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.

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.

• General set theory
• Lawvere theory
• Natural number object
• Simplex category
• FinVect

• 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.