In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a surjective function (also known as surjection, or onto function) is a
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 ...
that every element can be mapped from element so that . In other words, every element of the function's
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 ...
is the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of one element of its
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 ...
.
It is not required that be
unique; the function may map one or more elements of to the same element of .
The term ''surjective'' and the related terms ''
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
'' and ''
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
'' were introduced by
Nicolas Bourbaki, a group of mainly
French 20th-century
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
s who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word ''
sur'' means ''over'' or ''above'', and relates to the fact that the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of the domain of a surjective function completely covers the function's codomain.
Any function induces a surjection by
restricting its codomain to the image of its domain. Every surjective function has a
right inverse assuming the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, and every function with a right inverse is necessarily a surjection. The
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
Definition
A surjective function is a
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 ...
whose
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
is equal to its
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 ...
. Equivalently, a function
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 ...
and codomain
is surjective if for every
in
there exists at least one
in
with
.
Surjections are sometimes denoted by a two-headed rightwards arrow (),
as in
.
Symbolically,
:If
, then
is said to be surjective if and only if
:
.
Examples
![Codomain2](https://upload.wikimedia.org/wikipedia/commons/6/64/Codomain2.SVG)
* For any set ''X'', the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
id
''X'' on ''X'' is surjective.
* The function defined by ''f''(''n'') = ''n''
mod
Mod, MOD or mods may refer to:
Places
* Modesto City–County Airport, Stanislaus County, California, US
Arts, entertainment, and media Music
* Mods (band), a Norwegian rock band
* M.O.D. (Method of Destruction), a band from New York City, US ...
2 (that is,
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
** Even language, a language spoken by the Evens
* Odd and Even, a solitaire game w ...
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s are mapped to 0 and
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
integers to 1) is surjective.
* The function defined by ''f''(''x'') = 2''x'' + 1 is surjective (and even
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
), because for every
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''y'', we have an ''x'' such that ''f''(''x'') = ''y'': such an appropriate ''x'' is (''y'' − 1)/2.
* The function defined by ''f''(''x'') = ''x''
3 − 3''x'' is surjective, because the pre-image of any
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''y'' is the solution set of the cubic polynomial equation ''x''
3 − 3''x'' − ''y'' = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
(and hence not
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
), since, for example, the pre-image of ''y'' = 2 is . (In fact, the pre-image of this function for every ''y'', −2 ≤ ''y'' ≤ 2 has more than one element.)
* The function defined by ''g''(''x'') = ''x''
2 is ''not'' surjective, since there is no real number ''x'' such that ''x''
2 = −1. However, the function defined by (with the restricted codomain) ''is'' surjective, since for every ''y'' in the nonnegative real codomain ''Y'', there is at least one ''x'' in the real domain ''X'' such that ''x''
2 = ''y''.
* The
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
function is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, if defined with the set of real numbers as the domain, is not surjective (as its range is the set of positive real numbers).
* The
matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
is not surjective when seen as a map from the space of all ''n''×''n''
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
to itself. It is, however, usually defined as a map from the space of all ''n''×''n'' matrices to the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of degree ''n'' (that is, the
group
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 ...
of all ''n''×''n''
invertible matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
* The
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
from a
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 ...
to one of its factors is surjective, unless the other factor is empty.
* In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function.
Properties
A function is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
if and only if it is both surjective and
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
.
If (as is often done) a function is identified with its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
, then surjectivity is not a property of the function itself, but rather a property of the
mapping. This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.
Surjections as right invertible functions
The function is said to be a
right inverse of the function if ''f''(''g''(''y'')) = ''y'' for every ''y'' in ''Y'' (''g'' can be undone by ''f''). In other words, ''g'' is a right inverse of ''f'' if the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of ''g'' and ''f'' in that order is the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on the domain ''Y'' of ''g''. The function ''g'' need not be a complete
inverse of ''f'' because the composition in the other order, , may not be the identity function on the domain ''X'' of ''f''. In other words, ''f'' can undo or "''reverse''" ''g'', but cannot necessarily be reversed by it.
Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.
If is surjective and ''B'' is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of ''Y'', then ''f''(''f''
−1(''B'')) = ''B''. Thus, ''B'' can be recovered from its
preimage .
For example, in the first illustration above, there is some function ''g'' such that ''g''(''C'') = 4. There is also some function ''f'' such that ''f''(4) = ''C''. It doesn't matter that ''g''(''C'') can also equal 3; it only matters that ''f'' "reverses" ''g''.
Surjections as epimorphisms
A function is surjective if and only if it is
right-cancellative
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
An ...
:
given any functions , whenever ''g''
o ''f'' = ''h''
o ''f'', then ''g'' = ''h''. This property is formulated in terms of functions and their
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
and can be generalized to the more general notion of the
morphisms of 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) ...
and their composition. Right-cancellative morphisms are called
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
s. Specifically, surjective functions are precisely the epimorphisms in 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 o ...
. The prefix ''epi'' is derived from the Greek preposition ''ἐπί'' meaning ''over'', ''above'', ''on''.
Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse ''g'' of a morphism ''f'' is called a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of ''f''. A morphism with a right inverse is called a
split epimorphism
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism.
