Lets define a category $X_{\mathbb Z^+}$ from $\mathbb Z^+$ and the division relationship like this:
- The objects of $X_{\mathbb Z^+}$ are the elements of $\mathbb Z^+$
- If $a,b \in $$\mathbb Z^+$, i.e, $a,b$ are positive integers, we define the following morphism:
$Hom_{X_{\mathbb Z^+}}(a,b)= \left\{ \begin{array}{lcc} \{(a,b)\} & if & a|b\\ \\ \emptyset & if & a\nmid b \\ \end{array} \right.$
a)Verify that with these definitions, $X_{\mathbb Z^+}$ satisfies the axioms of a category
b)Prove that the product and coproduct of any finite family of objects $\{a_1,....a_n\} $ exist in $X_{\mathbb Z^+}$.
I know that in the positive integers, the relationship of division is an order relationship, i.e, it is refelexive, transitive and antisymmetric. I also know that the axioms of the categories are:
- Associativity: If $f:A\rightarrow B, g: B\rightarrow C, h: C\rightarrow D$ are morphisms of the class $X_{\mathbb Z^+}$, then $h\circ(g\circ f)= (h \circ g)\circ f$
- Identity: For each object $B$ of $X_{\mathbb Z^+}$ there exists a morphism $1_B:B\rightarrow B$ such that for any $f:A\rightarrow B, g: B \rightarrow C,$ $1_B\circ f=f$ and $g\circ1_B=g$
Can I get some help, please?