https://cattheory.subwiki.org/w/index.php?action=history&feed=atom&title=Opposite_categoryOpposite category - Revision history2026-07-29T21:07:15ZRevision history for this page on the wikiMediaWiki 1.41.2https://cattheory.subwiki.org/w/index.php?title=Opposite_category&diff=37&oldid=prevVipul: New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C}</math> is a category. The '''opposite category''' to <math>\mathcal{C}</math>, denoted <math>\mathcal{C}^{op}</math>, is defined as ...2008-12-09T01:32:36Z<p>New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C}</math> is a category. The '''opposite category''' to <math>\mathcal{C}</math>, denoted <math>\mathcal{C}^{op}</math>, is defined as ...</p>
<p><b>New page</b></p><div>{{basicdef}}<br />
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==Definition==<br />
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Suppose <math>\mathcal{C}</math> is a category. The '''opposite category''' to <math>\mathcal{C}</math>, denoted <math>\mathcal{C}^{op}</math>, is defined as follows:<br />
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* The objects of this category are the same as the objects of <math>\mathcal{C}</math>: <math>\operatorname{Ob}\mathcal{C}^{op} = \operatorname{Ob}\mathcal{C}</math>.<br />
* For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, <math>\mathcal{C}^{op}(A,B) = \mathcal{C}(A,B)</math>.<br />
* The identity maps remain the same.<br />
* For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C)</math>, the composite <math>f \circ g</math> in <math>\mathcal{C}^{op}</math> equals the composite <math>g \circ f</math> in <math>\mathcal{C}</math>.</div>Vipul