FinSet
Encyclopedia
In the mathematical field of category theory
, FinSet is the category
whose objects are all finite sets and whose morphism
s are all function
s between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them.
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. As such, FinSet and FinOrd are equivalent categories.
. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product
, the categorical sum is given by the disjoint union
, and the exponential object
BA 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 , the categorical sum is given by the ordinal sum , and the exponential object
is given by the ordinal exponentiation nm. The subobject classifier
in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, FinSet is the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
whose objects are all finite sets and whose morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s are all function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s 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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, each ordinal is the well-ordered set of all smaller ordinals. Unlike Set and FinSet, FinOrd is a small category.
FinOrd is a skeleton
Skeleton (category theory)
In mathematics, a skeleton of a category is a subcategory which, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalent category which captures all "categorical properties". In fact, two categories are equivalent...
of FinSet. As such, FinSet and FinOrd are equivalent categories.
Topoi
Like Set, FinSet and FinOrd are topoiTopos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
, the categorical sum is given by the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
, and the exponential object
Exponential object
In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...
BA is given by the set of all functions with domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
A and codomain
Codomain
In mathematics, the codomain or target set 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...
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 , and the exponential object
Exponential object
In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...
is given by the ordinal exponentiation nm. The subobject classifier
Subobject classifier
In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements...
in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.