I have just started my course in Category Theory and I have the following problem.
Let $(P,\leq)$ be a poset and $Q$ the category which it defines:
a)Under what conditions does $Q$ have a initial element?
b)Give a characterization of monorphism and isomorphism in $Q$
By now I am little lost in this theory but I imagine that:
a)If there exists $p\in{P}$ such that $p\leq{s}\;\forall{s\in{P}}$ then $p$ is a initial object
b)I think that the characterisation must be related to $\leq$, for example that $\leq$ is a total order.
These are only my conjectures about the problem. I don't know if they are true. Could someone help me to prove them?
Thanks.
As others have stated, $Q$ has an intial object if $P$ has a least element $s$ - meaning no other element is less than $s$.
In a poset, if $a\leq b$ and $b\leq a$, then $a=b$ as stated. This implies that isomorphisms in $Q$ are just equality, or the identity morphism. This has nothing to do if $Q$ is a total order - there are plenty of posets where this is true (something this is taken as the definition of equality in a poset).
A morphism $f:A\rightarrow B$ is a monomorphism if for all morphisms $g_1,g_2:C\rightarrow A$, $$f\circ g_1 = f\circ g_2 \Rightarrow g_1 = g_2$$
In $Q$, there exists at most 1 morphism between two objects. This makes every morphism a monomorphism.