I am trying to teach myself category theory and, as a begginer, I am looking for examples that I have a hands-on experience with.
Almost every introductory text in category theory contains following facts.
- The class of all posets with isotone maps is a category (called $Pos$).
- Every individual poset $P$ is a category, with comparable pairs $x\leq y$ as arrows. This can be called "a categorified poset".
Sometimes, product and coproduct in $Pos$ is characterized and (less frequently) it is pointed out that a Galois connection can be characterized as a pair of adjoint functors of categorified posets. Also, it is sometimes mentioned that products and coproducts in categorified posets are joins $\vee$ and meets $\wedge$, so a categorified poset has products and coproducts iff it is a latttice.
Moreover, some texts contain the fact that the category of finite posets and the category of finite distributive lattices are dually equivalent -- this is extremely useful.
But I cannot find much more. For example, when I tried to search for equalizers and coequalizers in $Pos$, the only source I found is the PhD thesis by Pietro Codara, available here, which is from 2004 and characterizes (co)equalizers in $Pos$.
Does anyone know about some other sources of information about $Pos$ from the category-theoretic viewpoint? Specifically, I am looking for things like
- useful functors to $Pos$ and from $Pos$,
- pullbacks, pushouts and other universal constructions in $Pos$,
- examples of adjoint functors, applications of Yoneda lemma etc.