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The free category on the graph shown here: $$ \tag{3.6} \textbf{3} := \textbf{Free}( [ v_1 \overset{f_1}{\rightarrow} v_2 \overset{f_2}{\rightarrow} v_3 ] ) $$ It has three objects and six morphisms: the three vertices and six paths in the graph.
Create six names, one for each of the six morphisms in 3. Write down a six-by-six table, label the rows and columns by the six names you chose.
Fill out the table by writing the name of the composite in each cell.
Where are the identities?
Comments
$$ \begin{array}{ c l c c c c c c} hom_3 & f_1 & f_2 & \text{?} & \text{?} & \text{?} & \text{?} \\ \hline f_1 & ? & ? & ? & ? & ? & ? \\ f_2 & ? & ? & ? & ? & ? & ? \\ \text{?} & ? & ? & ? & ? & ? & ? \\ \text{?} & ? & ? & ? & ? & ? & ? \\ \text{?} & ? & ? & ? & ? & ? & ? \\ \text{?} & ? & ? & ? & ? & ? & ? \end{array} $$
$$ \begin{array}{ c l c c c c c c} hom_3 & f_1 & f_2 & \text{?} & \text{?} & \text{?} & \text{?} \\\\ \hline f_1 & ? & ? & ? & ? & ? & ? \\\\ f_2 & ? & ? & ? & ? & ? & ? \\\\ \text{?} & ? & ? & ? & ? & ? & ? \\\\ \text{?} & ? & ? & ? & ? & ? & ? \\\\ \text{?} & ? & ? & ? & ? & ? & ? \\\\ \text{?} & ? & ? & ? & ? & ? & ? \end{array} $$$$ \begin{array}{ c l c c c c c c} \text{Hom}_\mathbf{3} & f_1 & f_2 & f_3 & \text{id}_1 & \text{id}_2 & \text{id}_3 \\ \hline f_1 & NA & f_3 & NA & NA & f_1 & NA \\ f_2 & NA & NA & NA & NA & NA & f_2 \\ f_3 & NA & NA & NA & NA & NA & f_3 \\ \text{id}_1 & f_1 & NA & f_3 & \text{id}_1 & NA & NA \\ \text{id}_2 & NA & f_2 & NA & NA & \text{id}_2 & NA \\ \text{id}_3 & NA & NA & NA & NA & NA & \text{id}_3\end{array}$$
1. Define \\(f_2\circ f_1:=f_3\\). Entries will be of the form \\(\text{Top}\circ\text{Side}\\), or \\(NA\\) if no such composition is possible. $$ \begin{array}{ c l c c c c c c} \text{Hom}_\mathbf{3} & f_1 & f_2 & f_3 & \text{id}_1 & \text{id}_2 & \text{id}_3 \\\\ \hline f_1 & NA & f_3 & NA & NA & f_1 & NA \\\\ f_2 & NA & NA & NA & NA & NA & f_2 \\\\ f_3 & NA & NA & NA & NA & NA & f_3 \\\\ \text{id}_1 & f_1 & NA & f_3 & \text{id}_1 & NA & NA \\\\ \text{id}_2 & NA & f_2 & NA & NA & \text{id}_2 & NA \\\\ \text{id}_3 & NA & NA & NA & NA & NA & \text{id}_3\end{array}$$ 2. The identities are on the diagonal. This is because the only morphisms with inverses in this category are the identity morphisms.