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Version 2 |
| A category $\mathcal{C}$ is said to be a \emph{complete category} if every small diagram has a limit, that is, a limiting cone exists over every small diagram (diagram such that collections of objects and morphisms are sets). |
A category $\mathcal{C}$ is said to be a \emph{complete category} if every small diagram has a limit, that is, a limiting cone exists over every small diagram (diagram such that collections of objects and morphisms are sets). |
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| Of course, in a complete category, a product exists for any given set of objects. Also, a set of morphisms with common domain and codomain has an equalizer. Conversely, we have |
Of course, in a complete category, a product exists for any given set of objects. Also, a set of morphisms with common domain and codomain has an equalizer. Conversely, we have |
| \begin{quote} |
\begin{quote} |
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in a category $\mathcal{C}$, if the product exists for an arbitrary set of objects, and the equalizer exists for any pair of morphisms with common domain and codomain, then $\mathcal{C}$ is complete.
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in a category $\mathcal{C}$, if a product exists for an arbitrary set of objects, and an equalizer exists for an arbitrary set of morphisms with common domain and codomain, then $\mathcal{C}$ is complete.
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| \end{quote} |
\end{quote} |
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| \textbf{Examples} |
\textbf{Examples} |
| \begin{itemize} |
\begin{itemize} |
| \item \textbf{Set} is complete. |
\item \textbf{Set} is complete. |
| \item \textbf{Group} is complete. |
\item \textbf{Group} is complete. |
| \item \textbf{Vector Space} is complete |
\item \textbf{Vector Space} is complete |
| \item \textbf{R-module} is complete for a given unital ring $R$. |
\item \textbf{R-module} is complete for a given unital ring $R$. |
| \item \textbf{Topological Space} is complete. |
\item \textbf{Topological Space} is complete. |
| \end{itemize} |
\end{itemize} |
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| A category $\mathcal{C}$ is said to be \emph{finitely complete} if every finite diagram (sets of objects and morphisms are finite) has a limit. |
A category $\mathcal{C}$ is said to be \emph{finitely complete} if every finite diagram (sets of objects and morphisms are finite) has a limit. |
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| A similar sufficient condition for a category $\mathcal{C}$ to be finitely complete is for $\mathcal{C}$ to possess a terminal object and that a pullback exists for every pair of morphisms with common codomain. |
A similar sufficient condition for a category $\mathcal{C}$ to be finitely complete is for $\mathcal{C}$ to possess a terminal object and that a pullback exists for every pair of morphisms with common codomain. |
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| \textbf{Examples} |
\textbf{Examples} |
| \begin{itemize} |
\begin{itemize} |
| \item Any complete category is clearly finitely complete. |
\item Any complete category is clearly finitely complete. |
| \item The subcategories of the above examples consisting of all objects with finite cardinality are finitely complete (but not complete). |
\item The subcategories of the above examples consisting of all objects with finite cardinality are finitely complete (but not complete). |
| \end{itemize} |
\end{itemize} |
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| \textbf{Remark}. The dual notion of a complete category is that of a \emph{cocomplete category}, and the dual of a finitely complete category is called a \emph{finitely cocomplete category}. |
\textbf{Remark}. The dual notion of a complete category is that of a \emph{cocomplete category}, and the dual of a finitely complete category is called a \emph{finitely cocomplete category}. |
