From Posets to Categories. Part 1

A poset (partially ordered set) is a pair (P, ≤), where

P is a set,

≤ is a binary relation on P satisfying the three axioms of partial order:

(i) Reflexive: ∀x ∈ P, x ≤ x

(ii) Antisymmetric: ∀x ,y ∈ P, x ≤ y & y ≤ x ⇒ x = y

(iii) Transitive: ∀x, y, z ∈ P, x ≤ y & y ≤ z ⇒ x ≤ z.

And what does this have to do with category theory?

“x ≤ y” ⇐⇒ “ x → y” “x = y” ⇐⇒ “ x ↔ y”

Given x, y ∈ P,

we say that u ∈ P is a least upper bound of x, y ∈ P if we have x → u & y → u, and for all z ∈ P satisfying x → z & y → z we must have u → z. It is more convenient to express this definition with a picture. We say that u ∈ P is a least upper bound of x, y if for all z ∈ P the following picture holds:

Dually, we say that l ∈ P is a greatest lower bound of x, y if for all z ∈ P the following picture holds:

Now suppose that u₁, u₂ ∈ P are two least upper bounds for x, y. Applying the defininition in both directions gives

u₁ → u₂ and u₂ → u₁,

and then from antisymmetry it follows that u₁ = u₂, which just means that u₁ and u₂ are indistinguishable within the structure of P. For this reason we can speak of the least upper bound (or “join”) of x, y. If it exists, we denote it by

x ∨ y

Dually, if it exists, we denote the greatest lower bound (or “meet”) by

x ∧ y

The definitions of meet and join are called “universal properties”. Whenever an object defined by a universal property exists, it is automatically unique in a certain canonical sense. However, since the object might not exist, maybe it is better to refer to a universal property as a “characterization,” or a “prescription,” rather than a “definition.”

Let P be a poset. We say that t ∈ P is a top element

if for all z ∈ P the following picture holds:

z —> t

Dually, we say that b ∈ P is a bottom element if for all z ∈ P the following picture holds:

b —> z

For any subset of elements of a poset S ⊆ P we say that the element ⋁ S ∈ P is its join if for all z ∈ P the following diagram is satisfied:

Dually, we say that ⋀ S ∈ P is the meet of S if for all z ∈ P the following diagram is satisfied:

If the objects ⋁ S and ⋀ S exist then they are uniquely characterized by their universal properties.

The universal properties in these diagrams will be called the “limit” and “colimit” properties when we move from posets to categories. Note that a limit/colimit diagram looks like a “cone over S”. This is one example of the link between category theory and topology.

Note that all definitions so far are included in this single (pair of) definition(s):

⋁ {x, y} = x∨ y & ⋀ {x, y} = x ∧ y

⋁∅ = 0 & ⋀ ∅ = 1.

Advertisement