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\begin{document}

%-------------------------------------------------------------------


\section*{diagram}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{CategoryShapedDiagrams}{Diagrams shaped like categories}\dotfill \pageref*{CategoryShapedDiagrams} \linebreak
\noindent\hyperlink{CategoryShapedDiagramFunctorially}{Functorial definition}\dotfill \pageref*{CategoryShapedDiagramFunctorially} \linebreak
\noindent\hyperlink{CategoryShapedDiagramInComponents}{Component definition}\dotfill \pageref*{CategoryShapedDiagramInComponents} \linebreak
\noindent\hyperlink{ShapeOfDirectedGraph}{Diagrams shaped like directed graphs}\dotfill \pageref*{ShapeOfDirectedGraph} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{commutative_diagrams}{Commutative diagrams}\dotfill \pageref*{commutative_diagrams} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Informally, a \textbf{diagram} in a [[category]] $C$ consists of some [[objects]] of $C$ connected by some [[morphisms]] of $C$. Frequently when doing category theory, we ``draw diagrams'' such as

\begin{displaymath}
\itexarray{A & \overset{f}{\to} & B\\
  ^h\downarrow && \downarrow^k\\
  C& \underset{g}{\to} & D}
\end{displaymath}
by drawing some objects (or dots labeled by objects) connected by arrows labeled by morphisms.

This terminology is often used when speaking about [[limits]] and [[colimits]]; that is, we speak about ``the limit or colimit of a diagram.''

There are two natural ways to give the notion of ``diagram'' a formal definition. One is to say that a diagram is a [[functor]], usually one whose domain is a (very) [[small category]]. This level of generality is sometimes convenient.

On the other hand, a more direct representation of what we draw on the page, when we ``draw a diagram,'' only involves labeling the vertices and edges of a [[directed graph]] (or [[quiver]]) by objects and morphisms of the category. This sort of diagram can be identified with a functor whose domain is a [[free category]], and this is the most common context when we talk about diagrams ``commuting.''

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{CategoryShapedDiagrams}{}\subsubsection*{{Diagrams shaped like categories}}\label{CategoryShapedDiagrams}

We discuss here diagrams of the ``shape of a small catgeory'', as well as the concept of [[cones]]/[[cocones]] over these and [[limit|limiting]]/[[colimit|colimiting]] (co-)cones. There is a quick abstract functorial definition

\begin{itemize}%
\item \emph{\hyperlink{CategoryShapedDiagramFunctorially}{functorial definition}}

\end{itemize}
and there is a more long-winded but more explicit definition in terms of components

\begin{itemize}%
\item \emph{\hyperlink{CategoryShapedDiagramInComponents}{component definition}}.

\end{itemize}
\hypertarget{CategoryShapedDiagramFunctorially}{}\paragraph*{{Functorial definition}}\label{CategoryShapedDiagramFunctorially}

We state the concise functorial definition of diagrams of the shape of categories.

\begin{defn}
\label{AbstractDefinition}\hypertarget{AbstractDefinition}{}
\textbf{(functorial definition)}

Let $\mathcal{C}$ be a [[category]] and let $\mathcal{I}$ [[small category]],

Then

\begin{enumerate}%
\item a \emph{diagram $X$ of shape $\mathcal{I}$ in $\mathcal{C}$} is a [[functor]] of the form

\begin{displaymath}
X \;\colon\; \mathcal{I} \longrightarrow \mathcal{C}
  \,,
\end{displaymath}

\item the \emph{category of $\mathcal{I}$-shaped diagrams in $\mathcal{C}$} is the [[functor category]] $Funct(\mathcal{I}, \mathcal{C})$;


\item a diagram $X \colon \mathcal{I} \to \mathcal{C}$ is \emph{constant} if it is a [[constant functor]]

\begin{displaymath}
const_{\tilde X}
    \;\colon\;  
  \mathcal{I} 
    \overset{\exists!}{\longrightarrow} 
  \ast
    \overset{\tilde X}{\longrightarrow}
  \mathcal{C}
\end{displaymath}
in which case it is given by the data of a single object $\tilde X$;


\item a \emph{[[cone]]} $C$ over a diagram $X \colon \mathcal{I} \to \mathcal{C}$ with \emph{tip} an object $\tilde X \in \mathcal{C}$ is a [[natural transformation]] from the constant diagram $const_{\tilde X} \colon \mathcal{I} \to \ast \to \mathcal{C}$ to $X$:

\begin{displaymath}
C \;\colon\; const_{\tilde X} \Rightarrow X
\end{displaymath}

