Endomorphisms of the symmetric 2-rig of finite sets

TL;DR

This paper characterizes the endomorphisms of a symmetric rig category of finite sets, showing they are trivial, which supports the idea that this category is a categorical analog of the natural numbers.

Contribution

It proves that the endomorphisms of the symmetric rig category of finite sets form a terminal groupoid, establishing a key categorical property.

Findings

Endomorphisms of the category are equivalent to the terminal category.

Provides an explicit description of endomorphisms using a semistrict skeletal version.

Supports the analogy between the category and the natural numbers.

Abstract

Let $\widehat{\mathbb{F}\mathbb{S}et}$ be the groupoid of finite sets and bijections between them equipped with the canonical symmetric rig category structure given by the disjoint union and the cartesian product of finite sets. We prove that the category (in fact, groupoid) of endomorphisms of $\widehat{\mathbb{F}\mathbb{S}et}$ is equivalent to the terminal category, thus providing some evidence that $\widehat{\mathbb{F}\mathbb{S}et}$ is the right categorical analog of the commutative rig $\mathbb{N}$ of nonnegative integers. This is shown using a particular semistrict skeletal version of $\widehat{\mathbb{F}\mathbb{S}et}$ for which the endomorphisms can be described very explicitly.