![Venn A subset B](https://upload.wikimedia.org/wikipedia/commons/b/b0/Venn_A_subset_B.svg)
In
mathematics, if
is a
subset of
then the inclusion map (also inclusion function, insertion, or canonical injection) is the
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 sends each element
of
to
treated as an element of
A "hooked arrow" ()
is sometimes used in place of the function arrow above to denote an inclusion map; thus:
(However, some authors use this hooked arrow for any
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
.)
This and other analogous
injective functions
from
substructures are sometimes called natural injections.
Given any
morphism between
objects
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 ...
and
, if there is an inclusion map into the
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 ...
then one can form the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
of
In many instances, one can also construct a canonical inclusion into the
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 ...
known as the
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
of
Applications of inclusion maps
Inclusion maps tend to be
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s of
algebraic structures; thus, such inclusion maps are
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
s. More precisely, given a substructure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for some binary operation
to require that
is simply to say that
is consistently computed in the sub-structure and the large structure. The case of a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
is similar; but one should also look at
nullary
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. ...
operations, which pick out a ''constant'' element. Here the point is that
closure means such constants must already be given in the substructure.
Inclusion maps are seen in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
where if
is a
strong deformation retract of
the inclusion map yields an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between all
homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
(that is, it is a
homotopy equivalence
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
).
Inclusion maps in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
come in different kinds: for example
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
s of
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s.
Contravariant objects (which is to say, objects that have
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
s; these are called
covariant in an older and unrelated terminology) such as
differential forms ''restrict'' to submanifolds, giving a mapping in the ''other direction''. Another example, more sophisticated, is that of
affine schemes, for which the inclusions
and
may be different
morphisms, where
is a
commutative ring and
is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of
See also
*
*
References
{{DEFAULTSORT:Inclusion Map
Basic concepts in set theory
Functions and mappings