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2 a typo

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.

1. The class of all posets with isotone maps is a category (called $Pos$).
2. 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 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.

# The category of posets

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.

1. The class of all posets with isotone maps is a category (called $Pos$).
2. 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 it 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.