Stuck again. Not even a page through :-(
Reading Awodey's Category Theory [p.17] he says (this is after the definition of what a slice category $\boldsymbol{C}/C$ is):
If $C=P$ is a poset category and $p\in P$, then $P/p\cong\downarrow (p)$ the slice category $P/p$ is just the "principal ideal" $\downarrow(p)$ of elements $q\in P$ with $q\leq p$.
I have a few questions:
- What is "principal ideal" and why is it in quotes? Is that not an offical name?
- Why is "it" isomorphic to a slice category $P/p$? I can't find any inverses here...
- What do elements $q\in P$ have to do with anything if it's $\downarrow (p)$ and not $\downarrow(q)$?
(Probably question #3 is really silly, considering I don't understand 1 and 2.)
Any help appreciated, thank you.
P.S.: My math level: newbie