Transport of structure

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. 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 "formula_1" and "formula_2" are vector spaces with formula_3 being an inner product on formula_2, such that there is an isomorphism formula_5 from "formula_1" to "formula_2", then one can define an inner product formula_8 on "formula_1" by the following rule:
Although the equation makes sense even when formula_5 is not an isomorphism, it only defines an inner product on "formula_1" when formula_5 is, since otherwise it will cause formula_14 to be degenerate. The idea is that formula_5 allows one to consider "formula_1" and "formula_2" 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 is involved: if formula_18 is such a manifold, and if "formula_19" is any topological space which is homeomorphic to "formula_18", then one can consider "formula_19" as a smooth manifold as well. That is, given a homeomorphism formula_22, one can define coordinate charts on "formula_19" by "pulling back" coordinate charts on "formula_18" through formula_5. Recall that a coordinate chart on formula_18 is an open set "formula_27" together with an injective map
for some natural number "formula_29"; to get such a chart on "formula_19", one uses the following rules:
Furthermore, it is required that the charts cover "formula_18" (the fact that the transported charts cover "formula_19" follows immediately from the fact that formula_5 is a bijection). Since "formula_18" is a "smooth" manifold, if "U" and "V", with their maps formula_28 and formula_38, are two charts on "formula_18", then the composition, the "transition map"
is smooth. To verify this for the transported charts on "formula_19", notice that
and therefore
Thus the transition map for formula_46 and formula_47 is the same as that for "formula_27" and "formula_1", hence smooth. That is, "formula_19" is a smooth manifold via transport of structure. This is a special case of transport of structures in general.
The second example also illustrates why "transport of structure" is not always desirable. Namely, one can take "formula_18" to be the plane, and "formula_19" to be an infinite one-sided cone. By "flattening" the cone, a homeomorphism of "formula_19" and "formula_18" can be obtained, and therefore the structure of a smooth manifold on "formula_19", but the cone is not "naturally" a smooth manifold. That is, one can consider "formula_19" 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 formula_57, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonical isomorphism between the two objects.