0,8]}$? 2. Why is $[2,6]\notin o_{[0,5]}\cup o_{[4,8]}$?

Sources: 7 Sketches, Exercise 7.76 and Solution A.7.

Solution: Solution 7.76

Exercise 7.77

Show that $U\subseteq \mathbb{R}$ is open in the subspace topology of $\mathbb{R}\subseteq \mathbb{IR}$ (Interval Domain) iff it is open in the usual topology.

Sources: 7 Sketches, Exercise 7.77 and Solution A.7.

Solution: Solution 7.77

Exercise 7.80

Fix a Topological Space $X$ and a subset $R\subseteq \mathbb{IR}$; define $H_X(U):=\{f:U\cap R\to X\text{ continuous}\}$. 1. Is $H_X$ a Presheaf? What are the restriction maps? 2. Is it a Sheaf?

Sources: 7 Sketches, Exercise 7.80 and Solution A.7.

Solution: Solution 7.80