Category of Standard Finite Sets

The category of standard finite sets $\mathbb{F}$ is a category whose objects are Finite Ordinals $[n]$, and whose morphisms are functions. This is in constrast with the Augmented Simplex Category, which has the same objects, but requires maps $[m]\to[n]$ to be Monotonic.

Properties

• The category of standard finite sets has Finite Products and Finite Coproducts.

• The category of standard finite sets is the walking Commutative Monoid Object in a Symmetric Monoidal Category: For any other Symmetric Monoidal Category $(\mathcal{C},\otimes,1)$ equipped with a Commutative Monoid Object $(X,\eta,\mu)$, there exists a unique Symmetric Monoidal Functor $F:\mathbb{F}\to\mathcal{C}$ with $F([1])=X$.