# Structure(2)

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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)

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 $( {\mathcal C}, U)$ consisting of a category ${\mathcal C}$ and a faithful functor $U: {\mathcal C} \rightarrow \mathop{\rm Set}$. Because $U$ is faithful, $f$ can be identified with $Uf$, and an object $C$ of a concrete category is a set $U( C)$ with extra structure while a morphism $f$ 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 ${\mathcal C} ( C, D)$ 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

$$\begin{array}{lcr} {} & C &{} \\ {} _ {f} \swarrow &{} &\searrow _ {g} \\ A &{} & B \\ \end{array}$$

Two $( A, B)$- spans $( f, g)$ and $( f ^ { \prime } , g ^ \prime )$ are equivalent if for all pairs of morphisms $( p: A \rightarrow D, q: B \rightarrow D)$ either both diagrams

$$\begin{array}{lcr} {} & C &{} \\ {} _ {f} \swarrow &{} &\searrow _ {g} \\ A &{} & B \\ {} _ {p} \searrow &{} &\swarrow _ {q} \\ {} & D &{} \\ \end{array} \ \ \begin{array}{lcr} {} & C &{} \\ {} _ {f ^ { \prime } } \swarrow &{} &\searrow _ {g ^ \prime } \\ A &{} & B \\ \searrow _ {p} &{} &\swarrow _ {q} \\ {} & D &{} \\ \end{array}$$

commute or both do not commute. A category satisfies the Isbell condition if for all objects $( A, B)$ there exists a set of $( A, B)$- spans $M _ {A,B }$ such that each $( A, B)$- span is equivalent to precisely one element of $M _ {A,B }$.

A structure is also an obsolete term for lattice.

#### 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 [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 $M$. Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on the point $x$ of the manifold $M$, but also on the choice of the coframe — an infinitesimal system of coordinates at the point $x$( see Chart).

More precisely, let $\mathop{\rm GL} ^ {k} ( n)$ be the general differential group of order $k$( the group of $k$- jets at zero of transformations of the space $\mathbf R ^ {n}$ that preserve the origin), and let $M _ {k}$ be the manifold of coframes of order $k$ of an $n$- dimensional manifold $M$( i.e. the manifold of $k$- jets $j _ {x} ^ {k} ( u)$ of local charts $u: M \supset U \rightarrow \mathbf R ^ {n}$ with origin at the point $x = u ^ {-} 1 ( 0)$). The group $\mathop{\rm GL} ^ {k} ( u)$ acts from the left on $M _ {k}$ by

$$j _ {0} ^ {k} ( \phi ) j _ {0} ^ {k} ( u) = \ j _ {x} ^ {k} ( \phi \circ u),\ \ j _ {0} ^ {k} ( \phi ) \in \mathop{\rm GL} ^ {k} ( n),\ \ j _ {x} ^ {k} ( u) \in M _ {k} ,$$

and this action defines on $M _ {k}$ the structure of a principal $\mathop{\rm GL} ^ {k} ( n)$- bundle $\pi _ {k} : M _ {k} \rightarrow M$, which is called the bundle of coframes of order $k$. Let $W$ be an arbitrary $\mathop{\rm GL} ^ {k} ( n)$- manifold, i.e. a manifold with a left action of the group $\mathop{\rm GL} ^ {k} ( n)$. Finally, let $W( M)$ be the orbit space of the left action of $\mathop{\rm GL} ^ {k} ( n)$ on $M _ {k} \times W$, while $\pi _ {W}$ is its natural projection onto $M$. The bundle $\pi _ {W} : W( M) \rightarrow M$( associated with $M _ {k}$ and $W$) is called a bundle of geometric structures of order $\leq k$ and of type $W$, while its sections are called structures of type $W$. Structures of type $W$ are in a natural one-to-one correspondence with $\mathop{\rm GL} ^ {k} ( n)$- equivariant mappings $S: M _ {k} \rightarrow W$. Thus, a structure of type $W$ can be seen as a $W$- valued function $S$ on the manifold $M _ {k}$ of $k$- frames that satisfies the following condition of equivariance:

$$S( gu ^ {k} ) = gS( u ^ {k} ),\ \ g \in \mathop{\rm GL} ^ {k} ( n),\ \ u ^ {k} \in M _ {k} .$$

The bundle $\pi _ {W}$ of geometric objects is a natural bundle in the sense that the diffeomorphism group of $M$ acts as the automorphism group of $\pi _ {W}$.

If $W$ is a vector space with a linear (or affine) action of $\mathop{\rm GL} ^ {k} ( n)$, then a structure of type $W$ 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 $V = \mathbf R ^ {n}$, $V ^ \star = \mathop{\rm Hom} ( V, \mathbf R )$ and let $V _ {q} ^ {p} = ((\otimes ^ {p} V)) \otimes ((\otimes ^ {q} V ^ \star ))$ be the space of tensors of type $( p, q)$ with the natural tensor representation of $\mathop{\rm GL} ^ {1} ( n) = \mathop{\rm GL} ( n)$. A structure of type $V _ {q} ^ {p}$ is called a tensor field of type $( p, q)$. It can be regarded as a vector function on the manifold of coframes $M _ {1}$ which assigns to the coframe $\theta = j _ {x} ^ {1} ( u) = ( du ^ {1} \dots du ^ {n} )$ the set of coordinates $S( \theta ) _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }$ of the tensor $S( \theta ) \in V _ {q} ^ {p}$, relative to the standard basis

$$\{ e _ {i _ {1} } \otimes \dots \otimes e _ {i _ {p} } \otimes e ^ {\star j _ {1} } \otimes {} \dots \otimes e ^ {\star j _ {q} } \}$$

of $V _ {q} ^ {p}$. Given a linear transformation of coframes $\theta \rightarrow g \theta = ( g _ {a} ^ {i} du ^ {a} )$, the coordinates $S _ {j _ {1} {} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }$ are transformed in accordance with the tensor representation:

$$S _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } ( g \theta ) = \ g _ {a _ {1} } ^ {i _ {1} } \dots g _ {a _ {p} } ^ {i _ {p} } ( g ^ {-} 1 ) _ {j _ {1} } ^ {b _ {1} } \dots ( g ^ {-} 1 ) _ {j _ {q} } ^ {b _ {q} } S ( \theta ) _ {b _ {1} \dots b _ {q} } ^ {a _ {1} \dots a _ {p} } .$$

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 $V _ {(} 2) ^ {1}$, where $V _ {(} 2) ^ {1} \approx V \otimes S ^ {2} V ^ \star$ is the kernel of the natural homomorphism $\mathop{\rm GL} ^ {2} ( n) \rightarrow \mathop{\rm GL} ^ {1} ( n)$, considered as a vector space with the natural action of $\mathop{\rm GL} ^ {2} ( n) = \mathop{\rm GL} ( n) V _ {(} 2) ^ {1}$. A large and important class of structures is the class of infinitesimally-homogeneous structures or $G$- structures (cf. $G$- structure) — structures of type $W$, where $W = \mathop{\rm GL} ^ {k} ( n)/G$ is a homogeneous space of the group $\mathop{\rm GL} ^ {k} ( n)$.

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 $G$- structures that are principal bundles with a fixed homomorphism onto a $G$- 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 infinitésimales et des pseudo-groupes de Lie" , Géométrie Diff. Coll. Internat. C.N.R.S. (1953) pp. 97–110

D.V. Alekseevskii