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 ...
and is an element of called the base point, also spelled basepoint.Maps between pointed sets and – called based maps, pointed maps, or point-preserving maps – are functions from to that map one basepoint to another, i.e. a map such that . This is usually denoted
: .
Pointed sets are very simple
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s. In the sense of universal algebra, a pointed set is a set together with a single nullary operation which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures.
The
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
of all pointed sets together with the class of all based maps form a
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)
*C ...
zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
s. There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is not a pointed set because it has no element that can be chosen as the basepoint.
The category of pointed sets and based maps is equivalent to the category of sets and
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
s. The base point serves as a "default value" for those arguments for which the partial function is not defined. One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in topology (
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Ale ...
) and in
theoretical computer science
computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumscribe the ...
."
The category of pointed sets and pointed maps is isomorphic to the
coslice category
In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...
(), where is (a functor selecting) a singleton set, and (the identity functor of) the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of ...
. This coincides with the algebraic characterization, since the unique map extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.
The category of pointed sets and pointed maps has both products and
coproduct
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 coprodu ...
s, but it is not a
distributive category
In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects A,B,C, the canonical map
: mathit_A \times\iota_1, \mathit_A \times\iota_2: A\!\times\!B \,+ A\!\times\!C \to A ...
. It is also an example of a category where is not isomorphic to .
Many algebraic structures are pointed sets in a rather trivial way. For example,
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
are pointed sets by choosing the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
as the basepoint, so that group homomorphisms are point-preserving maps. This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.
A pointed set may be seen as a pointed space under the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroid ...