In mathematics, particularly in universal algebra and category theory, transport of structure refers to the process whereby a mathematical object 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.
[1] Definitions by transport of structure are regarded as canonical.
Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them.
by the following rule: Although the equation makes sense even when
is not an isomorphism, it only defines an inner product on
A more elaborated example comes from differential topology, in which the notion of smooth manifold is involved: if
is any topological space which is homeomorphic to
, one can define coordinate charts on
by "pulling back" coordinate charts on
Recall that a coordinate chart on
together with an injective map for some natural number
, one uses the following rules: Furthermore, it is required that the charts cover
(the fact that the transported charts cover
is a smooth manifold, if U and V, with their maps
, then the composition, the "transition map" is smooth.
To verify this for the transported charts on
, notice that and therefore Thus the transition map for
is a smooth manifold via transport of structure.
This is a special case of transport of structures in general.
[2] The second example also illustrates why "transport of structure" is not always desirable.
to be an infinite one-sided cone.
By "flattening" the cone, a homeomorphism of
can be obtained, and therefore the structure of a smooth manifold on
, but the cone is not "naturally" a smooth manifold.
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, which states that there are exactly 28 smooth manifolds which are homeomorphic but not diffeomorphic to
Thus, transport of structure is most productive when there exists a canonical isomorphism between the two objects.