\item a \emph{[[cocone]]} $C$ under a diagram $X \colon \mathcal{I} \to \mathcal{C}$ is a [[natural transformation]] to a constant diagram $const_{\tilde X} \colon \mathcal{I} \to \ast \to \mathcal{C}$ from $X$:

\begin{displaymath}
C \;\colon\; X \Rightarrow const_{\tilde X}
\end{displaymath}

\item the \emph{limiting cone} (or \emph{[[limit]]}, for short) over a diagram $X$ is, if it exists, the [[terminal object]] in the [[category]] of [[cones]] over $X$, which means that it is a cone $C_{lim}$ with tip denoted $\underset{\longleftarrow}{\lim}_i X_i$ such that for every other cone $C$ with tip $\tilde X$ there is a unique [[natural transformation]] $\phi \colon const_{\tilde X} \Rightarrow const_{\underset{\longleftarrow}{\lim}_i X_i}$ such that

\begin{displaymath}
C = C_{lim} \circ \phi
\end{displaymath}

\item the \emph{colimiting cone} (or \emph{[[colimit]]}, for short) under a diagram $X$ is, if it exists, the [[initial object]] in the [[category]] of [[cocones]] under $X$, which means that it is a co-cone $C_{lim}$ with tip denoted $\underset{\longrightarrow}{\lim}_i X_i$ such that for every other cocone $C$ with tip $\tilde X$ there is a unique [[natural transformation]] $\phi \colon  const_{\underset{\longrightarrow}{\lim}_i X_i} \Rightarrow const_{\tilde X}$ such that

\begin{displaymath}
C = \phi \circ C_{lim}
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\hypertarget{CategoryShapedDiagramInComponents}{}\paragraph*{{Component definition}}\label{CategoryShapedDiagramInComponents}

We state an explicit component-based definition of diagrams of the shape of categories.

\begin{defn}
\label{Diagram}\hypertarget{Diagram}{}
\textbf{([[diagram]] in a [[category]])}

A [[diagram]] $X_\bullet$ in a [[category]] is

\begin{enumerate}%
\item a [[set]] $\{ X_i \}_{i \in I}$ of [[objects]] in the category;


\item for every [[pair]] $(i,j) \in I \times I$ of labels of objects a [[set]] $\{ X_i \overset{ f_\alpha }{\longrightarrow} X_j\}_{\alpha \in I_{i,j}}$ of [[morphisms]] between these objects;


\item for every label $i \in I$ a choice of element $\epsilon_i \in I_{i,i}$;


\item for each [[triple]] $i,j,k \in I$ a [[function]]

\begin{displaymath}
comp_{i,j,k} \;\colon\; I_{i,j} \times I_{j,k} \longrightarrow I_{i,k}
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item the pairing $comp$ is [[associativity|associative]] and [[unital]] with the $f_{\epsilon_i}$-s the [[neutral elements]];


\item for every $i \in I$ then $f_{\epsilon_i} = id_{X_i}$ is the [[identity morphism]] on the $i$-th obect;


\item for every composable pair of morphisms



\end{enumerate}
\begin{displaymath}
X_i 
      \overset{f_{\alpha} }{\longrightarrow} 
    X_j
      \overset{ f_{\beta} }{\longrightarrow}
    X_k
\end{displaymath}
then the [[composition|composite]] of these two morphisms equals the morphism of the diagram that is labeled by the value of $comp_{i,j,k}$ on their labels:

\begin{displaymath}
f_{\beta} \circ f_\alpha 
    \,=\,
    f_{comp_{i,j,k}( \alpha, \beta )}
    \,.
\end{displaymath}
The last condition we depict as follows:

\begin{displaymath}
\itexarray{ 
    && X_j
    \\
    & {}^{\mathllap{f_{\alpha}}}\nearrow && \searrow^{\mathrlap{f_{\beta}}}
    \\
    X_i && \underset{ comp_{i,j,k}(\alpha,\beta) }{\longrightarrow} && X_k
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{Cone}\hypertarget{Cone}{}
\textbf{([[cone]] over a [[diagram]])}

Consider a [[diagram]]

\begin{displaymath}
X_\bullet
    \,=\,
  \left(
     \left\{
       X_i \overset{f_\alpha}{\longrightarrow} X_j
     \right\}_{i,j \in I, \alpha \in I_{i,j}}
     \,,\,
     \mathrm{comp}
  \right)
\end{displaymath}
in some [[category]] (def. \ref{Diagram}). Then

\begin{enumerate}%
\item a \emph{[[cone]]} over this diagram is

\begin{enumerate}%
\item an [[object]] $\tilde X$ in the category;


