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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201202.png" /> be directed graphs (also called oriented graphs, diagram schemes or pre-categories; cf. also [[Graph, oriented|Graph, oriented]]). A diagram of shape (also called a diagram of type) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201203.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201204.png" /> is a morphism of graphs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201205.png" />; i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201207.png" /> are given by
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
<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/d120/d120120/d1201208.png" /></td> </tr></table>
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Out of 75 formulas, 63 were replaced by TEX code.-->
  
<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/d120/d120120/d1201209.png" /></td> </tr></table>
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Let $G$ and $G ^ { \prime }$ be directed graphs (also called oriented graphs, diagram schemes or pre-categories; cf. also [[Graph, oriented|Graph, oriented]]). A diagram of shape (also called a diagram of type) $G$ in $G ^ { \prime }$ is a morphism of graphs $d : G \rightarrow G ^ { \prime }$; i.e. if $G$ and $G ^ { \prime }$ are given by
  
(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012011.png" /> denote, respectively, a set of objects and a set of arrows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012012.png" />), then a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012013.png" /> is a pair of mappings
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\begin{equation*} G : A G \overset{\text{dom}_G}{\underset{\underset{\text{codom}_G}{\rightarrow} }{\rightarrow}} O G, \end{equation*}
  
<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/d120/d120120/d12012014.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d1201209.png"/></td> </tr></table>
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012016.png" />.
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(here $O G$ and $A G$ denote, respectively, a set of objects and a set of arrows of $G$), then a morphism $d$ is a pair of mappings
  
A diagram is called finite if its shape is a finite [[Graph|graph]], i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012018.png" /> are finite sets. A diagram in a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012019.png" /> is defined as a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012021.png" /> denotes the underlying graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012022.png" /> (with the same objects and arrows, forgetting which arrows are composites and which are identities).
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\begin{equation*} d _ { 0 } : O G \rightarrow O G ^ { \prime } , \quad d _ { A } : A G \rightarrow A G ^ { \prime } \end{equation*}
  
Every [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012023.png" /> is also a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012024.png" /> between the corresponding graphs. This observation defines the forgetful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012025.png" /> from small categories to small graphs (cf. also [[Functor|Functor]]).
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with $\operatorname {dom}_{G^{\prime}} \circ d _ { A } = d _ { 0 } \circ \operatorname {dom}_{G}$, $\operatorname{codom}_{G'} \circ d _ { A } = d _ { 0 } \circ \operatorname{codom}_{G}$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012026.png" /> be two diagrams of the same shape <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012027.png" /> in the same category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012028.png" />. A morphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012030.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012031.png" /> that carries each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012032.png" /> of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012033.png" /> to an arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012034.png" />, such that for any arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012036.png" /> the diagram
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A diagram is called finite if its shape is a finite [[Graph|graph]], i.e. $O G$ and $A G$ are finite sets. A diagram in a [[Category|category]] $\mathcal{C}$ is defined as a diagram $G \rightarrow U \mathcal{C}$, where $U \cal C$ denotes the underlying graph of $\mathcal{C}$ (with the same objects and arrows, forgetting which arrows are composites and which are identities).
  
<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/d120/d120120/d12012037.png" /></td> </tr></table>
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Every [[Functor|functor]] $F : \mathcal{C} \rightarrow \mathcal{C} ^ { \prime }$ is also a diagram $U F : U \mathcal C \rightarrow U \mathcal C ^ { \prime }$ between the corresponding graphs. This observation defines the forgetful functor $U : \operatorname{Cat} \rightarrow \operatorname{Graph}$ from small categories to small graphs (cf. also [[Functor|Functor]]).
 +
 
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Let $d , d ^ { \prime } : G \rightarrow \mathcal{C}$ be two diagrams of the same shape $G$ in the same category $\mathcal{C}$. A morphism between $d$ and $d ^ { \prime }$ is a mapping $\Phi : O G \rightarrow A \mathcal{C}$ that carries each object $g$ of the graph $G$ to an arrow $\Phi g : d g \rightarrow d ^ { \prime } g$, such that for any arrow $a : g \rightarrow g ^ { \prime }$ of $G$ the diagram
 +
 
 +
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012037.png"/></td> </tr></table>
  
 
commutes.
 
commutes.
  
