Finite sets

Finite sets provide the smallest concrete model in the catalogue and are useful for experimenting with universal constructions before linear algebra enters. The constructor is intended to model the category of finite sets and all maps described in Etingof et al. (2015), Example 2.3.1, p. 26, subject to the current limitations below. The implementation represents both the sets and their maps by ordinary Julia data.

Sets() models finite sets and functions. An object SetObject([1,2,3]) stores a Julia set. For objects $X$ and $Y$, SetMorphism(X,Y,f) constructs a map from a Julia function f, storing its values in a dictionary.

product(X,Y) forms the Cartesian product, and coproduct(X,Y) forms a disjoint union. Passing true as the third argument returns the object together with its projections or inclusions, respectively. These are categorical products and coproducts; this model has no linear structure over a coefficient field.

The cited example also carries its Cartesian symmetric monoidal structure. The present model implements product and coproduct, but it does not supply the package's monoidal interface: tensor product, tensor unit, associator, and braiding are unavailable.