Products and coproducts in posets

Question

While reading Bartosz' excellent Category theory for Programmers, I got stuck in the second exercise, which deals with products in posets. Given a poset,

    b   e
  ↗   ⤭   ↘
a → c   f → h
  ↘   ⤭   ↗
    d   g

how can I define a product in the categorical sense? What is categorized by the product of two objects? And what about the coproduct?

Answer 1

Let's have a look at the definition of a product first:

A product of objects a and b is the object c equipped with morphisms p :: c -> a and q :: c -> b exist, such that for any other object c' (with morphisms p' :: c' -> a and q' :: c' -> b), there exists a morphism m :: c' -> c such that p' = p . m and q' = q . m.

Remember that a morphism in a poset basically describes the relationship "less than or equal to".

Now the product c between two objects a and b must be an object less than or equal to both a and b. As an example, lets choose a as e and b as g from your graph:

    b   e -- this one is a
  ↗   ⤭   ↘
a → c   f → h
  ↘   ⤭   ↗
    d   g -- this one is b

Trivially, the first object that comes to mind which is always less than or equal to any other object is the smallest object, in this case a.

Now is a a valid candidate for the product of e and g? Let's check the definition of product:

Is there a morphism from a to e? Yes, this exists and can be written as pₐ = ce . ac (read as: "first the arrow from a to c, then the arrow from c to e").

Is there a morhism from a to g? Yes, this exists too and can be written as qₐ = cg . ac.

So far so good, the only question left is whether this is the 'best' candidate in the sense that no other object exists such that we can construct a unique isomorphism between a and the other candidate?

Looking at the graph, we can see that the object c also fulfills the required criteria, with p = ce and q = cg.

All that is left to do is rank these two objects according to the above definition. We see that there exists a morphism from a to c. This means c has to be the best candidate since we now can define the morphism m = ac such that pₐ = p . m = ce . ac and qₐ = q . m = cg . ac.

So, a product of two objects in a poset is actually the greatest object which is both smaller than both (also called the greatest lower bound). It is worth noting that in a total ordering, this corresponds to the function min(a, b), since every object must be relateable to any other object (Wolfram call this trichotomy law).

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Analog to the product definition, the coproduct corresponds to the smallest object greater than or equal to both a and b. In a total ordering, this corresponds to the maximum of both objects. You can work this one out for yourself.