Namespaces
Variants
Actions

Difference between revisions of "Diagram"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
(tex done, several typos corrected)
Line 20: Line 20:
 
for which
 
for which
  
$$  
+
$$ D (I)  \subset   \mathop{\rm Ob} (C) ,\  D (U)  \subset   \mathop{\rm Mor} (C) , $$
D ( I)  \subset   \mathop{\rm Ob} ( C) ,\  D ( U)  \subset   \mathop{\rm Mor} ( C) ,
 
$$
 
  
and  $  D ( u) \in \mathop{\rm Hom} ( D ( i) , D ( j) ) $
+
and  $  D (u) \in {\mathop{\rm Hom}\nolimits} ( D (i) , D (j) ) $
 
if the edge  $  u \in U $
 
if the edge  $  u \in U $
 
has source (origin)  $  i $
 
has source (origin)  $  i $
Line 36: Line 34:
 
with source  $  i $
 
with source  $  i $
 
and target  $  j $,  
 
and target  $  j $,  
i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let  $  D ( \phi ) :  D ( i) \rightarrow D ( j) $
+
i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let  $  D ( \phi ) :  D (i) \mathop \rightarrow \limits D (j) $
 
denote the composition of morphisms
 
denote the composition of morphisms
  
$$  
+
$$ D (u _ {n} )  \circ \dots \circ  D (u _ {1} ) . $$
D ( u _ {n} )  \circ \dots \circ  D ( u _ {1} ) .
 
$$
 
  
 
The diagram  $  D $
 
The diagram  $  D $
is said to be commutative if  $  D ( \phi ) = D ( \phi ^  \prime  ) $
+
is said to be commutative if  $  D ( \phi ) = D ( \phi ^  \prime  ) $
 
for any two oriented chains  $  \phi $
 
for any two oriented chains  $  \phi $
and  $  \phi ^  \prime  $
+
and  $  \phi ^  \prime  $
 
with identical source and target.
 
with identical source and target.
  
 
The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form
 
The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form
  
$$  
+
$$ \mathop \cdot \limits _ {i _ {1}} \mathop \rightarrow \limits ^ {u _ {1}} \mathop \cdot \limits _ {i _ {2}} \mathop \rightarrow \limits  \dots \mathop \rightarrow \limits \mathop \cdot \limits _ {i _ {n-1}} \mathop \rightarrow \limits ^ {u _ {n-1}} \mathop \cdot \limits _ {i _ {n}} . $$
\cdot _ { i _ {1} } \rightarrow ^ { {u _ 1 } } \dots \rightarrow \cdot _ {i _ {n-} 1 }
 
\rightarrow ^ { {u _ n-} 1 } \cdot _ {i _ {n} } .
 
$$
 
  
 
The corresponding diagram is represented as follows:
 
The corresponding diagram is represented as follows:
  
$$  
+
$$ A _ {1}  \mathop \rightarrow \limits ^ {f _ {1}}  A _ {2}  \mathop \rightarrow \limits \dots \mathop \rightarrow \limits A _ {n-1} \mathop \rightarrow \limits ^ {f _ {n-1}} A _ {n} , $$
A _ {1}  \rightarrow ^ { {f _ 1} }  A _ {2}  \rightarrow \dots \rightarrow  A _ {n-} \mathop \rightarrow \limits ^ { {f _ {n-} 1 }}   A _ {n} ,
 
$$
 
  
 
where the  $  A _ {k} = D ( i _ {k} ) $
 
where the  $  A _ {k} = D ( i _ {k} ) $
Line 70: Line 61:
 
corresponds to the graph
 
corresponds to the graph
  
$$  
+
$$   \begin{array}{lcr} {i _ {1}} \mathop \cdot \limits &\mathop \rightarrow \limits ^ {u _ {1}}  &\mathop \cdot \limits {i _ {2}} \\ {} _ {u _ {3}}  \searrow &{}  &\swarrow _ {u _ {2}\\ {} &\mathop \cdot \limits _ {i _ {3}}  &{}  \\ \end{array}   $$
 
 
\begin{array}{lcr}
 
{}  &i _ {1}  & \mathop \rightarrow \limits _ { {u _ {3} }\\
 
{} _ {u _ {1} }  \nearrow &i _ {2}  &\searrow _ {u _ {2}  }   \\
 
i _ {1}  &{}  \\
 
\end{array}
 
 
 
$$
 
  
 
and is represented as follows:
 
and is represented as follows:
  
$$  
+
$$   \begin{array}{lcr} A _ {1}  &\mathop \rightarrow \limits ^ {f _ {1}}  &A _ {2}  \\ {} _ {f _ {3}}  \searrow  &{}  &\swarrow _ {f _ {2}}  \\ {}  &A _ {3}  &{}  \\ \end{array}   $$
 
 
\begin{array}{lcr}
 
A _ {1}  &\rightarrow ^ { {f _ 1} }  &A _ {2}  \\
 
{} _ {f _ {3} }  \searrow  &{}  &\swarrow _ {f _ {2} }  \\
 
{}  &A _ {3}  &{}  \\
 
\end{array}
 
 
 
$$
 
  
 
Commutativity of this diagram means that  $  f _ {3} = f _ {2} \circ f _ {1} $.
 