\item for each $i \in I$ a morphism $\tilde X \overset{p_i}{\longrightarrow} X_i$ in the category



\end{enumerate}
such that

\begin{itemize}%
\item for all $(i,j) \in I \times I$ and all $\alpha \in I_{i,j}$ then the condition

\begin{displaymath}
f_{\alpha} \circ p_i = p_j
\end{displaymath}
holds, which we depict as follows:

\begin{displaymath}
\itexarray{
    && \tilde X
    \\
    & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p'_j}}
    \\
    X_i 
  && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
\end{displaymath}


\end{itemize}

\item a \emph{[[co-cone]]} over this diagram is

\begin{enumerate}%
\item an [[object]] $\tilde X$ in the category;


\item for each $i \in I$ a morphism $q_i \colon X_i \longrightarrow \tilde X$ in the category



\end{enumerate}
such that

\begin{itemize}%
\item for all $(i,j) \in I \times I$ and all $\alpha \in I_{i,j}$ then the condition

\begin{displaymath}
q_j \circ f_{\alpha} = q_i
\end{displaymath}
holds, which we depict as follows:

\begin{displaymath}
\itexarray{
    X_i && \overset{f_\alpha}{\longrightarrow} && X_j
    \\
    & {}_{\mathllap{q_i}}\searrow && \swarrow_{\mathrlap{q_j}}
    \\
    && \tilde X
  }
  \,.
\end{displaymath}


\end{itemize}


\end{enumerate}
\end{defn}
\begin{defn}
\label{LimitingCone}\hypertarget{LimitingCone}{}
Consider a [[diagram]]

\begin{displaymath}
X_\bullet
    \,=\,
  \left(
     \left\{
       X_i \overset{f_\alpha}{\longrightarrow} X_j
     \right\}_{i,j \in I, \alpha \in I_{i,j}}
     \,,\,
     \mathrm{comp}
  \right)
\end{displaymath}
in some [[category]] (def. \ref{Diagram}). Then

\begin{enumerate}%
\item its \emph{[[limit|limiting cone]]} (or just \emph{[[limit]]} for short) is, if it exists, [[generalized the|the]] [[cone]]

\begin{displaymath}
\left\{
  \itexarray{
    && \underset{\longleftarrow}{\lim}_i X_i
    \\
    & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}}
    \\
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
  \right\}
\end{displaymath}
over this diagram (def. \ref{Cone}) which is \emph{universal} or \emph{initial} among all possible cones, in that it has the property that for

\begin{displaymath}
\left\{
  \itexarray{
    && \tilde X
    \\
    & {}^{\mathllap{p'_i}}\swarrow && \searrow^{\mathrlap{p'_j}}
    \\
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
  \right\}
\end{displaymath}
any other [[cone]], then there is a unique morphism

\begin{displaymath}
\phi \;\colon\; \tilde X \overset{}{\longrightarrow} \underset{\longrightarrow}{\lim}_i X_i
\end{displaymath}
that factors the given cone through the limiting cone, in that for all $i \in I$ then

\begin{displaymath}
p'_i = p_i \circ \phi
\end{displaymath}
which we depict as follows:

\begin{displaymath}
\itexarray{
    \tilde X
    \\
    {}^{\mathllap{\phi}}\downarrow & \searrow^{\mathrlap{p_i}}
    \\
    \underset{\longrightarrow}{\lim}_i X_i
    &\underset{p_i}{\longrightarrow}& 
    X_i
  }
\end{displaymath}

\item its \emph{[[colimit|colimiting cocone]]} (or just \emph{[[colimit]]} for short) is, if it exists, [[generalized the|the]] [[cocone]]

\begin{displaymath}
\left\{
  \itexarray{
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
    \\
    & {}^{\mathllap{q_i}}\searrow && \swarrow^{\mathrlap{q_j}}
    \\
    \\
    && \underset{\longrightarrow}{\lim}_i X_i
  }
  \right\}
\end{displaymath}
under this diagram (def. \ref{Cone}) which is \emph{universal} or \emph{terminal} among all possible co-cones, in that it has the property that for

\begin{displaymath}
\left\{
  \itexarray{
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
    \\
    & {}^{\mathllap{q'_i}}\searrow && \swarrow_{\mathrlap{q'_j}}
    \\
    && \tilde X
  }
  \right\}
\end{displaymath}
any other [[cocone]], then there is a unique morphism

\begin{displaymath}
\phi \;\colon\;  \underset{\longrightarrow}{\lim}_i X_i \overset{}{\longrightarrow} \tilde X
\end{displaymath}
that factors the given co-cone through the co-limiting cocone, in that for all $i \in I$ then

\begin{displaymath}
q'_i = \phi \circ q_i
\end{displaymath}
which we depict as follows:

\begin{displaymath}
\itexarray{
    X_i
    &\overset{q_i}{\longrightarrow}&
    \underset{\longrightarrow}{\lim}_i X_i
    \\
    {}^{\mathllap{\phi}}\downarrow & \swarrow^{\mathrlap{q'_i}}
    \\
    \tilde X
  }
\end{displaymath}


