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''in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315601.png" />''
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315602.png" /> of an oriented graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315603.png" /> with set of vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315604.png" /> and set of edges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315605.png" /> into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315606.png" /> for which
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{{TEX|auto}}
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{{TEX|done}}
  
<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/d/d031/d031560/d0315607.png" /></td> </tr></table>
+
''in a category  $  C $''
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315608.png" /> if the edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315609.png" /> has source (origin) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156010.png" /> and target (end) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156011.png" />. The concept of a diagram in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156012.png" /> may also be defined as the image of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156013.png" />, in order to obtain a better visualization of diagrams.
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A mapping  $  D $
 +
of an oriented graph  $  \Gamma $
 +
with set of vertices  $  I $
 +
and set of edges  $  U $
 +
into the category  $  C $
 +
for which
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156014.png" /> be an oriented chain of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156015.png" /> with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156016.png" /> and target <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156017.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156018.png" /> denote the composition of morphisms
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$$  D (I)  \subset  \mathop{\rm Ob} (C) ,\  D (U)  \subset  \mathop{\rm Mor} (C) , $$
  
<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/d/d031/d031560/d03156019.png" /></td> </tr></table>
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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.
  
The diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156020.png" /> is said to be commutative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156021.png" /> for any two oriented chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156023.png" /> with identical source and target.
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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
 
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
  
<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/d/d031/d031560/d03156024.png" /></td> </tr></table>
+
$$  \mathop \cdot \limits _ {i _ {1}} \mathop \rightarrow \limits ^ {u _ {1}} \mathop \cdot \limits _ {i _ {2}} \mathop \rightarrow \limits  \dots \mathop \rightarrow \limits \mathop \cdot \limits _ {i _ {n-1}} \mathop \rightarrow \limits ^ {u _ {n-1}} \mathop \cdot \limits _ {i _ {n}} . $$
  
 
The corresponding diagram is represented as follows:
 
The corresponding diagram is represented as follows:
  
<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/d/d031/d031560/d03156025.png" /></td> </tr></table>
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$$  A _ {1}  \mathop \rightarrow \limits ^ {f _ {1}}  A _ {2}  \mathop \rightarrow \limits \dots \mathop \rightarrow \limits  A _ {n-1}  \mathop \rightarrow \limits ^ {f _ {n-1}}  A _ {n} ,  $$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156026.png" /> are objects in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156027.png" />, while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156028.png" /> are morphisms in this category.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156029.png" /> corresponds to the graph
+
A triangular diagram in a category $  C $
 +
corresponds to the graph
  
<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/d/d031/d031560/d03156030.png" /></td> </tr></table>
+
$$  \begin{array}{lcr} {i _ {1}} \mathop \cdot \limits  &\mathop \rightarrow \limits ^ {u _ {1}}  &\mathop \cdot \limits {i _ {2}}  \\ {} _ {u _ {3}}  \searrow  &{}  &\swarrow _ {u _ {2}}  \\ {}  &\mathop \cdot \limits _ {i _ {3}}  &{}  \\ \end{array}  $$
  
 
and is represented as follows:
 
and is represented as follows:
  
<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/d/d031/d031560/d03156031.png" /></td> </tr></table>
+
$$  \begin{array}{lcr} A _ {1}  &\mathop \rightarrow \limits ^ {f _ {1}}  &A _ {2}  \\ {} _ {f _ {3}}  \searrow  &{}  &\swarrow _ {f _ {2}}  \\ {}  &A _ {3}  &{}  \\ \end{array}  $$
  
Commutativity of this diagram means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156032.png" />.
+
Commutativity of this diagram means that $  f _ {3} = f _ {2} \circ f _ {1} $.
  
 
A square diagram corresponds to the graph
 
A square diagram corresponds to the graph
  
<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/d/d031/d031560/d03156033.png" /></td> </tr></table>
+
$$  \begin{array}{rcl}  i _ {1} \mathop \cdot \limits  &\mathop \rightarrow \limits ^ {u _ {1}}  &\mathop \cdot \limits i _ {2}  \\ \scriptsize {u _ {4}} \downarrow  &{}  &\downarrow \scriptsize {u _ {2}}  \\  i _ {4} \mathop \cdot \limits  &\mathop \rightarrow \limits _ {u _ {3}}  &\mathop \cdot \limits i _ {3}  \\ \end{array}  $$
  
 
and is represented as follows:
 
and is represented as follows:
  
<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/d/d031/d031560/d03156034.png" /></td> </tr></table>
+
$$  \begin{array}{lcl}  A _ {1}  &\mathop \rightarrow \limits ^ {f _ {1}} &A _ {2}  \\ \scriptsize {f _ {4}} \downarrow  &{}  &\downarrow \scriptsize {f _ {2}}  \\  A _ {4} &\mathop \rightarrow \limits _ {f _ {3}} &A _ {3} \\ \end{array}  $$
 
 
Commutativity of this diagram means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156035.png" />.
 
 
 
The class of diagrams with a given graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156036.png" /> forms a category. A morphism of a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156037.png" /> into a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156038.png" /> is taken to be a family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156040.png" /> runs through the set of vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156041.png" />, so that for any edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156042.png" /> with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156043.png" /> and target <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156044.png" /> the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156045.png" /> is met. In particular, one may speak of isomorphic diagrams. The graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156046.png" /> is sometimes referred to as the scheme of a diagram in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156047.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck,  "Sur quelques points d'algèbre homologique" ''Tohoku Math. J.'' , '''9''' (1957) pp. 119–221</TD></TR></table>
 
 
 
 
 
  
====Comments====
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> S. MacLane, "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR>
 +
</table>

Latest revision as of 10:06, 11 November 2023


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
[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=14728
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article