All diagrams of the shape <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012039.png" /> and all morphisms between them form a category.
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All diagrams of the shape $G$ in $\mathcal{C}$ and all morphisms between them form a category.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012040.png" /> be a diagram in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012041.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012042.png" /> be a finite sequence of arrows of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012043.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012045.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012046.png" />. A diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012047.png" /> is called commutative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012048.png" /> for any finite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012050.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012054.png" />.
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Let $d : G \rightarrow \mathcal C$ be a diagram in the category $\mathcal{C}$ and let $\alpha = ( a _ { 1 } , \dots , a _ { n } )$ be a finite sequence of arrows of the graph $G$ with $\operatorname{dom} a_{i+1}=\operatorname{codom} a_i$, $i = 1 , \dots , n - 1$. Put $d \alpha = d a _ { n } \circ \ldots \circ d a _ { 1 }$. A diagram $d$ is called commutative if $d \alpha = d \alpha'$ for any finite sequence $\alpha ^ { \prime } = ( a ^ { \prime_1 }  , \ldots , a ^ { \prime_m } )$ in $G$ with $\operatorname{dom} \alpha_{j+1}^{\prime} = \operatorname { codom } \alpha _ { j } ^ { \prime }$, $j = 1 , \dots , m - 1$, $\operatorname{dom} a_1= \operatorname { dom } a_1'$, $\operatorname{codom}a_n=\operatorname{codom}a_m'$.
  
A sequence is a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012056.png" /> is of the form
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A sequence is a diagram $d : G \rightarrow \mathcal C$, where $G$ is 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/d/d120/d120120/d12012057.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012057.png"/></td> </tr></table>
  
 
The corresponding diagram is represented by
 
The corresponding diagram is represented 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/d/d120/d120120/d12012058.png" /></td> </tr></table>
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\begin{equation*} c _ { 1 } \stackrel { \phi _ { 1 } } { \rightarrow } \ldots \stackrel { \phi _ { n - 1 } } { \rightarrow } c _ { n }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012059.png" /> are objects and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012060.png" /> are arrows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012061.png" />.
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where $c _ { k } = d ( g _ { k } )$ are objects and $\phi _ { k } = d ( a _ { k } )$ are arrows of $\mathcal{C}$.
  
A triangle diagram in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012062.png" /> is a diagram with shape graph
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A triangle diagram in the category $\mathcal{C}$ is a diagram with shape 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/d120/d120120/d12012063.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012063.png"/></td> </tr></table>
  
 
and is represented as
 
and is represented as
  
<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/d120/d120120/d12012064.png" /></td> </tr></table>
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\begin{equation*} \left. \begin{array} { c c c } { \square } &amp; { c _ { 2 } } &amp; { \square } \\ { \square } &amp; { \square } &amp; { \searrow ^ { \phi _ { 2 } } } \\ { \square ^ { \phi _ { 1 } } \nearrow } &amp; { \vec { \phi _ { 3 } } }  &amp;{c_3} \end{array} \right. . \end{equation*}
  
Commutativity means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012065.png" />.
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Commutativity means that $\phi _ { 2 } \circ \phi _ { 1 } = \phi _ { 3 }$.
  
A quadratic diagram (also called a square diagram) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012066.png" /> corresponds to the graph
+
A quadratic diagram (also called a square diagram) in $\mathcal{C}$ 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/d120/d120120/d12012067.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012067.png"/></td> </tr></table>
  
 
and is represented as
 
and is represented as
  
<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/d120/d120120/d12012068.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012068.png"/></td> </tr></table>
  
Commutativity means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120120/d12012069.png" />.
+
Commutativity means $\phi _ { 2 } \circ \phi _ { 1 } = \phi _ { 3 } \circ \phi _ { 4 }$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Gabriel,   M. Zisman,  "Calculus of fractions and homotopy theory" , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grothendieck,   "Sur quelques points d'algebre homologique"  ''Tôhoku Math. J. Ser. II'' , '''9'''  (1957)  pp. 120–221</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Maclane,   "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top"> P. Gabriel, M. Zisman,  "Calculus of fractions and homotopy theory" , Springer  (1967)</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique"  ''Tôhoku Math. J. Ser. II'' , '''9'''  (1957)  pp. 120–221</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top"> S. Maclane, "Categories for the working mathematician" , Springer  (1971)</td></tr>
 +
</table>

Latest revision as of 19:26, 11 November 2023

Let $G$ and $G ^ { \prime }$ be directed graphs (also called oriented graphs, diagram schemes or pre-categories; cf. also Graph, oriented). A diagram of shape (also called a diagram of type) $G$ in $G ^ { \prime }$ is a morphism of graphs $d : G \rightarrow G ^ { \prime }$; i.e. if $G$ and $G ^ { \prime }$ are given by

\begin{equation*} G : A G \overset{\text{dom}_G}{\underset{\underset{\text{codom}_G}{\rightarrow} }{\rightarrow}} O G, \end{equation*}

(here $O G$ and $A G$ denote, respectively, a set of objects and a set of arrows of $G$), then a morphism $d$ is a pair of mappings

\begin{equation*} d _ { 0 } : O G \rightarrow O G ^ { \prime } , \quad d _ { A } : A G \rightarrow A G ^ { \prime } \end{equation*}

with $\operatorname {dom}_{G^{\prime}} \circ d _ { A } = d _ { 0 } \circ \operatorname {dom}_{G}$, $\operatorname{codom}_{G'} \circ d _ { A } = d _ { 0 } \circ \operatorname{codom}_{G}$.

