## 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 *K*_{1}, *K*_{2}, to say *K*_{1} ⇒ *K*_{2} is to say *K*_{1} is *a species of* *K*_{2}.

This is formalized as follows:

For kinds *K*_{1} = (*A*_{1},*X*_{1}) and *K*_{2} = (*A*_{2},*X*_{2}), saying that *K*_{1} is *a species of* *K*_{2} amounts to:

- (
*A*_{1}⊆*A*_{2}) - (
*X*_{2}⊆*X*_{1})

That is,

- every
*K*_{1}individual is also a*K*_{2}individual - every
*K*_{2}trait is also a*K*_{1}trait.

We have the following theorem (T1), for natural kinds *K*_{1} = (*A*_{1},*X*_{1}) and *K*_{2} = (*A*_{2},*X*_{2}):

*K*_{1} ⇒ *K*_{2} iff *A*_{1} ⊆ *A*_{2} and *X*_{2} ⊆ *X*_{1}.

# 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 *K*_{1} = (*A*_{1},*X*_{1}), *K*_{2} = (*A*_{2},*X*_{2}), and *K*_{3} = (*A*_{3},*X*_{3}) be natural kinds and suppose *K*_{1} ⇒ *K*_{2} and *K*_{2} ⇒ *K*_{3}.

By the theorem (T1) in the previous section, *K*_{1} ⇒ *K*_{2} iff *A*_{1} ⊆ *A*_{2} and *X*_{2} ⊆ *X*_{1}, and *K*_{2} ⇒ *K*_{3} iff *A*_{2} ⊆ *A*_{3} and *X*_{3} ⊆ *X*_{2}.

If *A*_{1} ⊆ *A*_{2} and *A*_{2} ⊆ *A*_{3}, then *A*_{1} ⊆ *A*_{3}, and if *X*_{3} ⊆ *X*_{2} and *X*_{2} ⊆ *X*_{1}, then *X*_{3} ⊆ *X*_{1}, so *K*_{1} ⇒ *K*_{3} on the assumption that *K*_{1} ⇒ *K*_{2} and *K*_{2} ⇒ *K*_{3}.

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.