complete categoryCompleteCategory2006-09-17 17:35:472009-01-17 11:13:25Definitionfinitely complete categorycocomplete categoryfinitely cocomplete category\usepackage{amssymb,amscd}
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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).
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}
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.
\end{quote}
\textbf{Examples}
\begin{itemize}
\item \textbf{Set} is complete.
\item \textbf{Group} is complete.
\item \textbf{Vector Space} is complete
\item \textbf{R-module} is complete for a given unital ring $R$.
\item \textbf{Topological Space} is complete.
\end{itemize}
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 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.
\textbf{Examples}
\begin{itemize}
\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).
\end{itemize}
\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}.