In mathematics, if is a subset of then the inclusion map is the function that sends each element of to treated as an element of
An inclusion map may also referred to as an inclusion function, an insertion,[1] or a canonical injection.
A "hooked arrow" (U+21AA↪RIGHTWARDS ARROW WITH HOOK)[2] 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.)
This and other analogous injective functions[3] from substructures are sometimes called natural injections.
Given any morphism between objects and , if there is an inclusion map into the domain , then one can form the restriction of In many instances, one can also construct a canonical inclusion into the codomain known as the range of
^MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2. Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.
A {\displaystyle A} is a subset of B , {\displaystyle B,} then the inclusionmap is the function ι {\displaystyle \iota } that sends each element x {\displaystyle...
Boolean analogue to the subset relation Inclusionmap, or inclusion function, or canonical injection Inclusion (logic), the concept that all the contents...
S} which itself has the structure of a manifold, and for which the inclusionmap S → M {\displaystyle S\rightarrow M} satisfies certain properties. There...
always continuous. The identity function is idempotent. Identity matrix Inclusionmap Indicator function Knapp, Anthony W. (2006), Basic algebra, Springer...
exotic inclusionmap S5 → S6 as a transitive subgroup (the obvious inclusionmap Sn → Sn+1 fixes a point and thus is not transitive) and, while this map does...
model of dimension n. In the definition above ι: H1(n) R → Mn+1 is the inclusionmap and the superscript star denotes the pullback. The present purpose is...
is also Hamiltonian, with momentum map the composition of the inclusionmap with M {\displaystyle M} 's momentum map. Suppose that the action of a Lie...
interpreted by means of the fiber product of schemes, applied to f and the inclusionmap of Y ′ {\displaystyle Y'} into Y. For the second, the idea is that morphisms...
{\displaystyle X} and any subset S ⊆ X , {\displaystyle S\subseteq X,} the inclusionmap S → X {\displaystyle S\to X} (which sends any element s ∈ S {\displaystyle...
Y → X {\displaystyle i:Y\to X} be the inclusionmap. Then for any topological space Z {\displaystyle Z} a map f : Z → Y {\displaystyle f:Z\to Y} is continuous...
of monoids, Mon, the inclusionmap N → Z is a non-surjective epimorphism. To see this, suppose that g1 and g2 are two distinct maps from Z to some monoid...
normal subgroup of G. In particular, if H is a subgroup of G, then the inclusionmap i from H to G is a monomorphism, and will be normal if and only if H...
function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusionmaps. The final topology is also the...
x} is said to be well-pointed if the inclusionmap x→X{\displaystyle {x}\to X} is a cofibration. The inclusionmap Sn−1→Dn{\displaystyle S^{n-1}\to D^{n}}...
f:X\hookrightarrow Y.} (On the other hand, this notation is sometimes reserved for inclusionmaps.) Given X {\displaystyle X} and Y {\displaystyle Y} , several different...
Lie group that is a subset of G {\displaystyle G} and such that the inclusionmap from H {\displaystyle H} to G {\displaystyle G} is an injective immersion...
three-dimensional submanifold, called the compact core or Scott core, such that its inclusionmap induces an isomorphism on fundamental groups. In particular, this means...
linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g : S(V) → A such that f = g ∘ i, where i is the inclusionmap of V...
following commutative diagram: Here i is the inclusionmap and ΦX, ΦY are the maps obtained by composing the quotient map with the canonical injections into the...
general convention, and the latter notation is more commonly used for inclusionmaps or embeddings.[citation needed] Specifically, for a partial function...
if caught, would be unable to explain the inclusion of the "trap street" on their map as innocent. On maps that are not of streets, other "trap" features...
are as above, we say that U{\displaystyle U} can be excised if the inclusionmap of the pair (X∖U,A∖U){\displaystyle (X\setminus U,A\setminus U)} into...
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