{{Short description|Category whose objects are finite sets and whose morphisms are functions}} 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 {{nowrap|''A'' × ''B''}}, the categorical sum is given by the disjoint union {{nowrap|''A'' + ''B''}}, and the exponential object ''B''<sup>''A''</sup> 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 {{nowrap|''n'' · ''m''}}, the categorical sum is given by the ordinal sum {{nowrap|''n'' + ''m''}}, and the exponential object is given by the ordinal exponentiation ''n''<sup>''m''</sup>. The subobject classifier in '''FinSet''' and '''FinOrd''' is the same as in '''Set'''. '''FinOrd''' is an example of a PRO.
== See also == * General set theory * Lawvere theory * Natural number object * Simplex category * FinVect
== 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 [https://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3 online] at [http://www.mcs.vuw.ac.nz/~rob/ Robert Goldblatt's homepage].
Category:Categories in category theory