\end{enumerate}
\end{defn}
\hypertarget{ShapeOfDirectedGraph}{}\subsubsection*{{Diagrams shaped like directed graphs}}\label{ShapeOfDirectedGraph}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{([[free diagram]])}

If $J$ is a [[directed graph]] with [[free category]] $F(J)$, then a \textbf{diagram} in $C$ of shape $J$ is a functor $D\colon F(J) \to C$, or equivalently a graph morphism $\bar{D}\colon J \to U(C)$.

\end{defn}
Here $F\colon Quiv \to Cat$ denotes the [[free category]] on a quiver and $U\colon Cat \to Quiv$ the underlying quiver of a category, which form a pair of [[adjoint functors]]. These are the sorts of diagrams which we ``draw on a page'' --- we draw a quiver, and then label its vertices with objects of $C$ and its edges with morphisms in $C$, thereby forming a graph morphism $J\to U(C)$.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item For either sort of diagram, $J$ may be called the \textbf{shape}, \textbf{scheme}, or \textbf{index} category or graph.


\item Note that given a diagram $D:J\to C$, the image of the shape $J$ is not necessarily a [[subcategory]] of $C$, even if $J$ is itself taken to be a category. This is because the functor $D$ could identify objects of $J$, thereby producing new potential composites which do not exist in $J$. (Sometimes one talks about the ``image'' of a functor as a subcategory, but this really means the subcategory \emph{generated} by the image in the literal objects-and-morphisms sense.)


\item $C$ must be a [[strict category]] to make sense of $U(C)$; however, $F(J)$ always makes sense.



\end{itemize}
\hypertarget{commutative_diagrams}{}\subsection*{{Commutative diagrams}}\label{commutative_diagrams}

If $J$ is a category, then a diagram $J\to C$ is \textbf{[[commutative diagram|commutative]]} if it factors through a [[thin category]]. Equivalently, a diagram of shape $J$ commutes iff any two morphisms in $C$ that are assigned to any pair of [[parallel morphisms]] in $J$ (i.e., with same source and target in $J$) are equal.

If $J$ is a quiver, as is more common when we speak about ``commutative'' diagrams, then a diagram of shape $J$ commutes if the functor $F(J) \to C$ factors through a thin category. Equivalently, this means that given any two parallel \emph{paths} of arbitrary finite length (including zero) in $J$, their images in $C$ have equal composites.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The shape of the \textbf{empty diagram} is the [[initial object|initial category]] with no object and no morphism.

Every category $C$ admits a unique diagram whose shape is the empty ([[initial object|initial]]) category, which is called the \textbf{empty diagram} in $C$.


\item The shape of the \textbf{terminal diagram} is the [[terminal category]] $J = \{*\}$ consisting of a single object and a single morphism (the identity morphism on that object).

Specifying a diagram in $C$ whose shape is $\{*\}$ is the same as specifying a single [[object]] of $C$, the image of the unique object of $1$. (See [[global element]])


\item A diagram of the shape $\{a \to b\}$ in $C$ is the choice of any one [[morphism]] $D_{a b} : X_a \to X_b$ in $C$.

Notice that strictly speaking this counts as a \emph{commuting diagram} , but is a degenerate case of a commuting diagram, since there is only a single morphism involved, which is necessarily equal to itself.


\item If $J$ is the quiver with one object $a$ and one endo-edge $a\to a$, then a diagram of shape $J$ in $C$ consists of a single [[endomorphism]] in $C$. Since $a\to a$ and the zero-length path are parallel in $J$, such a diagram only commutes if the endomorphism is an identity. Note, in particular, that a single endomorphism can be considered as a diagram with more than one shape (this one and the previous one), and that whether this diagram ``commutes'' depends on the chosen shape.