Commutativity of this diagram means that  $  f _ {3} = f _ {2} \circ f _ {1} $.
Line 96: Line 71:
 
A square diagram corresponds to the graph
 
A square diagram corresponds to the graph
  
$$  
+
$$   \begin{array}{rcl}  i _ {1} \mathop \cdot \limits &\mathop \rightarrow \limits ^ {u _ {1}} &\mathop \cdot \limits i _ {2}  \\ \scriptsize {u _ {4}} \downarrow  &{}  &\downarrow \scriptsize {u _ {2}}  \\  i _ {4} \mathop \cdot \limits &\mathop \rightarrow \limits _ {u _ {3}}  &\mathop \cdot \limits i _ {3}  \\ \end{array}   $$
 
 
\begin{array}{lcl}
 
  i _ {1}  &\cdot \rightarrow ^ { {\phi _ 1} } \cdot &i _ {2}  \\
 
size - 3 {\phi _ {4} } \downarrow  &{}  &\downarrow size - 3 {\phi _ {2} }  \\
 
  i _ {4} &\cdot  \mathop \rightarrow \limits _ { {\phi _ {3} }}  \cdot &i _ {3}  \\
 
\end{array}
 
 
 
$$
 
  
 
and is represented as follows:
 
and is represented as follows:
  
$$  
+
$$   \begin{array}{lcl}  A _ {1}  &\mathop \rightarrow \limits ^ {f _ {1}}  &A _ {2}  \\ \scriptsize {f _ {4}} \downarrow  &{}  &\downarrow \scriptsize {f _ {2}}  \\  A _ {4}  &\mathop \rightarrow \limits _ {f _ {3}} &A _ {3}  \\ \end{array}   $$
 
 
\begin{array}{lcl}
 
  A _ {1}  &\rightarrow ^ { {f _ 1} }  &A _ {2}  \\
 
size - 3 {f _ {4} } \downarrow  &{}  &\downarrow size - 3 {f _ {2} }  \\
 
  A _ {4}  & \mathop \rightarrow \limits _ { {f _ {3} }&A _ {3}  \\
 
\end{array}
 
 
 
$$
 
  
 
Commutativity of this diagram means that  $  f _ {2} \circ f _ {1} = f _ {3} \circ f _ {4} $.
 
Commutativity of this diagram means that  $  f _ {2} \circ f _ {1} = f _ {3} \circ f _ {4} $.
Line 123: Line 82:
 
forms a category. A morphism of a diagram  $  D $
 
forms a category. A morphism of a diagram  $  D $
 
into a diagram  $  D _ {1} $
 
into a diagram  $  D _ {1} $
is taken to be a family of morphisms  $  \nu _ {i} :  D ( i) \rightarrow D _ {1} ( i) $,  
+
is taken to be a family of morphisms  $  \nu _ {i} :  D (i) \mathop \rightarrow \limits D _ {1} (i) $,  
 
where  $  i $
 
where  $  i $
 
runs through the set of vertices of  $  \Gamma $,  
 
runs through the set of vertices of  $  \Gamma $,  
Line 129: Line 88:
 
with source  $  i $
 
with source  $  i $
 
and target  $  j $
 
and target  $  j $
the condition  $  D _ {1} ( u) \circ \nu _ {i} = \nu _ {j} \circ D ( u) $
+
the condition  $  D _ {1} (u) \circ \nu _ {i} = \nu _ {j} \circ D (u) $
 
is met. In particular, one may speak of isomorphic diagrams. The graph  $  \Gamma $
 
is met. In particular, one may speak of isomorphic diagrams. The graph  $  \Gamma $
 
is sometimes referred to as the scheme of a diagram in  $  C $.
 
is sometimes referred to as the scheme of a diagram in  $  C $.

