# FinSet

In the mathematical field of category theory, **FinSet** is the category whose objects are all finite sets and whose morphisms are all functions between them. **FinOrd** is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between
them.

## Properties

**FinSet** is a full subcategory of **Set**, the category whose objects are all sets and whose morphisms are all functions. Like **Set**, **FinSet** is a large category.

**FinOrd** is a full subcategory of **FinSet** as by the standard definition, suggested by John von Neumann, each ordinal is the well-ordered set of all smaller ordinals. Unlike **Set** and **FinSet**, **FinOrd** is a small category.

**FinOrd** is a skeleton of **FinSet**. Therefore, **FinSet** and **FinOrd** are equivalent categories.

## Topoi

Like **Set**, **FinSet** and **FinOrd** are topoi. As in **Set**, in **FinSet** the categorical product of two objects *A* and *B* is given by the cartesian product *A* × *B*, the categorical sum is given by the disjoint union *A* + *B*, and the exponential object *B*^{A} is given by the set of all functions with domain *A* and codomain *B*. In **FinOrd**, the categorical product of two objects *n* and *m* is given by the ordinal product *n* · *m*, the categorical sum is given by the ordinal sum *n* + *m*, and the exponential object is given by the ordinal exponentiation *n*^{m}. The subobject classifier in **FinSet** and **FinOrd** is the same as in **Set**. **FinOrd** is an example of a PRO.

## References

- Robert Goldblatt (1984).
*Topoi, the Categorial Analysis of Logic*(Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications, and available online at Robert Goldblatt's homepage.