# FinSet

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{{Short description|Category whose objects are finite sets and whose morphisms are functions}}
In the mathematical field of [category theory](/source/category_theory), '''FinSet''' is the [category](/source/category_(mathematics)) whose [object](/source/object_(category_theory))s are all [finite set](/source/finite_set)s and whose [morphism](/source/morphism)s are all [function](/source/function_(mathematics))s between them. '''FinOrd''' is the category whose objects are all [finite ordinal number](/source/finite_ordinal_number)s and whose morphisms are all functions between 
them.

== Properties ==
'''FinSet''' is a [full subcategory](/source/full_subcategory) of '''[Set](/source/Set_(category))''', the category whose objects are all sets and whose morphisms are all functions. Like '''Set''', '''FinSet''' is a [large category](/source/large_category).

'''FinOrd''' is a full subcategory of '''FinSet''' as by the standard definition, suggested by [John von Neumann](/source/John_von_Neumann), each ordinal is the [well-ordered set](/source/well-ordered_set) of all smaller ordinals. Unlike '''Set''' and '''FinSet''', '''FinOrd''' is a [small category](/source/small_category).

'''FinOrd''' is a [skeleton](/source/skeleton_(category_theory)) of '''FinSet'''. Therefore, '''FinSet''' and '''FinOrd''' are [equivalent categories](/source/equivalent_categories).

== Topoi ==
Like '''Set''', '''FinSet''' and '''FinOrd''' are [topoi](/source/topos). As in '''Set''', in '''FinSet''' the [categorical product](/source/categorical_product) of two objects ''A'' and ''B'' is given by the [cartesian product](/source/cartesian_product) {{nowrap|''A'' × ''B''}}, the [categorical sum](/source/categorical_sum) is given by the [disjoint union](/source/disjoint_union) {{nowrap|''A'' + ''B''}}, and the [exponential object](/source/exponential_object) ''B''<sup>''A''</sup> is given by the set of all functions with [domain](/source/domain_of_a_function) ''A'' and [codomain](/source/codomain) ''B''. In '''FinOrd''', the categorical product of two objects ''n'' and ''m'' is given by the [ordinal product](/source/product_of_ordinals) {{nowrap|''n'' · ''m''}}, the categorical sum is given by the [ordinal sum](/source/ordinal_addition) {{nowrap|''n'' + ''m''}}, and the [exponential object](/source/exponential_object) is given by the [ordinal exponentiation](/source/ordinal_exponentiation) ''n''<sup>''m''</sup>. The [subobject classifier](/source/subobject_classifier) in '''FinSet''' and '''FinOrd''' is the same as in '''Set'''. '''FinOrd''' is an example of a [PRO](/source/PRO_(category_theory)).

== See also ==
* [General set theory](/source/General_set_theory)
* [Lawvere theory](/source/Lawvere_theory)
* [Natural number object](/source/Natural_number_object)
* [Simplex category](/source/Simplex_category)
* [FinVect](/source/FinVect)

== References ==
* [Robert Goldblatt](/source/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

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Adapted from the Wikipedia article [FinSet](https://en.wikipedia.org/wiki/FinSet) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/FinSet?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.