nLab
FinSet

Definition

$FinSet$ is the category of finite sets and all functions between them: the full subcategory of Set on finite sets.

(For constructive purposes, take the strictest sense of ‘finite’.)

It is easy (and thus common) to make $FinSet$ skeletal; there is one object for each natural number $n$ (including $n=0$), and a morphism from $m$ to $n$ is an $m$-tuple $(f_0,\dots ,f_{m-1})$ of numbers satisfying $0\le f_i<n$. This amounts to identifying $n$ with the set $\{0,\dots ,n-1\}$. (Sometimes $\{1,\dots ,n\}$ is used instead.)

Subcategories of $\mathrm{FinSet}$

The simplex category $\Delta$ embeds into $FinSet$ as a category with the same objects but fewer morphisms. The category of cyclic sets introduced by Connes lies in between. All the three are special cases of extensions of $\Delta$ by a group in a particularly nice way. Full classification of allowed “skewsimplical groups” has been given by Krasauskas and independently by Loday and Fiedorowicz.

In topos theory

The category $\mathrm{FinSet}$ is a topos and the inclusion $\mathrm{FinSet}\hookrightarrow \mathrm{Set}$ is a logical morphism of toposes. (Elephant, example 2.1.2).

Mathematics done within or about $FinSet$ is finite mathematics.

A presheaf of sets on $FinSet$ is a symmetric set; one generally uses the skeletal version of $FinSet$ for this.

The copresheaf category $[\mathrm{FinSet},\mathrm{Set}]$ is the classifying topos for the theory of objects (the empty theory over the signature with one sort and no primitive symbols except equality). (Elephant, D3.2).

category: category