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In mathematics, particularly in
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of stu ...
and category theory, transport of structure refers to the process whereby a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
acquires a new structure and its canonical definitions, as a result of being isomorphic to (or otherwise identified with) another object with a pre-existing structure. Definitions by transport of structure are regarded as canonical. Since mathematical structures are often defined in reference to an underlying
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
, many examples of transport of structure involve spaces and mappings between them. For example, if ''V'' and ''W'' are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s with (\cdot,\cdot) being an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on W, such that there is 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 ...
\phi from ''V'' to ''W'', then one can define an inner product cdot, \cdot/math> on ''V'' by the following rule: : _1, v_2= (\phi(v_1), \phi(v_2)) Although the equation makes sense even when \phi is not an isomorphism, it only defines an inner product on ''V'' when \phi is, since otherwise it will cause cdot,\cdot/math> to be
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
. The idea is that \phi allows one to consider ''V'' and ''W'' as "the same" vector space, and by following this analogy, then one can transport an inner product from one space to the other. A more elaborated example comes from differential topology, in which the notion of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is involved: if M is such a manifold, and if ''X'' is any
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
which is homeomorphic to ''M'', then one can consider ''X'' as a smooth manifold as well. That is, given a homeomorphism \phi \colon X \to M, one can define coordinate charts on ''X'' by "pulling back" coordinate charts on ''M'' through \phi. Recall that a coordinate chart on M is an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
''U'' together with an injective map :c \colon U \to \mathbb^n for some
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
''n''; to get such a chart on ''X'', one uses the following rules: :U' = \phi^(U) and c' = c \circ \phi. Furthermore, it is required that the charts
cover Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of co ...
''M'' (the fact that the transported charts cover ''X'' follows immediately from the fact that \phi is a bijection). Since ''M'' is a ''smooth'' manifold, if ''U'' and ''V'', with their maps c \colon U \to \mathbb^n and d \colon V \to \mathbb^n, are two charts on ''M'', then the composition, the "transition map" :d \circ c^ \colon c(U \cap V) \to \mathbb^n (a self-map of \mathbb^n) is smooth. To verify this for the transported charts on ''X'', notice that :\phi^(U) \cap \phi^(V) = \phi^(U \cap V), and therefore :c'(U' \cap V') = (c \circ \phi)(\phi^(U \cap V)) = c(U \cap V), and :d' \circ (c')^ = (d \circ \phi) \circ (c \circ \phi)^ = d \circ (\phi \circ \phi^) \circ c^ = d \circ c^. Thus the transition map for U' and V' is the same as that for ''U'' and ''V'', hence smooth. That is, ''X'' is a smooth manifold via transport of structure. This is a special case of transport of structures in general., Chapter IV, Section 5 "Isomorphism and transport of structures". The second example also illustrates why "transport of structure" is not always desirable. Namely, one can take ''M'' to be the plane, and ''X'' to be an infinite one-sided cone. By "flattening" the cone, a homeomorphism of ''X'' and ''M'' can be obtained, and therefore the structure of a smooth manifold on ''X'', but the cone is not "naturally" a smooth manifold. That is, one can consider ''X'' as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of exotic spheres, discovered by
Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition ''not''
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...
) to S^7, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
isomorphism between the two objects.


See also

* List of mathematical jargon * Equivalent definitions of mathematical structures#Transport of structures; isomorphism


References

{{DEFAULTSORT:Transport Of Structure Mathematical terminology Mathematical structures