Revision as of 10:00, 21 June 2020


in a category $ C $

A mapping $ D $ of an oriented graph $ \Gamma $ with set of vertices $ I $ and set of edges $ U $ into the category $ C $ for which

$$ D (I) \subset \mathop{\rm Ob} (C) ,\ D (U) \subset \mathop{\rm Mor} (C) , $$

and $ D (u) \in {\mathop{\rm Hom}\nolimits} ( D (i) , D (j) ) $ if the edge $ u \in U $ has source (origin) $ i $ and target (end) $ j $. The concept of a diagram in $ C $ may also be defined as the image of the mapping $ D $, in order to obtain a better visualization of diagrams.

Let $ \phi = ( u _ {1} \dots u _ {n} ) $ be an oriented chain of the graph $ \Gamma $ with source $ i $ and target $ j $, i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let $ D ( \phi ) : D (i) \mathop \rightarrow \limits D (j) $ denote the composition of morphisms

$$ D (u _ {n} ) \circ \dots \circ D (u _ {1} ) . $$

The diagram $ D $ is said to be commutative if $ D ( \phi ) = D ( \phi ^ \prime ) $ for any two oriented chains $ \phi $ and $ \phi ^ \prime $ with identical source and target.

The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form

$$ \mathop \cdot \limits _ {i _ {1}} \mathop \rightarrow \limits ^ {u _ {1}} \mathop \cdot \limits _ {i _ {2}} \mathop \rightarrow \limits \dots \mathop \rightarrow \limits \mathop \cdot \limits _ {i _ {n-1}} \mathop \rightarrow \limits ^ {u _ {n-1}} \mathop \cdot \limits _ {i _ {n}} . $$

The corresponding diagram is represented as follows:

$$ A _ {1} \mathop \rightarrow \limits ^ {f _ {1}} A _ {2} \mathop \rightarrow \limits \dots \mathop \rightarrow \limits A _ {n-1} \mathop \rightarrow \limits ^ {f _ {n-1}} A _ {n} , $$

where the $ A _ {k} = D ( i _ {k} ) $ are objects in the category $ C $, while the $ f _ {k} = D ( u _ {k} ) $ are morphisms in this category.

A triangular diagram in a category $ C $ corresponds to the graph

$$ \begin{array}{lcr} {i _ {1}} \mathop \cdot \limits &\mathop \rightarrow \limits ^ {u _ {1}} &\mathop \cdot \limits {i _ {2}} \\ {} _ {u _ {3}} \searrow &{} &\swarrow _ {u _ {2}} \\ {} &\mathop \cdot \limits _ {i _ {3}} &{} \\ \end{array} $$

and is represented as follows:

$$ \begin{array}{lcr} A _ {1} &\mathop \rightarrow \limits ^ {f _ {1}} &A _ {2} \\ {} _ {f _ {3}} \searrow &{} &\swarrow _ {f _ {2}} \\ {} &A _ {3} &{} \\ \end{array} $$

Commutativity of this diagram means that $ f _ {3} = f _ {2} \circ f _ {1} $.

A square diagram corresponds to the graph

$$ \begin{array}{rcl} i _ {1} \mathop \cdot \limits &\mathop \rightarrow \limits ^ {u _ {1}} &\mathop \cdot \limits i _ {2} \\ \scriptsize {u _ {4}} \downarrow &{} &\downarrow \scriptsize {u _ {2}} \\ i _ {4} \mathop \cdot \limits &\mathop \rightarrow \limits _ {u _ {3}} &\mathop \cdot \limits i _ {3} \\ \end{array} $$

and is represented as follows:

$$ \begin{array}{lcl} A _ {1} &\mathop \rightarrow \limits ^ {f _ {1}} &A _ {2} \\ \scriptsize {f _ {4}} \downarrow &{} &\downarrow \scriptsize {f _ {2}} \\ A _ {4} &\mathop \rightarrow \limits _ {f _ {3}} &A _ {3} \\ \end{array} $$

Commutativity of this diagram means that $ f _ {2} \circ f _ {1} = f _ {3} \circ f _ {4} $.

The class of diagrams with a given graph $ \Gamma $ forms a category. A morphism of a diagram $ D $ into a diagram $ D _ {1} $ is taken to be a family of morphisms $ \nu _ {i} : D (i) \mathop \rightarrow \limits D _ {1} (i) $, where $ i $ runs through the set of vertices of $ \Gamma $, so that for any edge $ u $ with source $ i $ and target $ j $ the condition $ D _ {1} (u) \circ \nu _ {i} = \nu _ {j} \circ D (u) $ is met. In particular, one may speak of isomorphic diagrams. The graph $ \Gamma $ is sometimes referred to as the scheme of a diagram in $ C $.

References

[1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221

Comments

References

[a1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[a2] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Diagram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram&oldid=46644
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article