I am trying to teach myself category theory and, as a beginner, 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.
Best Answer
Here are some basic remarks and examples: (Caution. This answer refers to preorders; but many of the remarks also apply to partially ordered sets aka posets)
Many concepts of category theory have a nice illustration when applied to preorders; but also the other way round: Many concepts familiar from preorders carry over to categories (for example suprema motivate colimits; see also below).
This is partially justified by the following observation: An arbitrary category is a sort of a preorder but where you have to specify in addition a reason why $x \leq y$, in form of an arrow $x \to y$. The axioms for a category tell you: For every $x$ there is a distinguished reason for $x \leq x$, and whenever you have a reason for $x \leq y$ and for $y \leq z$, you also get a reason for $x \leq z$.
A preorder is a category such that every diagram commutes.
In a preorder, the limit of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't matter.
When $f^* : P \to Q$ is a cocontinuous functor between preorders, where $P$ is complete, then $f^*$ has a right adjoint $f_*$; you can write it down explicitly: $f_*(q)$ is the infimum of the $p$ with $f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add the solution set condition, so that the a priori big limit can be replaced by a small one and therefore exists.
Let $f : X \to Y$ be a map of sets. Then the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ between the power sets is right adjoint to image functor $\mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way.
The inclusion functor $\mathrm{Pre} \to \mathrm{Cat}$ has a left adjoint: It sends every category to its set of objects with the order $x \leq y$ if there is a morphism $x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in $\mathrm{Cat}$ are constructed "pointwise". Thus, the same is true for limits in $\mathrm{Pre}$ (which one could equally well see directly). For example, the pullback of $f : P \to Q$ and $g : P' \to Q$ is the pullback of sets $P \times_Q P'$ equipped with the order $(a,b) \leq (c,d)$ iff $a \leq c$ and $b \leq d$. If we apply this to difference kernels, we see that $f : P \to Q$ is a monomorphism iff the underlying map of $f$ is injective.
The forgetful functor $\mathrm{Pre} \to \mathrm{Set}$ creates coproducts: Take the disjoint union $\coprod_i P_i$ and take the order $a \leq b$ iff $a,b$ lie in the same $P_i$, and with respect to that preorder we have $a \leq_i b$.
The construction of coequalizers seems to be more delicate; see this SE discussion.
I don't have a reference for all these observations, but they are easy. A general reference for basic category-theoretic constructions (and it surely says something about preorders and posets) is the book "Abstract and Concrete Categories - The Joy of Cats" by Adamek, Herrlich, Strecker which you can find online.
EDIT: Here is something not so basic: Sefi Ladkani studied the notion of derived equivalent posets. Two posets $X,Y$ are called (universally) derived equivalent if for some specific (every) abelian category $\mathcal{A}$ the diagram categories $\mathcal{A}^X$, $\mathcal{A}^Y$ are derived equivalent.