In the mathematical field of category theory, FinSet is the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

whose objects are all

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...

s and whose

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

s are all functions between them. FinOrd is the category whose objects are all

finite ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the lea ...

s and whose morphisms are all functions between them.

Properties

FinSet is a

full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

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 John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

, each ordinal is the

well-ordered set In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...

of all smaller ordinals. Unlike Set and FinSet, FinOrd is a

small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows as ...

. FinOrd is a

skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...

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 In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rin ...

of two objects ''A'' and ''B'' is given by the

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...

, the categorical sum is given by the

disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...

, and the

exponential object In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed ca ...

''B''^''A'' is given by the set of all functions with

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...

''A'' and

codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...

''B''. In FinOrd, the categorical product of two objects ''n'' and ''m'' is given by the ordinal product , the categorical sum is given by the

ordinal sum In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

{{nowrap, ''n'' + ''m'', and the

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 __notoc__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His most popular books are ''Logics of Time and Computatio ...

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

Properties

Topoi

See also

References