In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y ...
.
Surjections as binary relations
Any function with domain ''X'' and codomain ''Y'' can be seen as a
left-total
In mathematics, a binary relation ''R'' ⊆ ''X''×''Y'' between two sets ''X'' and ''Y'' is total (or left total) if the source set ''X'' equals the domain . Conversely, ''R'' is called right total if ''Y'' equals the range .
When ''f'': ''X'' ...
and
right-unique binary relation between ''X'' and ''Y'' by identifying it with its
function graph
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset o ...
. A surjective function with domain ''X'' and codomain ''Y'' is then a binary relation between ''X'' and ''Y'' that is right-unique and both left-total and
right-total.
Cardinality of the domain of a surjection
The
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If is a surjective function, then ''X'' has at least as many elements as ''Y'', in the sense of
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s. (The proof appeals to the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
to show that a function
satisfying ''f''(''g''(''y'')) = ''y'' for all ''y'' in ''Y'' exists. ''g'' is easily seen to be injective, thus the
formal definition
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for formal events
* Semi-formal attir ...
of , ''Y'', ≤ , ''X'', is satisfied.)
Specifically, if both ''X'' and ''Y'' are
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
with the same number of elements, then is surjective if and only if ''f'' is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
.
Given two sets ''X'' and ''Y'', the notation is used to say that either ''X'' is empty or that there is a surjection from ''Y'' onto ''X''. Using the axiom of choice one can show that and together imply that , ''Y'', = , ''X'', , a variant of the
Schröder–Bernstein theorem In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function .
In terms of the cardinality of the two sets, this classically implies that if ...
.
Composition and decomposition
The
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of surjective functions is always surjective: If ''f'' and ''g'' are both surjective, and the codomain of ''g'' is equal to the domain of ''f'', then is surjective. Conversely, if is surjective, then ''f'' is surjective (but ''g'', the function applied first, need not be). These properties generalize from surjections in 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 o ...
to any
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
s in any
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) ...
.
Any function can be decomposed into a surjection and an
injection
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...
: For any function there exist a surjection and an injection such that ''h'' = ''g''
o ''f''. To see this, define ''Y'' to be the set of
preimages where ''z'' is in . These preimages are
disjoint and
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
''X''. Then ''f'' carries each ''x'' to the element of ''Y'' which contains it, and ''g'' carries each element of ''Y'' to the point in ''Z'' to which ''h'' sends its points. Then ''f'' is surjective since it is a projection map, and ''g'' is injective by definition.
Induced surjection and induced bijection
Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection can be factored as a projection followed by a bijection as follows. Let ''A''/~ be the
equivalence classes of ''A'' under the following
equivalence relation: ''x'' ~ ''y'' if and only if ''f''(''x'') = ''f''(''y''). Equivalently, ''A''/~ is the set of all preimages under ''f''. Let ''P''(~) : ''A'' → ''A''/~ be the
projection map
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projectio ...
which sends each ''x'' in ''A'' to its equivalence class
'x''sub>~, and let ''f''
''P'' : ''A''/~ → ''B'' be the well-defined function given by ''f''
''P''(
'x''sub>~) = ''f''(''x''). Then ''f'' = ''f''
''P'' o ''P''(~).
Space of surjections
Given fixed and , one can form the set of surjections . The
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of this set is one of the twelve aspects of Rota's
Twelvefold way
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of ...
, and is given by
, where
denotes a
Stirling number of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to Partition of a set, partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) ...
.
Gallery
File:Surjective composition.svg, Surjective composition: the first function need not be surjective.
File:Non-surjective function2.svg, alt=, Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements ''y'' in ''Y'' do have a value ''x'' in ''X'' such that ''y'' = ''f''(''x''), some parts are not. Left: There is ''y''0 in ''Y'', but there is no ''x''0 in ''X'' such that ''y''0 = ''f''(''x''0). Right: There are ''y''1, ''y''2 and ''y''3 in ''Y'', but there are no ''x''1, ''x''2, and ''x''3 in ''X'' such that ''y''1 = ''f''(''x''1), ''y''2 = ''f''(''x''2), and ''y''3 = ''f''(''x''3).
File:Surjective function.svg, alt=, Interpretation for surjective functions in the Cartesian plane, defined by the mapping ''f'' : ''X'' → ''Y'', where ''y'' = ''f''(''x''), ''X'' = domain of function, ''Y'' = range of function. Every element in the range is mapped onto from an element in the domain, by the rule ''f''. There may be a number of domain elements which map to the same range element. That is, every ''y'' in ''Y'' is mapped from an element ''x'' in ''X'', more than one ''x'' can map to the same ''y''. Left: Only one domain is shown which makes ''f'' surjective. Right: two possible domains ''X''1 and ''X''2 are shown.
See also
*
Bijection, injection and surjection
In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which '' arguments'' (input expressions from the domain) and '' images'' (output expressions from the codomain) are related or ''m ...
*
Cover (algebra)
*
Covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
*
Enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (fo ...
*
Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
*
Index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
*
Section (category theory)
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism.
In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y is ...
References
Further reading
*
{{Mathematical logic
Functions and mappings
Basic concepts in set theory
Mathematical relations
Types of functions