Tree of Life

example

The set of animal classifications (‘tiger’, ‘mammal’, ‘sapiens’, ‘carnivore’, …) is ordered by specificity: $\mathsf{tiger}\le \mathsf{mammal}$ since a tiger is a type of mammal. The result is a Preorder which is a tree, the tree of life. The taxonomic ranks $\mathsf{species}\le \mathsf{genus}\le \mathsf{family}\le \mathsf{order}\le \mathsf{class}\le \mathsf{phylum}\le \mathsf{kingdom}$ form a Total Order, and the map $F$ sending each classification to its rank is a Monotone Map: whenever there is a path $x\to y$ upstairs there is a path $F(x)\to F(y)$ downstairs.

Sources: 7 Sketches Example 1.61; CTfS Exercises 3.4.4.2–3.4.4.4

The map $F$ (tiger, lion, habilis, sapiens $\mapsto$ species; panthera, homo $\mapsto$ genus; …) is a monotone map from the upper preorder to the lower one. It is far from an isomorphism: it collapses many classifications to one rank. It is a typical observation in the sense of generative effects: it forgets information but respects the order.

Taxonomies in Category Theory for Scientists (Exercises 3.4.4.2–3.4.4.3): should a taxonomy be a preorder, a partial order or a linear order? A partial order — and, if every taxon has a unique parent, a tree. In the tree of life most pairs of species have a Join (their most specific common taxon) but no Meet (nothing is both a tiger and a lion); a taxonomy is also a natural Olog whose arrows all read “is”.