I'm just getting started in category theory, and I'm not understanding the basic definitions. For example, a common example of a category is a poset. So suppose I have a trivial poset $P$ of 10 elements:
- Can I take the 10 elements of $P$ to be the objects of a category $A$?
- If so, does $A$ have one morphism representing the partial order, or are there many morphisms, one between each pair of qualifying elements?
- Can I alternatively take $P$ to be the only object of a category $B$?
- If so, does $B$ just have one morphism $P \rightarrow P$?
- Given a second poset $Q$, can I take $P$ and $Q$ to be the only two objects of a category $C$?
- If so, are there now 3 morphisms: $P \rightarrow P, Q \rightarrow Q, P \rightarrow Q$?
So far I find it easy to work with concepts like functors, subcategories, isomorphism, equivalence, etc. What I can't figure out is how to correlate these concepts with concrete data. Apologies if these questions are extremely trivial, but I have spent a lot of time reading and searching, and have yet to encounter a thorough explanation of these terms.