View source for 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Eléments d'histoire des mathématiques" , Hermann  (1960)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)</TD></TR></table>



====Comments====
Sets endowed with a given structure plus mappings of sets which preserve this structure together form a [[Category|category]]. Such categories are called concrete (cf. also [[Category|Category]]; [[Sets, category of|Sets, category of]]). More precisely, a concrete category is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906701.png" /> consisting of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906702.png" /> and a faithful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906703.png" />. Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906704.png" /> is faithful, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906705.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906706.png" />, and an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906707.png" /> of a concrete category is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906708.png" /> with extra structure while a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s0906709.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067010.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067011.png" /></td> </tr></table>

Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067012.png" />-spans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067014.png" /> are equivalent if for all pairs of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067015.png" /> either both diagrams

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067016.png" /></td> </tr></table>

commute or both do not commute. A category satisfies the Isbell condition if for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067017.png" /> there exists a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067018.png" />-spans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067019.png" /> such that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067020.png" />-span is equivalent to precisely one element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067021.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Adamek,   "Theory of mathematical structures" , Reidel  (1983)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. 26</TD></TR></table>

A structure is also an obsolete term for [[Lattice|lattice]].

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Kurosh,   "Theory of groups" , '''2''' , Chelsea, reprint  (1955)  pp. 85  (Translated from Russian)</TD></TR></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067022.png" />. Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067023.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067024.png" />, but also on the choice of the coframe — an infinitesimal system of coordinates at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067025.png" /> (see [[Chart|Chart]]).

More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067026.png" /> be the general differential group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067027.png" /> (the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067028.png" />-jets at zero of transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067029.png" /> that preserve the origin), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067030.png" /> be the manifold of coframes of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067031.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067032.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067033.png" /> (i.e. the manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067034.png" />-jets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067035.png" /> of local charts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067036.png" /> with origin at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067037.png" />). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067038.png" /> acts from the left on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067039.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067040.png" /></td> </tr></table>

and this action defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067041.png" /> the structure of a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067042.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067043.png" />, which is called the bundle of coframes of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067045.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067046.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067047.png" />-manifold, i.e. a manifold with a left action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067048.png" />. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067049.png" /> be the orbit space of the left action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067051.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067052.png" /> is its natural projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067053.png" />. The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067054.png" /> (associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067056.png" />) is called a bundle of geometric structures of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067058.png" /> and of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067059.png" />, while its sections are called structures of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067061.png" />. Structures of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067062.png" /> are in a natural one-to-one correspondence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067063.png" />-equivariant mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067064.png" />. Thus, a structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067065.png" /> can be seen as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067066.png" />-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067067.png" /> on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067068.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067069.png" />-frames that satisfies the following condition of equivariance:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067070.png" /></td> </tr></table>

The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067071.png" /> of geometric objects is a natural bundle in the sense that the diffeomorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067072.png" /> acts as the automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067073.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067074.png" /> is a vector space with a linear (or affine) action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067075.png" />, then a structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067076.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067080.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067081.png" /> be the space of tensors of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067082.png" /> with the natural tensor representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067083.png" />. A structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067084.png" /> is called a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067086.png" />. It can be regarded as a vector function on the manifold of coframes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067087.png" /> which assigns to the coframe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067088.png" /> the set of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067089.png" /> of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067090.png" />, relative to the standard basis

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067091.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067092.png" />. Given a linear transformation of coframes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067093.png" />, the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067094.png" /> are transformed in accordance with the tensor representation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067095.png" /></td> </tr></table>

The most important examples of tensor structures are a [[Vector field|vector field]], a [[Riemannian metric|Riemannian metric]], a [[Differential form|differential form]], a [[Symplectic structure|symplectic structure]], a [[Complex structure|complex structure]], and most commonly, an [[Affinor|affinor]]. All linear structures (of whatever order) are exhausted by Rashevskii super-tensors (see [[#References|[4]]]). An example of an affine structure of order two is an [[Affine connection|affine connection]] without torsion, which can be regarded as a structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067097.png" /> is the kernel of the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067098.png" />, considered as a vector space with the natural action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s09067099.png" />. A large and important class of structures is the class of infinitesimally-homogeneous structures or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670100.png" />-structures (cf. [[G-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670101.png" />-structure]]) — structures of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670103.png" /> is a homogeneous space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670104.png" />.

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|spinor structure]], a symplectic spinor structure, etc. A natural generalization is to study generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670105.png" />-structures that are principal bundles with a fixed homomorphism onto a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090670/s090670106.png" />-structure, and sections of associated bundles.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Rashevskii,   "Caractères tensoriels de l'espace sousprojectif"  ''Trudy Sem. Vektor. i Tenzor. Anal.'' , '''1'''  (1933)  pp. 126–142</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Veblen,   J.H.C. Whitehead,   "The foundations of differential geometry" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>

''D.V. Alekseevskii''

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Cartan,   "La théorie des groupes et les récherches récentes de géometrie différentielle"  ''Enseign. Math.'' , '''24'''  (1925)  pp. 5–18</TD></TR></table>