A diagram is called finite if its shape is a finite graph, i.e. $O G$ and $A G$ are finite sets. A diagram in a category $\mathcal{C}$ is defined as a diagram $G \rightarrow U \mathcal{C}$, where $U \cal C$ denotes the underlying graph of $\mathcal{C}$ (with the same objects and arrows, forgetting which arrows are composites and which are identities).

Every functor $F : \mathcal{C} \rightarrow \mathcal{C} ^ { \prime }$ is also a diagram $U F : U \mathcal C \rightarrow U \mathcal C ^ { \prime }$ between the corresponding graphs. This observation defines the forgetful functor $U : \operatorname{Cat} \rightarrow \operatorname{Graph}$ from small categories to small graphs (cf. also Functor).

Let $d , d ^ { \prime } : G \rightarrow \mathcal{C}$ be two diagrams of the same shape $G$ in the same category $\mathcal{C}$. A morphism between $d$ and $d ^ { \prime }$ is a mapping $\Phi : O G \rightarrow A \mathcal{C}$ that carries each object $g$ of the graph $G$ to an arrow $\Phi g : d g \rightarrow d ^ { \prime } g$, such that for any arrow $a : g \rightarrow g ^ { \prime }$ of $G$ the diagram

commutes.

All diagrams of the shape $G$ in $\mathcal{C}$ and all morphisms between them form a category.

Let $d : G \rightarrow \mathcal C$ be a diagram in the category $\mathcal{C}$ and let $\alpha = ( a _ { 1 } , \dots , a _ { n } )$ be a finite sequence of arrows of the graph $G$ with $\operatorname{dom} a_{i+1}=\operatorname{codom} a_i$, $i = 1 , \dots , n - 1$. Put $d \alpha = d a _ { n } \circ \ldots \circ d a _ { 1 }$. A diagram $d$ is called commutative if $d \alpha = d \alpha'$ for any finite sequence $\alpha ^ { \prime } = ( a ^ { \prime_1 } , \ldots , a ^ { \prime_m } )$ in $G$ with $\operatorname{dom} \alpha_{j+1}^{\prime} = \operatorname { codom } \alpha _ { j } ^ { \prime }$, $j = 1 , \dots , m - 1$, $\operatorname{dom} a_1= \operatorname { dom } a_1'$, $\operatorname{codom}a_n=\operatorname{codom}a_m'$.

A sequence is a diagram $d : G \rightarrow \mathcal C$, where $G$ is of the form

The corresponding diagram is represented by

\begin{equation*} c _ { 1 } \stackrel { \phi _ { 1 } } { \rightarrow } \ldots \stackrel { \phi _ { n - 1 } } { \rightarrow } c _ { n }, \end{equation*}

where $c _ { k } = d ( g _ { k } )$ are objects and $\phi _ { k } = d ( a _ { k } )$ are arrows of $\mathcal{C}$.

A triangle diagram in the category $\mathcal{C}$ is a diagram with shape graph

and is represented as

\begin{equation*} \left. \begin{array} { c c c } { \square } & { c _ { 2 } } & { \square } \\ { \square } & { \square } & { \searrow ^ { \phi _ { 2 } } } \\ { \square ^ { \phi _ { 1 } } \nearrow } & { \vec { \phi _ { 3 } } } &{c_3} \end{array} \right. . \end{equation*}

Commutativity means that $\phi _ { 2 } \circ \phi _ { 1 } = \phi _ { 3 }$.

A quadratic diagram (also called a square diagram) in $\mathcal{C}$ corresponds to the graph

and is represented as

Commutativity means $\phi _ { 2 } \circ \phi _ { 1 } = \phi _ { 3 } \circ \phi _ { 4 }$.

References

[a1] P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)
[a2] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. Ser. II , 9 (1957) pp. 120–221
[a3] S. Maclane, "Categories for the working mathematician" , Springer (1971)
How to Cite This Entry:
Diagram(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram(2)&oldid=12484
This article was adapted from an original article by T. Datuashvili (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article