https://citizendium.org/wiki/index.php?title=Abelian_category&feed=atom&action=historyAbelian category - Revision history2024-03-28T20:34:53ZRevision history for this page on the wikiMediaWiki 1.39.5https://citizendium.org/wiki/index.php?title=Abelian_category&diff=375514&oldid=previmported>David E. Volk: subpages, workgroup moved to metadata page2008-01-21T23:20:40Z<p>subpages, workgroup moved to metadata page</p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The theory of abelian categories was developed simultaneously (by [[Alexander Grothendieck]] and someone else) during the 1950s to develop a language with the aim of defining [[cohomology]] in a general framework. The foundations of abelian categories and [[homological algebra]] were outlined in the paper ''Sur quelques points d'algèbre homologique'' (often referred to as "Tohoku," owing to the name of the journal in which it was published) by Grothendieck. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{subpages}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The theory of <ins style="font-weight: bold; text-decoration: none;">'''</ins>abelian categories<ins style="font-weight: bold; text-decoration: none;">''' </ins>was developed simultaneously (by [[Alexander Grothendieck]] and someone else) during the 1950s to develop a language with the aim of defining [[cohomology]] in a general framework. The foundations of abelian categories and [[homological algebra]] were outlined in the paper ''Sur quelques points d'algèbre homologique'' (often referred to as "Tohoku," owing to the name of the journal in which it was published) by Grothendieck. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A surprising result due to Freyd and Mitchell states that every [[Category theory|small]] abelian category is [[Category of functors|equivalent]] to a category of [[Module|modules]]. This has one important consequence of allowing one to often prove propositions regarding (small) diagrams in any abelian category by assuming the diagram is in the category of modules, a procedure usually referred to as "diagram chasing."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A surprising result due to Freyd and Mitchell states that every [[Category theory|small]] abelian category is [[Category of functors|equivalent]] to a category of [[Module|modules]]. This has one important consequence of allowing one to often prove propositions regarding (small) diagrams in any abelian category by assuming the diagram is in the category of modules, a procedure usually referred to as "diagram chasing."</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div></td></tr>
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</table>imported>David E. Volkhttps://citizendium.org/wiki/index.php?title=Abelian_category&diff=375516&oldid=previmported>Paul Wormer: added cats2008-01-18T12:23:57Z<p>added cats</p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of modules over a ring <math>R</math>. In particular, the category of modules over <math>\mathbb{Z}</math> (which is equivalent to the category of [[Abelian group|abelian groups]]), is also an abelian category.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of modules over a ring <math>R</math>. In particular, the category of modules over <math>\mathbb{Z}</math> (which is equivalent to the category of [[Abelian group|abelian groups]]), is also an abelian category.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div></td></tr>
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</table>imported>Paul Wormerhttps://citizendium.org/wiki/index.php?title=Abelian_category&diff=375515&oldid=previmported>Giovanni Antonio DiMatteo at 23:35, 9 January 20082008-01-09T23:35:48Z<p></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The theory of abelian categories was developed simultaneously during the 1950s to develop a language with the aim of defining [[cohomology]] in a general framework. The foundations of abelian categories and [[homological algebra]] were outlined in the paper ''Sur quelques points d'algèbre homologique'' (often referred to as "Tohoku," owing to the name of the journal in which it was published) by <del style="font-weight: bold; text-decoration: none;">[[Alexander </del>Grothendieck<del style="font-weight: bold; text-decoration: none;">]]</del>. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The theory of abelian categories was developed simultaneously <ins style="font-weight: bold; text-decoration: none;">(by [[Alexander Grothendieck]] and someone else) </ins>during the 1950s to develop a language with the aim of defining [[cohomology]] in a general framework. The foundations of abelian categories and [[homological algebra]] were outlined in the paper ''Sur quelques points d'algèbre homologique'' (often referred to as "Tohoku," owing to the name of the journal in which it was published) by Grothendieck. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A surprising result due to Freyd and Mitchell states that every [[Category theory|small]] abelian category is [[Category of functors|equivalent]] to a category of [[Module|modules]]. This has one important consequence of allowing one to often prove propositions regarding (small) diagrams in any abelian category by assuming the diagram is in the category of modules, a procedure usually referred to as "diagram chasing."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A surprising result due to Freyd and Mitchell states that every [[Category theory|small]] abelian category is [[Category of functors|equivalent]] to a category of [[Module|modules]]. This has one important consequence of allowing one to often prove propositions regarding (small) diagrams in any abelian category by assuming the diagram is in the category of modules, a procedure usually referred to as "diagram chasing."</div></td></tr>
</table>imported>Giovanni Antonio DiMatteohttps://citizendium.org/wiki/index.php?title=Abelian_category&diff=375517&oldid=previmported>Giovanni Antonio DiMatteo: /* Definition and Examples */2008-01-09T23:34:56Z<p><span dir="auto"><span class="autocomment">Definition and Examples</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>An abelian category is an [[additive category]] satisfying the additional requirement that every monomorphism is a kernel and every epimorphism is a cokernel. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>An abelian category is an [[additive category]] satisfying the additional requirement <ins style="font-weight: bold; text-decoration: none;">that every morphism has a kernel and a cokernel, and </ins>that every monomorphism is a kernel <ins style="font-weight: bold; text-decoration: none;">of some morphism </ins>and every epimorphism is a cokernel <ins style="font-weight: bold; text-decoration: none;">of some morphism</ins>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of modules over a ring <math>R</math>. In particular, the category of modules over <math>\mathbb{Z}</math> (which is equivalent to the category of [[Abelian group|abelian groups]]), is also an abelian category.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of modules over a ring <math>R</math>. In particular, the category of modules over <math>\mathbb{Z}</math> (which is equivalent to the category of [[Abelian group|abelian groups]]), is also an abelian category.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div></td></tr>
</table>imported>Giovanni Antonio DiMatteohttps://citizendium.org/wiki/index.php?title=Abelian_category&diff=375518&oldid=previmported>Giovanni Antonio DiMatteo: Soit page2008-01-09T23:33:04Z<p>Soit page</p>
<p><b>New page</b></p><div>The theory of abelian categories was developed simultaneously during the 1950s to develop a language with the aim of defining [[cohomology]] in a general framework. The foundations of abelian categories and [[homological algebra]] were outlined in the paper ''Sur quelques points d'algèbre homologique'' (often referred to as "Tohoku," owing to the name of the journal in which it was published) by [[Alexander Grothendieck]]. <br />
<br />
A surprising result due to Freyd and Mitchell states that every [[Category theory|small]] abelian category is [[Category of functors|equivalent]] to a category of [[Module|modules]]. This has one important consequence of allowing one to often prove propositions regarding (small) diagrams in any abelian category by assuming the diagram is in the category of modules, a procedure usually referred to as "diagram chasing."<br />
<br />
==Definition and Examples==<br />
<br />
An abelian category is an [[additive category]] satisfying the additional requirement that every monomorphism is a kernel and every epimorphism is a cokernel. <br />
<br />
#The category of modules over a ring <math>R</math>. In particular, the category of modules over <math>\mathbb{Z}</math> (which is equivalent to the category of [[Abelian group|abelian groups]]), is also an abelian category.<br />
#The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian category.</div>imported>Giovanni Antonio DiMatteo