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, cate ...
, 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. Th ...
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 a ...
s are all
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
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. Intuitive ...
of
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
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 ass ...
.
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-ord ...
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 asso ...
.
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
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
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 ring ...
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\ti ...
, the
categorical sum
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
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 (th ...
, 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 c ...
''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 the ...
''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 r ...
{{nowrap, ''n'' + ''m'', 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 c ...
is given by the
ordinal exponentiation
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an ex ...
''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 retired f ...
.
See also
*
General set theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. ...
*
Lawvere theory In category theory, a Lawvere theory (named after United States, American mathematician William Lawvere) is a category (mathematics), category that can be considered a categorical counterpart of the notion of an equational theory.
Definition
Let \ ...
*
Natural number object In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO ''N'' is given by:
# a global element ''z'' : 1 → '' ...
*
Simplex category
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects.
Formal definition ...
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
onlinea
Robert Goldblatt's homepage
Categories in category theory