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Exercise 48 - Chapter 4

edited August 2018 in Exercises

\(\def\cat#1{{\mathcal{#1}}}\) \(\def\comp#1{{\widehat{#1}}}\) \(\def\conj#1{{\check{#1}}}\) \(\def\id{{\mathrm{id}}}\) \(\def\true{{\mathrm{true}}}\) \(\def\false{{\mathrm{false}}}\) \(\def\Cat#1{{\textbf{#1}}}\) \(\def\BB{{\mathbb{B}}}\) \(\def\ZZ{{\mathbb{Z}}}\) \(\def\tn#1{{\text{#1}}}\) Consider the monoidal category \((\Cat{Set},1,\times)\), together with the diagram

Suppose that \(A=B=C=D=F=G=\ZZ\) and \(E=\BB={\true,\false}\), and suppose that \(f_C(a)=|a|\), \(f_D(a)=a*5\), \(g_E(d,b)=d\leq b\), \(g_F(d,b)=d-b\), and \(h(c,e)=\tn{if }e\tn{ then }c\tn{ else }1-c\).

  1. What are \(g_E(5,3)\) and \(g_F(5,3)\)?
  2. What are \(g_E(3,5)\) and \(g_F(3,5)\)?
  3. What is \(h(5,\true)\)?
  4. What is \(h(-5,\true)\)?
  5. What is \(h(-5,\false)\)?

The whole diagram now defines a function \(A\times B\to G\times F\); call it \(q\).

  1. What are \(q_G(-2,3)\) and \(q_F(-2,3)\)?
  2. What are \(q_G(2,3)\) and \(q_F(2,3)\)?

Comments

  • 1.
    1. \(g_E(5,3)=\mathrm{false}\), \(g_F(5,3)=2\).
    2. \(g_E(3,5)=\mathrm{true}\), \(f_F(3,5)=-2\).
    3. \(h(5,\mathrm{true})=5\).
    4. \(h(-5,\mathrm{true})=-5\).
    5. \(h(-5,\mathrm{false})=6\).

    \(\)

    1. \(q_G(-2,3)=2\), \(q_F(-2,3)=-13\).
    2. \(q_G(2,3)=-1\), \(q_F(2,3)=7\).

    In general, \(q_G(a,b)=\begin{cases}\lvert a\rvert\,\,\text{if}\,5a\leq b\\1-\lvert a\rvert\,\,\text{otherwise}\end{cases}\) and \(q_F(a,b)=5a-b\).

    Comment Source:1. \\(g\_E(5,3)=\mathrm{false}\\), \\(g\_F(5,3)=2\\). 2. \\(g\_E(3,5)=\mathrm{true}\\), \\(f\_F(3,5)=-2\\). 3. \\(h(5,\mathrm{true})=5\\). 4. \\(h(-5,\mathrm{true})=-5\\). 5. \\(h(-5,\mathrm{false})=6\\). \\(\\) 1. \\(q\_G(-2,3)=2\\), \\(q\_F(-2,3)=-13\\). 2. \\(q\_G(2,3)=-1\\), \\(q\_F(2,3)=7\\). In general, \\(q\_G(a,b)=\begin{cases}\lvert a\rvert\,\,\text{if}\,5a\leq b\\\\1-\lvert a\rvert\,\,\text{otherwise}\end{cases}\\) and \\(q\_F(a,b)=5a-b\\).
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