Suppose I wanted to compare Linnaean classification, which arranges species by similarity in ranked taxa, to modern phylogenetic systematics, which appeals to descent with modification and branching trees. I want to represent the structure of each with category theory and then compare them so as to say how similar/dissimilar they are (fortunately for me, Baez has already formalized one phylogenetic model with operads: https://arxiv.org/abs/1512.03337).
Roughly, the Linnaean system looks like this:
There is class membership:
- individual organism (me) $\in$ lowest taxon (homosapien) $\in$ higher taxon (species)
and class inclusion where lower taxa are subsets of higher taxa:
- mammalia $\subset$ vertebrata
The classification is achieved through some similarity metric, where a certain amount of phenetic properties must be shared for two objects to belong to the same class.
How can this be represented categorically? Can we make the hierarchical relationships arrows, with the objects being the elements? Or would a higher structure be needed? I understand that the Baez operad paper uses topological operads, so would a metric space be more useful here if my aim is to compare the two? Or do I need higher structures? I'm a bit lost!
