The category of posets

Question

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.

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 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.

Answer 1

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.

Answer 2

Here is a fact that should be much more widely known than it is. The category of posets is isomorphic (not just equivalent) to the category of $T_0$ Alexandrof spaces. A topological space is said to be Alexandrof if arbitrary (not just finite) intersections of open sets are open. For example, every finite topological space is an Alexandrov space. Finite spaces are fascinating and the connection with algebraic topology is very close: finite spaces have associated finite simplicial complexes. Don't just look categorically: that takes the fun out of it! There are notes on my web page and there is a book by Barmak.

Answer 3

Probably not what you are looking for, but:

There is this recent paper by George Raptis:

Homotopy theory of posets, 2010

discussing about model category structures on the category of posets.

Answer 4

In a paper of Doubilet, Rota and Stanley , they see the incidence algebra of a poset as a contravariant functor from some subcategory of $Pos$ to the category of algebras over a field $K$. For instance, you may uniquely recover the poset from its incidence algebra.

I think this is quite interesting cos they then show how to arrive at some types of generating functions familiar from combinatorics.

Answer 5

Abstract simplicial complexes happen to be posets, and to every abstract simplicial complex, one may associate a topological space, its geometric realization. This is as functor $\mathrm{Pos} \rightarrow \mathrm{Top}$. (asc's are defined by the property that if $A \in \Delta$ and $B\subseteq A$, then $B \in \Delta$)

Also, you might want to take a look into the theory of Bruhat--Tits buildings. Basically, one associates to a simple algebraic group a certain simplicial complex $\Delta(G)$. However, I have one tried to figure out if that association is a functor without any success since given up.

Both example's seem to be more geometric/topological/algebraic than really category theoretical (viz. more focused on the objects than the functor), but maybe still instructive.

Answer 6

There is an example of a model category structure on a preorder of families of sets, defined as follows: $X\leq Y$ iff every element $x\in X$ is bounded $x\leq y_x$ by some element of $y_x\in Y$; families $X$ and $Y$ are weakly equivalent iff $X\leq Y$ and for every $y\in Y$ there is $x\in X$ such that $y$ and $x$ differ by finitely many elements; cofibrant objects are families of countable sets.

The example is also quite easy, however, it is set theoretic in nature. You may find the definitions in

A homotopy theory for set theory, I

A homotopy approach to set theory