Partitions and Selectors

April 22, 2020

Look at this interesting stuff

Let I be a finite set, X be another set

I -> X is a selector function X -> I is a partition function

for example, Let I = {0, 1}, X = N

f: {0, 1} -> N selects two natural numbers g: N -> {0, 1} partitions N into two subsets

now, what happenes if we categorify this notion

I becomes a discrete category – elements as objects, only identity morphisms X becomes a category of all sets

functions become functors

F : I -> Sets G : Sets -> I

but now, F does what g used to do -> it selects a sets, essentially selecting a partition for a particular set

the set is not guaranteed to be a subset of a given set

• what happens if we turn I into a poset category?
• what happens if we turn X into a poset category? (treating a set A as a poset, ordered by inclusion, with elements as objects and inclusions as morphisms)

let’s go even further – what we really did here is defined a functor H : Sets -> Cat turning each set into a category (I becomes a discrete category, X becomes a Sets category)