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Exercise 60 - Chapter 3

In Eq. (3.2), a graph is shown (forget the distinction between white and black nodes). Write down the corresponding Gr-instance. (Do not be concerned that you are in the primordial ooze.)

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Equation 3.2

Figure

Comments

  • 1.

    According to definition 3.59 a \(\mathcal{C}\)-instance functor from \(\mathcal{C}\) to Set. Gr is the free category on the graph with two vertices and two (parallel) arrows between them. Following the example in Seven Sketches, the Gr-instance corresponding to the graph above is \[\begin{matrix}\begin{array}{c|cc}\mathrm{Arrow}&\mathrm{Source}&\mathrm{Target}\\\hline \mathrm{Mngr}&\mathrm{Employee}&\mathrm{Employee}\\\mathrm{WorksIn}&\mathrm{Employee}&\mathrm{Department}\\\mathrm{Secr}&\mathrm{Department}&\mathrm{Employee}\\\mathrm{FName}&\mathrm{Employee}&\mathrm{string}\\\mathrm{DName}&\mathrm{Department}&\mathrm{string}\end{array}&\begin{array}{c|}\mathrm{Vertex}\\\hline \mathrm{Employee}\\\mathrm{Department}\\\mathrm{string}\end{array}\end{matrix}\].

    Comment Source:According to definition 3.59 a \\(\mathcal{C}\\)-instance functor from \\(\mathcal{C}\\) to **Set**. **Gr** is the free category on the graph with two vertices and two (parallel) arrows between them. Following the example in Seven Sketches, the **Gr**-instance corresponding to the graph above is \\[\begin{matrix}\begin{array}{c|cc}\mathrm{Arrow}&\mathrm{Source}&\mathrm{Target}\\\\\hline \mathrm{Mngr}&\mathrm{Employee}&\mathrm{Employee}\\\\\mathrm{WorksIn}&\mathrm{Employee}&\mathrm{Department}\\\\\mathrm{Secr}&\mathrm{Department}&\mathrm{Employee}\\\\\mathrm{FName}&\mathrm{Employee}&\mathrm{string}\\\\\mathrm{DName}&\mathrm{Department}&\mathrm{string}\end{array}&\begin{array}{c|}\mathrm{Vertex}\\\\\hline \mathrm{Employee}\\\\\mathrm{Department}\\\\\mathrm{string}\end{array}\end{matrix}\\].
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