\item A diagram of shape the [[poset]] indicated by

\begin{displaymath}
\left\{
    \itexarray{
    a &\to& b
    \\
    \downarrow && \downarrow
    \\
    b' &\to& c
    }
  \right\}
\end{displaymath}
is a \textbf{commuting square} in $C$: this is a choice of four (not necessarily distinct!) objects $X_a, X_b, X_{b'}, X_c$ in C, together with a choice of (not necessarily distinct) four morphisms $D_{a b} : X_a \to X_b$, $D_{b c} : X_b \to X_c$ and $D_{a b'} : X_a \to X_{b'}$, $D_{b' c} : X_{b'} \to X_c$ in $C$, such that the composite morphism $D_{b c}\circ D_{a b}$ equals the composite $D_{b' c}\circ D_{a b'}$.

One typically ``draws the diagram'' as

\begin{displaymath}
\itexarray{
    X_a &\stackrel{D_{a b}}{\to}& X_b
    \\
    {}^{\mathllap{D_{a b'}}}\downarrow && 
      \downarrow^{\mathrlap{D_{b c}}}
    \\
    X_{b'} &\stackrel{D_{b' c}}{\to}& X_{c}
    }
\end{displaymath}
in $C$ and says that \emph{the diagram commutes} if the above equality of composite morphisms holds.

Notice that the original poset had, necessarily, a morphism $a \to c$ and could have equivalently been depicted as

\begin{displaymath}
\left\{
    \itexarray{
    a &\to& b
    \\
    \downarrow &\searrow& \downarrow
    \\
    b' &\to& c
    }
  \right\}
\end{displaymath}
in which case we could more explicitly draw its image in $C$ as

\begin{displaymath}
\itexarray{
    X_a &\stackrel{D_{a b}}{\to}& X_b
    \\
    {}^{\mathllap{D_{a b'}}}\downarrow &\searrow^{\stackrel{D_{b c}\circ D_{a b}}{= D_{b' c}\circ D_{a b'}}}& 
      \downarrow^{\mathrlap{D_{b c}}}
    \\
    X_{b'} &\stackrel{D_{b' c}}{\to}& X_{c}
    }
\end{displaymath}

\item By contrast, a diagram whose shape is the \emph{quiver}

\begin{displaymath}
\left\{
    \itexarray{
    a &\to& b
    \\
    \downarrow && \downarrow
    \\
    b' &\to& c
    }
  \right\}
\end{displaymath}
is a not-necessarily-commuting square. The free category on this quiver differs from the poset in the previous example by having \emph{two} morphisms $a\to c$, one given by the composite $a\to b\to c$ and the other by the composite $a \to b'\to c$. But the poset in the previous category is the poset reflection of this $F(J)$, so a diagram of this shape commutes, in the sense defined above, iff it is a commuting square in the usual sense.


\item A pair of objects is a diagram whose shape is a [[discrete category]] with two objects.


\item A pair of [[parallel morphisms]] is a diagram whose shape is a category $J = \{a \stackrel{\to}{\to} b\}$ with two objects and two morphisms from one to the other.

Notice that if we required $\{a \stackrel{\to}{\to} b\}$ to be a [[poset]] this would necessarily make these two morphisms equal, and hence reduce this example to the one where $J = \{a \to b\}$. In other words, a diagram of this shape only \emph{commutes} if the two morphisms are equal.


\item A [[span]] is a diagram whose shape is a category with just three objects and single morphisms from one of the objects to the other two;

\begin{displaymath}
J = \left\{
    \itexarray{
      && a
      \\
      & \swarrow && \searrow
      \\
      b &&&& c
    }
  \right\}
\end{displaymath}
dually, a [[cospan]] is a diagram whose shape is [[opposite category|opposite]] to the shape of a span.

\begin{displaymath}
J = \left\{
    \itexarray{
      b &&&& c
      \\
      & \searrow && \swarrow
      \\
      && a
    }
  \right\}
\end{displaymath}

\item A [[transfinite composition]] diagram is one of the shape the [[poset]] indicated by

\begin{displaymath}
J = 
  \left\{
   \itexarray{
    a_0 &\to& a_1 &\to& \cdots 
    \\
    & \searrow & \downarrow & \swarrow & \cdots 
    \\
    && b
   }
  \right\}
  \,,
\end{displaymath}
where the indices may range over the [[natural number]]s or even some more general [[ordinal number]].

This is a non-finite commuting diagram.


\item [[tower diagram]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[commuting diagram]]


\item [[internal diagram]]


\item [[free diagram]]



\end{itemize}
[[!redirects diagram]] [[!redirects diagrams]]

[[!redirects small diagram]] [[!redirects small diagrams]]



\end{document}