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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.
 
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.
  
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<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>
 
<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  $  ( {\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}
  
====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.
+
 
 +
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
  
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>
+
\begin{array}{lcr}
 +
{}  & C  &{}  \\
 +
{} _ {f}  \swarrow  &{}  &\searrow _ {g}  \\
 +
A  &{}  & B  \\
 +
{} _ {p} \searrow  &{}  &\swarrow _ {q}  \\
 +
{}  & D  &{}  \\
 +
\end{array}
 +
\ \
  
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
+
\begin{array}{lcr}
 +
{}  & C  &{}  \\
 +
{} _ {f ^ { \prime }  } \swarrow  &{}  &\searrow _ {g  ^  \prime  }  \\
 +
A  &{}  & B  \\
 +
\searrow _ {p}  &{}  &\swarrow _ {q}  \\
 +
{}  & D  &{}  \\
 +
\end{array}
  
<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" />.
+
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 }  $.
  
 
====References====
 
====References====
Line 25: Line 80:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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]]).
+
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|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
+
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
  
<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>
+
$$
 +
j _ {0}  ^ {k} ( \phi ) j _ {0}  ^ {k} ( u)  = \
 +
j _ {x}  ^ {k} ( \phi \circ u),\ \
 +
j _ {0}  ^ {k} ( \phi ) \in  \mathop{\rm GL}  ^ {k} ( n),\ \
 +
j _ {x}  ^ {k} ( u) \in M _ {k} ,
 +
$$
  
and this action defines on <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:
+
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:
  
<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>
+
$$
 +
S( gu  ^ {k} )  = gS( u  ^ {k} ),\ \
 +
g \in  \mathop{\rm GL}  ^ {k} ( n),\ \
 +
u  ^ {k} \in M _ {k} .
 +
$$
  
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" />.
+
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 <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).
+
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 <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
+
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
  
<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>
+
$$
 +
\{ e _ {i _ {1}  } \otimes \dots \otimes e _ {i _ {p}  } \otimes e ^ {\star j _ {1} } \otimes
 +
{} \dots \otimes e ^ {\star j _ {q} } \}
 +
$$
  
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:
+
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:
  
<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>
+
$$
 +
S _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } ( g \theta )  = \
 +
g _ {a _ {1}  } ^ {i _ {1} } \dots g _ {a _ {p}  } ^ {i _ {p} } ( g  ^ {-} 1 ) _ {j _ {1}  } ^ {b _ {1} } \dots ( g  ^ {-} 1 ) _ {j _ {q}  } ^ {b _ {q} } S ( \theta ) _ {b _ {1}  \dots b _ {q} } ^ {a _ {1} \dots a _ {p} } .
 +
$$
  
The most important examples of tensor structures are a [[Vector field|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 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 $  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| $  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|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.
+
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 $  G $-
 +
structures that are principal bundles with a fixed homomorphism onto a $  G $-
 +
structure, and sections of associated bundles.
  
 
====References====
 
====References====

Revision as of 14:55, 7 June 2020


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 $ ( {\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 } $.

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 $ 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 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). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Structure(2)&oldid=49610
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article