September 13, 2021

## The Category of Natural Kinds

This is part 3 of the series on natural kinds. The previous post is here.

Having established what natural kinds are, we’re ready to examine maps between them and lift our discussion to the categorical setting. This short post establishes the categorical nature of natural kinds and maps between them.

# Onward

In the context of natural kinds, implication (remember, there’s a logic here!) is taken to be the species-genus relation: for natural kinds K1, K2, to say K1 ⇒ K2 is to say K1 is a species of K2.

This is formalized as follows:

For kinds K1 = (A1,X1) and K2 = (A2,X2), saying that K1 is a species of K2 amounts to:

• (A1A2)
• (X2X1)

That is,

• every K1 individual is also a K2 individual
• every K2 trait is also a K1 trait.

We have the following theorem (T1), for natural kinds K1 = (A1,X1) and K2 = (A2,X2):

K1 ⇒ K2 iff A1 ⊆ A2 and X2 ⊆ X1.

# Upward

Natural kinds, together with the species-genus relation, form a category:

Let K = (A,X) be any natural kind.

### Identity Arrow

By reflexivity of the subset relation, A ⊆ A and X ⊆ X, so X ⇒ X.

### Arrow Composition

Now let K1 = (A1,X1), K2 = (A2,X2), and K3 = (A3,X3) be natural kinds and suppose K1 ⇒ K2 and K2 ⇒ K3.

By the theorem (T1) in the previous section, K1 ⇒ K2 iff A1 ⊆ A2 and X2 ⊆ X1, and K2 ⇒ K3 iff A2 ⊆ A3 and X3 ⊆ X2.

If A1 ⊆ A2 and A2 ⊆ A3, then A1 ⊆ A3, and if X3 ⊆ X2 and X2 ⊆ X1, then X3 ⊆ X1, so K1 ⇒ K3 on the assumption that K1 ⇒ K2 and K2 ⇒ K3.

The subset relation is associative, so evidently this composition is too.

These facts tell us that natural kinds with the species-genus implication form a category.

In the next post, we’ll examine some of the categorical features of natural kinds, with an eye to building bridges to other parts of logic and mathematics.