I found an exercise which says:
If $P$ is a partially ordered set and $\mathscr P$ the defining category,
characterize monomorphism, epimorphism and isomorphism in $\mathscr P$.
So, if there's at most one morphism from $a$ to $b\;$ ($a, b \in P$), then should every morphism be a monomorphism and an epimorphism? And what about isomorphisms? Are morphisms with inverses isomorphisms?
Thank you.
Best Answer
Yes, because there is at most one morphism between two object $a, b$, any morphism is mono and epi. By definition an isomorphism is a morphism with an inverse. For an isomorphism, keep in mind that $m: a \to b$ means that $a \leq b$. If we have $m: a \to b$ and $n: b \to a$ such that $mn$ and $nm$ are identities (fun exercise: proof that $mn$ and $nm$ are always the identity), we have that $a \leq b$ and $b \leq a$, so $a = b$. This means that the only isomorphisms are the identity on some object.