In the mathematical field of

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

, 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

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

s 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. T ...

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, morphisms ...

s 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 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. Intuitively, ...

Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

, 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 c ...

, each ordinal is the well-ordered set of all smaller ordinals. Unlike Set and FinSet, FinOrd is a small category. 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 In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...

Topoi

Like Set, FinSet and FinOrd are

topoi In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a noti ...

. As in Set, in FinSet the categorical product 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\t ...

, 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 ''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 * ...

''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 th ...

''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 {{nowrap, ''n'' + ''m'', and the exponential object is given by the ordinal exponentiation ''n''^''m''. The

subobject classifier In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "true ...

in FinSet and FinOrd is the same as in Set. FinOrd is an example of a

PRO Pro is an abbreviation meaning " professional". Pro, PRO or variants thereof may also refer to: People * Miguel Pro (1891–1927), Mexican priest * Pro Hart (1928–2006), Australian painter * Mlungisi Mdluli (born 1980), South African retire ...

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 availabl
online
a
Robert Goldblatt's homepage
Categories in category theory

Properties

Topoi

See also

References