supplemental axioms for an Abelian categoryAxiomsForAnAbelianCategory2001-12-12 01:59:382004-04-07 05:10:44AxiomcompletecocompleteAbelian categorymonickernelcokernelcoproductproduct\usepackage{amssymb}
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\usepackage{xypic}These are axioms introduced by Alexandre Grothendieck for an Abelian category. The first two are satisfied by definition in an Abelian category, and others may or may not be.
\begin{itemize}
\item[(Ab1)] Every morphism has a kernel and a cokernel.
\item[(Ab2)] Every monic is the kernel of its cokernel.
\item[(Ab3)] Coproducts exist. (Coproducts are also called direct sums.) If this axiom is satisfied the category is often just called cocomplete.
\item[(Ab3*)] Products exist. If this axiom is satisfied the category is often just called complete.
\item[(Ab4)] Coproducts exist and the coproduct of monics is a monic.
\item[(Ab4*)] Products exist and the product of epics is an epic.
\item[(Ab5)] Coproducts exist and filtered colimits of exact sequences are exact.
\item[(Ab5*)] Products exist and filtered inverse limits of exact sequences are exact.
\end{itemize}
Grothendieck introduced these in his homological algebra paper \emph{Sur quelques points d'alg\`ebre homologique} in the T\^ohoku Math Journal (number 2, volume 9, 1957). They can also be found in Weibel's excellent book \emph{An introduction to homological algebra}, Cambridge Studies in Advanced Mathematics (Cambridge University Press, 1994).