In mathematics, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists. 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. For example, if V and W are vector spaces, and if
Although the equation makes sense even when
A more involved example comes from differential topology, in which we have the notion of a smooth manifold. If M is such a manifold, and if X is any topological space which is homeomorphic to M, we can consider X as a smooth manifold as well. That is, let
for some n; to get such a chart on X, we let
Furthermore, it is required that the charts cover M, we must check that the transported charts cover X, which follows immediately from the fact that
are two charts on M, then the composition, the "transition map"
is smooth. We must check this for our transported charts on X. We have
and therefore
Therefore the transition map for
Although the second example involved considerably more checking, the principle was the same, and any experienced mathematician would have no difficulty performing the necessary verifications. Therefore when such an operation is indicated, it is invoked merely as "transport of structure" and the details left to the reader, if desired.
The second example also illustrates why "transport of structure" is not always desirable. Namely, we can take M to be the plane, and we can take X to be an infinite one-sided cone. By "flattening" the cone we achieve a homeomorphism of X and M, and therefore the structure of a smooth manifold on X, but the cone is not "naturally" a smooth manifold. That is, we 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, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition not diffeomorphic) to