# Structure(2)

Also called mathematical structure. A generic name for unifying concepts whose general characteristic is that they can be applied to sets whose elements are of an indefinite nature. In order to define a structure, relations are given in which the elements of the set appear (the type characteristic of a structure), and it is then postulated that these relations satisfy certain conditions — axioms of the structure.

#### References

[1] | N. Bourbaki, "Eléments d'histoire des mathématiques" , Hermann (1960) |

[2] | N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French) |

#### Comments

Sets endowed with a given structure plus mappings of sets which preserve this structure together form a category. Such categories are called concrete (cf. also Category; Sets, category of). More precisely, a concrete category is a pair consisting of a category and a faithful functor . Because is faithful, can be identified with , and an object of a concrete category is a set with extra structure while a morphism is an actual mapping of sets that preserves the extra structure; composition of morphisms is accomplished by the usual composition of mappings of sets. Often the set of morphisms consists of all structure-preserving mappings of sets, but this need not be the case.

A category is concrete if and only if it satisfies the Isbell condition (the Freyd concreteness theorem). Here, the Isbell condition is the following. A span in a category is a diagram of the form

Two -spans and are equivalent if for all pairs of morphisms either both diagrams

commute or both do not commute. A category satisfies the Isbell condition if for all objects there exists a set of -spans such that each -span is equivalent to precisely one element of .

#### References

[a1] | J. Adamek, "Theory of mathematical structures" , Reidel (1983) pp. Chapt. 6 |

[a2] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. 26 |

A structure is also an obsolete term for lattice.

#### Comments

#### References

[a1] | A.G. Kurosh, "Theory of groups" , 2 , Chelsea, reprint (1955) pp. 85 (Translated from Russian) |

A structure on a manifold, a geometric quantity, a geometric object, or a field of geometric objects, is a section of a bundle associated with the principal bundle of coframes on the manifold . Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on the point of the manifold , but also on the choice of the coframe — an infinitesimal system of coordinates at the point (see Chart).

More precisely, let be the general differential group of order (the group of -jets at zero of transformations of the space that preserve the origin), and let be the manifold of coframes of order of an -dimensional manifold (i.e. the manifold of -jets of local charts with origin at the point ). The group acts from the left on by

and this action defines on the structure of a principal -bundle , which is called the bundle of coframes of order . Let be an arbitrary -manifold, i.e. a manifold with a left action of the group . Finally, let be the orbit space of the left action of on , while is its natural projection onto . The bundle (associated with and ) is called a bundle of geometric structures of order and of type , while its sections are called structures of type . Structures of type are in a natural one-to-one correspondence with -equivariant mappings . Thus, a structure of type can be seen as a -valued function on the manifold of -frames that satisfies the following condition of equivariance:

The bundle of geometric objects is a natural bundle in the sense that the diffeomorphism group of acts as the automorphism group of .

If is a vector space with a linear (or affine) action of , then a structure of type is said to be linear (or affine).

A basic example of a linear structure of order one is a tensor structure, or a tensor field. Let , and let be the space of tensors of type with the natural tensor representation of . A structure of type is called a tensor field of type . It can be regarded as a vector function on the manifold of coframes which assigns to the coframe the set of coordinates of the tensor , relative to the standard basis

of . Given a linear transformation of coframes , the coordinates are transformed in accordance with the tensor representation:

The most important examples of tensor structures are a vector field, a Riemannian metric, a differential form, a symplectic structure, a complex structure, and most commonly, an affinor. All linear structures (of whatever order) are exhausted by Rashevskii super-tensors (see [4]). An example of an affine structure of order two is an affine connection without torsion, which can be regarded as a structure of type , where is the kernel of the natural homomorphism , considered as a vector space with the natural action of . A large and important class of structures is the class of infinitesimally-homogeneous structures or -structures (cf. -structure) — structures of type , where is a homogeneous space of the group .

The above definition of a structure is not sufficiently general, and does not include a number of important geometric structures such as a spinor structure, a symplectic spinor structure, etc. A natural generalization is to study generalized -structures that are principal bundles with a fixed homomorphism onto a -structure, and sections of associated bundles.

#### References

[1] | P. Rashevskii, "Caractères tensoriels de l'espace sousprojectif" Trudy Sem. Vektor. i Tenzor. Anal. , 1 (1933) pp. 126–142 |

[2] | V. Vagner, "The theory of geometric objects and the theory of finite and infinite continuous transformation groups" Dokl. Akad. Nauk SSSR , 46 : 9 (1945) pp. 347–349 (In Russian) |

[3] | O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932) |

[4] | P.K. Rashevskii, "On linear representations of differential groups and Lie groups with nilpotent radical" Trudy Moskov. Mat. Obshch. , 6 (1957) pp. 337–370 (In Russian) |

[5] | S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) |

[6] | Ch. Ehresmann, "Introduction à la théorie des structures infinitésimals et des pseudo-groupes de Lie" , Géométrie Diff. Coll. Internat. C.N.R.S. (1953) pp. 97–110 |

*D.V. Alekseevskii*

#### Comments

Historically, E. Cartan was the first to introduce the concept of a structure.

#### References

[a1] | E. Cartan, "La théorie des groupes et les récherches récentes de géometrie différentielle" Enseign. Math. , 24 (1925) pp. 5–18 |

**How to Cite This Entry:**

Structure(2).

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