| Version 2 |
Version 1 |
| Let $\mathcal{C}$ be a category and $D$ a diagram in $\mathcal{C}$. A \emph{cone over} $D$ consists of the following: |
Let $\mathcal{C}$ be a category and $D$ a diagram in $\mathcal{C}$. A \emph{cone over} $D$ consists of the following: |
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|
| \begin{enumerate} |
\begin{enumerate} |
| \item an object $a$ of $\mathcal{C}$, |
\item an object $a$ of $\mathcal{C}$, |
| \item a morphism $f:a\to d$ for each object in $D$, |
\item a morphism $f:a\to d$ for each object in $D$, |
| \item a commutative triangle |
\item a commutative triangle |
| $$\xymatrix{ |
$$\xymatrix{ |
| & a \ar[dl]_{f_1} \ar[dr]^{f_2} & \\ |
& a \ar[dl]_{f_1} \ar[dr]^{f_2} & \\ |
| d_1 \ar[rr]^g && d_2 |
d_1 \ar[rr]^g && d_2 |
| } |
} |
| $$ |
$$ |
| for every morphism $f:d_1\to d_2$ in $D$. |
for every morphism $f:d_1\to d_2$ in $D$. |
| \end{enumerate} |
\end{enumerate} |
|
|
| A cone over $D$ is denoted by $\lbrace a\to d\mid d\in D\rbrace$, or simply $a\to D$. |
A cone over $D$ is denoted by $\lbrace a\to d\mid d\in D\rbrace$, or simply $a\to D$. |
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|
| A \emph{limiting cone} over $D$ is a cone over $D$, $c\to D$, such that for any cone $a\to D$, there is a unique morphism $h: a\to c$ such that the diagram |
A \emph{limiting cone} over $D$ is a cone over $D$, $c\to D$, such that for any cone $a\to D$, there is a unique morphism $h: a\to c$ such that the diagram |
| $$\xymatrix{ |
$$\xymatrix{ |
| a \ar[rr]^h \ar[dr]_{x} && c \ar[dl]^{y} \\ |
a \ar[rr]^h \ar[dr]_{x} && c \ar[dl]^{y} \\ |
| & d & |
& d & |
| } |
} |
| $$ |
$$ |
| is commutative for every object $d$ of $D$. If a diagram $D$ has a limiting cone, then $D$ is said to have a \emph{limit}. |
is commutative for every object $d$ of $D$. If a diagram $D$ has a limiting cone, then $D$ is said to have a \emph{limit}. |
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|
| \textbf{Remarks}. |
\textbf{Remarks}. |
| \begin{itemize} |
\begin{itemize} |
| \item Any two limiting cones of a diagram $D$ are isomorphic in the sense that if $c_1\to D$ and $c_2\to D$ are limiting cones, then there are morphisms $p:c_1\to c_2$ and $q:c_2\to c_1$ such that $pq=1_{c_2}$ and $qp=1_{c_1}$. |
\item Any two limiting cones of a diagram $D$ are isomorphic in the sense that if $c_1\to D$ and $c_2\to D$ are limiting cones, then there are morphisms $p:c_1\to c_2$ and $q:c_2\to c_1$ such that $pq=1_{c_2}$ and $qp=1_{c_1}$. |
| \item We may form a category from the collection of all cones over a diagram $D$ as follows: |
\item We may form a category from the collection of all cones over a diagram $D$ as follows: |
| \begin{itemize} |
\begin{itemize} |
| \item objects are cones over $D$, |
\item objects are cones over $D$, |
| \item a morphism from a cones $a\to D$ and a cone $b\to D$ is a morphism $h: a\to b$ such that |
\item a morphism from a cones $a\to D$ and a cone $b\to D$ is a morphism $h: a\to b$ such that |
| $$\xymatrix{ |
$$\xymatrix{ |
| a \ar[rr]^h \ar[dr]_{x} && b \ar[dl]^{y} \\ |
a \ar[rr]^h \ar[dr]_{x} && b \ar[dl]^{y} \\ |
| & d & |
& d & |
| } |
} |
| $$ |
$$ |
| is a commutative triangle. |
is a commutative triangle. |
| \end{itemize} |
\end{itemize} |
| Clearly, identity morphisms and compositions of morphisms can then be defined accordingly. |
Clearly, identity morphisms and compositions of morphisms can then be defined accordingly. |
| \item From the above construction of the category of cones over $D$, a limiting cone is just a terminal object in that category. |
\item From the above construction of the category of cones over $D$, a limiting cone is just a terminal object in that category. |
| \end{itemize} |
\end{itemize} |
|
|
| \textbf{Examples} |
\textbf{Examples} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $D$ consists of a single object $d$, then a limiting cone, which always exists, is the identity morphism $1_d:d\to d$. |
\item If $D$ consists of a single object $d$, then a limiting cone, which always exists, is the identity morphism $1_d:d\to d$. |
| \item If $D$ is the empty set (no objects and no morphisms), a limiting cone over $D$ is just a terminal object in $\mathcal{C}$. |
\item If $D$ is the empty set (no objects and no morphisms), a limiting cone over $D$ is just a terminal object in $\mathcal{C}$. |
| \item If $D$ consists of two objects $a,b$ without any morphisms, then the limiting cone is the product of the two objects $a\times b$. |
\item If $D$ consists of two objects $a,b$ without any morphisms, then the limiting cone is the product of the two objects $a\times b$. |
| \item If $D$ has the following diagram |
\item If $D$ has the following diagram |
| $$\xymatrix{ |
$$\xymatrix{ |
| & a\ar[d]^x \\ |
& a\ar[d]^x \\ |
| b\ar[r]^y & c |
b\ar[r]^y & c |
| }$$ |
}$$ |
| then the limiting cone is the pullback, written $a\times_c b$. If $c$ turns out to be a terminal object, then $a\times_c b \cong a\times b$. If $a=b$, then $a\times_c a$ is the equalizer of $x$ and $y$. |
then the limiting cone is the pullback, written $a\times_c b$. If $c$ turns out to be a terminal object, then $a\times_c b \cong a\times b$. If $a=b$, then $a\times_c a$ is the equalizer of $x$ and $y$. |
| \end{enumerate} |
\end{enumerate} |
|
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| \textbf{Remark.} If all arrows (morphisms) are reversed, we have a cone under a diagram $D$. A cone under a diagram $D$ is also known as a \emph{cocone} for $D$. The dual concept of a limiting cone is thus a limiting cocone, which is an initial object in the category of cocones. All of the examples cited above can be dualized, and the respective results are an identity morphism, an initial object, a coproduct, and a pushout. |
\textbf{Remark.} If all arrows (morphisms) are reversed, we have a cone under a diagram $D$. A cone under a diagram $D$ is also known as a \emph{cocone} for $D$. The dual concept of a limiting cone is thus a limiting cocone, which is an initial object in the category of cocones. All of the examples cited above can be dualized, and the respective results are an identity morphism, an initial object, a coproduct, and a pushout. |
